Marketing Communication and Branding Assignment Example | Topics and Well Written Essays - 750 words

Promoting Communication and Branding - Assignment Example This exposition exhibits that promotion banner is one of the best components of coordinated showcasing correspondence. Saving the destinations of the showcasing correspondence for Travel worth as a main priority, certain banners can be set up to use in the battle arranged by the organization. The banners for the organization would incorporate an alluring message to get the eyes of the watcher, which would assist the organization with getting clients. The primary target of Travel worth is to get explorers for their experience visits and so as to accomplish it; the banners ought to contain certain realities about the experience world and certain pictures identified with fun in the visits. The slogan for the banners can be ‘Payless, Have more, Do more’, which would speak to that the expense of the visit would be less, fun and experience would be more. There ought to be sure plans on the banner to pull in the clients, for example, ‘Be the first to profit the limits on the Tour, and be the last one to benefit the limits on the Fun’. These lines would likewise pull in the experience sweethearts to profit the plans offered by Travel worth. The banner ought to ideally be splendid in hues, for example, yellow or red, which would handily get the eyes of the clients. The banner ought to contain certain pictures of the past visits organized by the organization and the encounters that the voyagers had. Every one of these components would make the banner important just as compelling for the organization to draw in the clients and get their inclinations for the items offered by the specific company.... Comparable to the referenced targets being referred to number one, it tends to be expressed that the model would assist with recognizing the circumstance, the emergency in the market, and it would likewise assist with arranging a battle to draw in new or existing clients. The model would assist with taking various systems. To spread mindfulness about organization and its current Tour programs, the battle will be adequately utilized as a vital goal. So as to make an alternate picture of the association and to draw in the buyers, a strategic methodology will be taken by giving different plans and limits. Comparable to existing circumstance of financial downturn the organization can take a choice to propose certain limits or deal to their client, which would assist them with acquiring new travel darlings. With regards to the key perspective, up degree in the better news paper or in other media for notice of the organization would be made and which would be actualized through the activit ies. Travelworth would get plentiful degree to keep up the maintainability with the assistance of IMC and SOSTAC model. In the time of downturn in the European economy the organizations are fundamentally battling for the maintainability in the market, and to spare the current clients and the brand name. SOSTAC model would help Travelworth to be steady in the market and to increase new clients and revenue2. Question 3 Q.3. Notice banner is one of the best components of coordinated showcasing correspondence. Saving the goals of the promoting correspondence for Travelworth at the top of the priority list, certain banners can be set up to use in the crusade arranged by the organization. The banners for the organization would incorporate an alluring message to get the eyes of the watcher, which would assist the organization with getting

Situational Analysis of Share Dignity Organization-Free-Samples

Question: Talk about Share Dignity Organization. Answer: Presentation As of now, altruistic associations have flourished on the planet helping individuals of various needs. Effective administration of these organizations requests a superior comprehension of their condition of activity through situational examination. This encourages the administration to take the correct methodologies, for example, proper advanced techniques that are probably going to affect decidedly to the organization. This investigation of Share the Dignity association gives a clear comprehension of the associations condition and the present procedures that it has grasped along these lines helping the administration settle on the correct choices (Charles, 2012). Miniaturized scale condition The miniaturized scale condition of Share Dignity Organization can be best investigated through Strength Weaknesses Opportunities Threats examination instruments (SWOT Analysis). Right off the bat, the as a not-for-profit association, Share Dignity appreciates numerous focal points, for example, charge exclusion, which decreases their expense of running business, other than the association gets income from online gifts (Chermack Bernadette, 2007). Likewise, the volunteer staff causes the group to chop down the costs along these lines giving it an upper hand over others. A portion of the shortcomings of Share Dignity incorporate spending issues where the association needs to consolidate the compensation scale with benefit making organizations for its non-volunteer laborers. The open doors for the organization incorporate openness to awards and furthermore numerous potential accomplices. With respect to, the association endures dangers of financial emergencies since it depends on the b uyers money surges. Figure 1.0 Large scale Environment Offer Dignity's large scale condition can be best examined utilizing PESTLE investigation. Strategically, the association's activity is fundamentally influenced the political powers of nations where it works. The Economic State of the world influences the capacity of contributors which thusly influences the organization. Social factors such the convictions, social practices likewise sway the activity of the association in light of individuals, for example, volunteers and great wills. Mechanically, the organization needs to stay aware of the changing advancements for its endurance. Likewise, the lawful elements incorporate the current lawful framework has a control on the exercises of the association (Hanssan, 2012). Plus, the ecological variables influence what the association can do, for example, the green approach. In this manner, it needs to guarantee that it moderates the earth at constantly. Figure 1.2 Organization Analysis Offer Dignity is of the Australian no benefit making associations that have some expertise in helping ladies in Australia and different pieces of the world. The association bargains in furnishing the destitute and penniless ladies with the arrangement of clean items for use during their periods. The organization is continued by raising accounts from givers through gathering pledges, staff volunteers and furthermore the cooperative attitude individuals in the general public. Subsequently, Share Dignity works in altruistic exercises of giving sterile cushions to support the penniless, vagrants everything being equal. Contender Analysis' The three foundation players in Australia that contends the Share the Dignity association incorporate the UN Women, Global sisters, and the Global Women's Project. Right off the bat, there are the UN Women which bargains in comparable exercises to help ladies in the general public and fund-raises and is likewise financed correspondingly consequently; it contends Share the Dignity (Mankiw Taylor, 2011). Besides, there are Global Sisters one of the associations that have practical experience in helping ladies and has comparable financing procedures to Share the Dignity consequently, giving rivalry to Share the Dignity. Thirdly, there is the Global Womens Project which gives is a worldwide association that additionally has establishes in Australia, and this likewise offers an immediate rivalry to the Share Dignity Company since they have comparative clients (Blythe, 2009). To look at these players in the non-benefit industry, these contenders have similitudes as in they all help penniless and vagrants in Australia. In addition, they are likewise comparative in the way that their clients are the givers for raising support activities, the generosity individuals, and the volunteer specialists. With respect to, not at all like Share the Dignity which represents considerable authority in giving out the clean cushions explicitly, those associations participate in numerous different exercises. Along these lines, they can give rivalry to Share Dignity association. Shopper Analysis The objective market for Share the Dignity is in Australia and the whole world. In the market, the organization focusses on the legislatures which offer it the awards, other benefit, and non-benefit making associations that have an enthusiasm for supporting ladies. In addition, the overall population is likewise significant shoppers of this association since it frames the contributors for raising money, the positive attitude individuals and furthermore the volunteers who are happy to offer the administrations to the association. A gander at the organization's advanced system, it is clear that association has adjusted well the business with its media technique. Customers can be reached through the organization site; the internet based life handles other advanced openings such the email communicate and SEO. These techniques have a basic job to the association since they can arrive at the different objective market in Australia and the whole globe. Promoting Mix Analysis The fundamental showcasing methodologies that Share the Dignity organization utilizes can be summed up into the 4Ps (Product, Price, Place, and Promotion). With respect to, the association has demonstrated endeavors by guaranteeing that benefactors can without much of a stretch discover them for raising money through their site and different ways. At the cost, given that association is non-benefit making, it doesn't have any cutoff to its clients. The item, the association, takes care of the penniless and vagrants in Australia by furnishing them with sterile cushions (Karniel Reich, 2011). Offer the Dignity has embraced different limited time exercises to arrive at the greater part of the clients on the planet. This has experienced internet based life, print media and numerous others (Wooliscroft, 2011). Site Analysis As of now, a site is a basic part of the matter of an association. Offer the Dignity association has its site at www.sharethedignity.com.au. The site has assumed a basic job in arriving at the clients of this organization. It has been utilized an empowering agent for raising and arriving at the clients; benefactors, well-wishers, and volunteer laborers. The organizations have a coordinated installment framework where the contributors are capable effectively and helpfully to make their commitments to the association (James, 2014). Internet based life Audit The association likewise has its web-based social networking stages set up. This has assumed a basic job in contacting more individuals who are potential clients to the association. These internet based life stages the association has grasped incorporate Facebook, Twitter, LinkedIn, YouTube, Google+, Instagram, Snapchat and furthermore the portable application. The internet based life has been viable and figured out how to arrive at a huge number of the companys potential customers. Advanced Tactics of Share Dignity As of now, the vast majority are found at the social locales. Consequently, it has been a smart thought to have the social stages for the organization set up. In any case, the organization has likewise incorporated other media showcasing viewpoints in its business for viability. These incorporate email advertising, where it keeps up a rundown of clients messages for correspondence. Additionally, the Search Engine Optimization (SEO) positioning is significant on the grounds that it causes clients to rapidly arrive at the organization on the web. End A few issues have been distinguished in this situational examination, and they incorporate the administration of the internet based life for the venture. The web based life should keep clients refreshed with most recent data and advancements. Another issue is an improvement of the SEO of the organization site. There is have to enhance the web index for the venture's site. Furthermore, the association needs to receive the new advanced procedures, for example, joining of Google AdSense which will guarantee that many arrive at the gathering pledges activities of this association. All in all, fruitful administration of each business association requests a superior comprehension of the predominant situational factors. This involves a comprehension of the earth through devices, for example, the SWOT and Pestle examination, Competitors Analysis, Consumer Analysis. Other than a comprehension of rivals in the market and their degree of the grasp of the computerized, social strategies in the business. A comprehension of these elements is basic in helping the organization increment its efficiency. References Blythe, J. (2009). Key Concepts in Marketing. London: Sage Publications. ISBN978-1-84787-498-6. Charles, L. (2012). Fundamentals of Marketing (7ed.). Bricklayer, OH: South-Western Cengage Learning. Chermack, J. Bernadette K. (December 2007). The Use of and Misuse of SWOT examination and suggestions for HRD experts. Human Resource Development International.10(4): 383399. Hanssan, A. (2012). Assessment arrangement and enterprise: experimental proof from Sweden. Small Business Economics. James, W. (2014). Your Small Business Adventure: Finding Your Niche and Growing a Successful Business. ALA/Huron Street Press. ISBN978-1-937589-44-8. John A., Katherine E. Jocz (2012). All Business is Local: Why Place Matters More than at any other time in a Global, Virtual World. Penguin. Karniel, A. Reich, Y. (2011).Managing the Dynamic of New Product Development Processes. Another Product Lifecycle Management Paradigm. Springer. pp.13.ISBN978-0-85729-569-9. Mankiw, N. Taylor, M. (2011). Financial matters (second ed., reexamined Ed.). Andover: Cengage Learning. Wooliscroft, B. (2011). Advertising Theory as History. Showcasing Theory, 11(4), 499-501.

Challenging behavior Literature review Example | Topics and Well Written Essays - 1250 words

Testing conduct - Literature survey Example A kid may create various practices at various ages yet the difficult conduct is generally indispensable since it should be controlled as ahead of schedule as could be expected under the circumstances. A great many people accept that the achievement of a kid will rely upon his capacity to react to anticipation of testing practices. Youngsters who Challenging conduct is an anomalous conduct whose recurrence, event and term enormously risk the physical security of the person in question and that of other around him. This conduct as a rule denies the individual access to standard social and common offices .Challenging conduct could be shown by any individual, however is generally knowledgeable about individuals with formative handicaps, dementia, and psychosis and regularly by youngsters (Kaiser, 2007, p.13) Types of testing practices Challenging conduct can show itself in various manners relying upon the idea of the kid. Quiet youngsters will respond totally different from rough kids mu ch under similar conditions. The most significant thing is that the youngsters will respond adversely to those individuals who cause their issues. A typical trademark is self harming conduct particularly when a kid is irate. This includes taking part in practices that cause physical damage or wounds to oneself. Most kids will do this promptly they are affronted so as to look for consideration. For instance hitting oneself, gnawing oneself or even head butting. Most youngsters create forceful conduct towards those individuals who impact them to build up a difficult conduct. This includes all conduct that hurts others truly. They incorporate hitting others, gnawing others, spitting on others, shouting, punching and kicking others among others (Emerson, 1995, p.24) Some kids take part in early sexual conduct to communicate their emotions about certain things. This incorporates all demonstrations of irregular conduct including sexual acts. For instance open masturbation, assault, and vi ewing obscene materials. Property coordinated conduct hypothesis. In this hypothesis, the conduct of the kid is accepted to make obliteration property around the individual. The response of the kid is connected to the It incorporates tossing of articles, physical obliteration and in any event, taking (Emerson, 2011, p.32) Theories of testing practices Challenging conduct has different causes, the greater part of which are outside the ability to control of the individual showing the conduct. These could be social, organic, natural or mental among others. Social reason hypothesis: this hypothesis holds that the conduct of a youngster is credited by the general public in which he is raised. These are the social connections with others, companions and family members. This depends on the hypothesis that practices are improved and impacted by perception on others activities and practices. Weariness, social cooperation and social standards could trigger particular sort of practices in indi viduals. Friend gatherings could likewise impact one to build up certain irregular practices. The youngsters will embrace the practices that are regular in a specific culture. A general public that doesn't denounce a difficult conduct will urge the kids to be uncontrollable. (David, 1974, p.53) Biological causes hypothesis. This hypothesis depends on the hypothesis that conduct results from organic and biochemical procedures. Conditions, for example, torment and certain drug may cause testing practices. Certain acquired qualities from guardians likewise impact conduct. The kid takes in a large portion of the practices from their folks. The guardians ought to abstain from acting in a way that inclines the kids to a difficult beh

Converting Cubic Inches to Cubic Feet

Changing over Cubic Inches to Cubic Feet Changing over cubic creeps to cubic feet is a typical issue. Here is the change factor and a worked model. Change Factor 1728 cubic inches 1 cubic foot 0.000578704 cubic feet 1 cubic inch Straightforward Example Convert 54.9 cubic creeps into cubic feet. You can increase by the change factor: 54.9 cubic inches x 0.000578704 cubic feet per cubic inch 0.0318 cubic feet or on the other hand you could basically divideâ the esteem in cubic feet by 1728: 54.9 cubic inches/1728 cubic feet for each cubic inch 0.0318 cubic feet Worked Example You measure a case and discover it is 12 inches in length, 5 inches high, and 3 inches down. The volume of the crate is length x width x tallness so the volume of the case is: 12 x 5 x 3 volume in cubic inches 180 cubic inches Presently, to change over this to cubic feet, you know there are 1728 cubic creeps in 1 cubic foot: 180 cubic inches x (1 cubic foot/1728 cubic inches) volume in cubic feet 180 cubic inches x 0.000578704 cubic feet/cubic inch volume in cubic feet 0.1042 cubic feet

Alexander Hamilton and Thomas Jefferson Essay Example | Topics and Well Written Essays - 250 words

Alexander Hamilton and Thomas Jefferson - Essay Example As the President he took a shot at inversions of various Federalist strategies like the evacuation of the extract charge on bourbon, the cut back on military spending plan and decrease of the national obligation, made sure about nullification of the Naturalization Act, and furthermore chipped away at getting the Alien and Sedition Act to lapse and acquitted anybody influenced by it (Wright, 2008). Alexander Hamilton then again was brought into the world straightforward anyway he developed to turn into an exceptionally solid and perceived figure in the incredible focal government. He also joined the Cabinet as the secretary of the treasury when Washington turned into the primary President in 1789. He has end up being extremely gainful and has made sure about in the making of the national bank and managing the administration obligations (DiLorenzo, 2008). Despite the fact that he had been extremely useful to the nation and end up being an exceptionally effective pioneer, he was in an outrage of two-timing issues which drove him to be open scorned. Both these pioneers were similarly acceptable strategically and with the sum they gave to people in general, anyway Alexander Hamilton was the one in particular who was confronted with a great deal of open derision. Later Alexander expected to pick between two political rivals for the administration and this is the point at which the two, Hamilton and Jefferson dealt with a similar side to prevail upon Adam Burr, who was viewed as deceitful. Both Hamilton just as Jefferson were to incredibly principled and centered (Wright,

5 Ways to Tell If Youre Using the Wrong 529 Plan

According to our 3nd annual College Savings Survey, 72 percent of the respondents who intend to open a 529 plan don't know the steps involved. What's more, 50 percent said they don't know which investment option they would choose when they enroll. So what if your family is currently using a 529 plan? There are over 100 different plans available ï ¿ ½ how can you be sure you made the right decision and are using the best plan to suit your family's needs? Here are five red flags to look out for: 1. You're not happy with your plan's performance We're talking about investment performance, which determines how much your plan is worth. In other words, poor performance can eat away at your savings. One way to check your plans performance is to look at Savingforcollege.com's quarterly performance rankings. Of course, past performance does not guarantee future results, but the rankings will give you a good idea of how your plan compares to other 529 plans. RELATED: When should you switch 529 plans? 2. You're missing out on a state tax deduction or credit 35 states, including the District of Columbia, offer a tax deduction or credit for residents. If you're using an out-of-state plan, check to see if there would be any benefit to switching to your home state's plan. You can rollover your 529 plan balance into a different 529 plan one time in a 12-month period. RELATED: Should you use your employer-sponsored 529 plan? 3. You chose a static option years ago and have yet to adjust your allocations When you enroll in a 529 plan you selected your investment options and likely had to choose between static or age-based options. A static option is not programmed to change over time, which means as your child gets closer to college it's up to you to shift the underlying investments toward less risky options. If you're not comfortable managing your investments that way you can switch to an age-based option, which will automatically adjust your investments based on the age of the child you are saving for. Since age-based options do most of the work for you, they are typically the best option for most people. Yet according to our College Savings Survey only 20 percent of families who intend on opening a 529 plan said they would choose this option. RELATED: Investing trends of 529 college savers 4. You're paying high fees that are eroding your savings Our College Savings survey showed that families listed "low management fees" as a criteria for selecting their 529 plan. These fees, along with expenses of the underlying investments and sales charges if you're using an advisor-sold plan, can have a big impact on your savings. To see the plans with the lowest fees, check out Savingforcollege.com's 529 Fee Study. RELATED: 8 common 529 plan mistakes to avoid 5. You're not on track to meet your savings goal It's important to give your 529 plan regular check-ups to make sure you'll have the money you need when it's time for college. If you find that your plan's investment strategy isn't working in your favor, it might be time for a change. A tool like the College Savings Planner can help you identify your goals based on your child's age and real expected college costs. RELATED: The magic number for college savings According to our 3nd annual College Savings Survey, 72 percent of the respondents who intend to open a 529 plan don't know the steps involved. What's more, 50 percent said they don't know which investment option they would choose when they enroll. So what if your family is currently using a 529 plan? There are over 100 different plans available ï ¿ ½ how can you be sure you made the right decision and are using the best plan to suit your family's needs? Here are five red flags to look out for: 1. You're not happy with your plan's performance We're talking about investment performance, which determines how much your plan is worth. In other words, poor performance can eat away at your savings. One way to check your plans performance is to look at Savingforcollege.com's quarterly performance rankings. Of course, past performance does not guarantee future results, but the rankings will give you a good idea of how your plan compares to other 529 plans. RELATED: When should you switch 529 plans? 2. You're missing out on a state tax deduction or credit 35 states, including the District of Columbia, offer a tax deduction or credit for residents. If you're using an out-of-state plan, check to see if there would be any benefit to switching to your home state's plan. You can rollover your 529 plan balance into a different 529 plan one time in a 12-month period. RELATED: Should you use your employer-sponsored 529 plan? 3. You chose a static option years ago and have yet to adjust your allocations When you enroll in a 529 plan you selected your investment options and likely had to choose between static or age-based options. A static option is not programmed to change over time, which means as your child gets closer to college it's up to you to shift the underlying investments toward less risky options. If you're not comfortable managing your investments that way you can switch to an age-based option, which will automatically adjust your investments based on the age of the child you are saving for. Since age-based options do most of the work for you, they are typically the best option for most people. Yet according to our College Savings Survey only 20 percent of families who intend on opening a 529 plan said they would choose this option. RELATED: Investing trends of 529 college savers 4. You're paying high fees that are eroding your savings Our College Savings survey showed that families listed "low management fees" as a criteria for selecting their 529 plan. These fees, along with expenses of the underlying investments and sales charges if you're using an advisor-sold plan, can have a big impact on your savings. To see the plans with the lowest fees, check out Savingforcollege.com's 529 Fee Study. RELATED: 8 common 529 plan mistakes to avoid 5. You're not on track to meet your savings goal It's important to give your 529 plan regular check-ups to make sure you'll have the money you need when it's time for college. If you find that your plan's investment strategy isn't working in your favor, it might be time for a change. A tool like the College Savings Planner can help you identify your goals based on your child's age and real expected college costs. RELATED: The magic number for college savings

The Dos and Donts of Essay Topics for Macroeconomics

The Do's and Don'ts of Essay Topics for Macroeconomics The One Thing to Do for Essay Topics for Macroeconomics All these questions and lots of others will bug your mind if you would like to compose an outstanding paper in macroeconomics. One of the most difficult things about writing a paper is locating a good topic to write about. A term paper carries a great deal of weight in it when it regards your final year effects. A research paper isn't something that's written overnight. Or the discussion might concentrate on one specific group (for instance, millennials) and the way in which they affect the economy. It will be quite useful for analyzing the economies of any countries with the assistance of inflation and stagflation too. The association between the nation's politics and its economy. The function of the technological progress in the increase of financial crisis. The Rise of Essay Topics for Macroeconomics Always keep in mind that the success of your research is d ependent on it. Our main goal is to provide a personal approach to each client which uses our service and tailor our services to satisfy their needs. Additional advanced resource links consist of primary source materials that are available on the internet or graduate level readings. You could struggle to discover the research that aids your argument, and you may also be second-guessing your own understanding. What Everybody Dislikes About Essay Topics for Macroeconomics and Why The majority of the times, governments make some wonderful policies. Whether you decide to compose a paper about the financial effect of pollution or the effects of a different nation's financial decline, you ought to be in a position to craft a relevant, interesting paper that's supported by peer-reviewed data. It is necessary for humanity to use their understanding and effort in uplifting society through academe and societal institutions. Economic advantages caused by changes in climate. The War Ag ainst Essay Topics for Macroeconomics Imagine the caliber of writing you'll receive from a low-cost essay writing service. Thus, the best method to make certain you score as large marks as possible is by selecting the most outstanding subjects. Remember your prof may, obviously, have various ideas regarding what kind of assignment you ought to be writing. A student is made to dig in deeper into the information, and here, they've got a chance to grasp the concepts that they've learned and be in a position to apply them to a topic that's realistic. Ruthless Essay Topics for Macroeconomics Strategies Exploited You are able to also list reasons it happened and discuss the way the government is attempting to make sure that it doesn't happen again. It will be a lot less difficult to write about something you get a personal interest in. Therefore, an individual would talk about the subject of inflation and stagflation in detail to comprehend the function of these entities in the to pic of economics. To tell the truth, if you truly wish to write about a mutual subject that's been resolved, you can do so, but you'd better attempt to take a look at the problem from a new perspective. Writing a paper on macroeconomics can be a really intriguing job, if you're passionate about the discipline. If you order macroeconomics essay from us, you are certain to get a perfect bit of writing that is going to be absolutely original and faultless. Macroeconomics is an intriguing subject that provides you hundreds of good topics to select from when you decide what things to write about in your term paper. If you are fighting to discover a great topic for your macroeconomics paper, have a look at a few of these. Macroeconomics is an intriguing science, even supposing it isn't a specific science. It's quite near to the micro and macroeconomics too. In the majority of cases, you might have to compare and contrast macroeconomics and microeconomics. The end result of these 3 scenarios is constant stress, submitting poor assignments, which results in inadequate performance. For this reason, you must begin brainstorming for ideas once you're given the assignment. Occasionally it becomes tough to manage lots of college assignments given with the exact deadline. Do that by creating a strong thesis and back it up with important parts of information. Generally the scarce resources are among the most important subject of studying in economic researches. Creating a strong thesis really does depend on the degree of the topic you select. Finding an easy and effortless topic for economics thesis isn't that easy for the students which is the reason why they can take this completely free help easily to compose their thesis on the innovative topic. On the flip side, in the event the administration set up assessments. Type of Essay Topics for Macroeconomics Topic pretty well says everything. It isn't necessary that you enjoy each of these topics or use them as is. If you believe, selecting a perfect topic will take more than expected you can search for good topics online. Just choose a topic and write! Yet, the most intriguing discussions are the ones that are current and relevant. For instance, if you should go over the General Motors, you could discuss whether you thought it turned out to be an excellent idea dependent on the facts that you gather. If you are joyful that you'll have the ability to carry out any crucial research based on a topic which you like the sound of, then it could be appropriate to proceed. 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The Attack Essay - 918 Words

The Attack by Yasmina Khadra follows the story of Dr. Amin Jaafari, a successful Palestinian surgeon who resides in Israel. He lives with his beloved wife, Sihem, live alone in a beautiful house. Amin and Sihem live happily until an unimaginable event occurs, Sihem is killed in a suicide attack. The police discover that Sihem was not a victim, but the bomber. Amin desperately attempts to understand how his kind, loving wife could possibly carry out such an attack. He sets out on an incredible journey to investigate the secret life Sihem lived outside of their marriage. In the novel, The Attack, Yasmina Khadra illustrates the intense frustration that precedes suicide attacks and the intense physical and physiological results of†¦show more content†¦(158) The commander held Amin for a week so he could experience the same pain, suffering and frustration that the Commander and other extremists feel. As they continue to lose their homeland the desperation as a result of the conflict the deep seated hatred only grows deeper. The aftermath of a suicide attack has both physical and ideological consequences. The physical effects of the bombing are much more apparent. While working in the emergency room in the aftermath of Sihem’s suicide attack, Amin can see firsthand how devastating these attacks can be. In the course of fifteen minutes, the lobby of the emergency room is transformed into a battlefield. No fewer than a hundred wounded people are packed into this space, the majority of them lying on the floor. All the gurney are loaded with broken bodies, many horribly riddles with splinter and shard... The whole hospital echoes with wailing and screaming... (14-15) This tragic scene illuminates how many people can be hurt and ki lled in suicidal events. The author is trying to show what the terrible consequences of Sihem’s actions were. In another instance, Wissam Jaafari makes those around him suffer physical repercussions The officer informs me that as a consequence of the suicide operation carried out by Wissam Jaafari against an army checkpoint and in accordance with the instructions hes received from his superiors , we have a half of an hour to evacuate the dwellingShow MoreRelatedCyber Attacks, Intrusion Attacks And Network Attacks1285 Words   |  6 Pagesdifferent types of attacks that their organization faces. Some of the attacks include cyber- attacks, intrusion attacks, network attacks, and social engineering attacks. This paper will present research and comparison of methods of cyber-attacks, intrusion attacks and network attacks. Comparison of Cyber-Attacks, Intrusion Attacks, and Network Attacks Cyberterrorism, cyber campaigns, and cyber-warfare are all considered forms of cyber-attacks. 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The Paris and Brussels attacks are proof thatRead MoreCorrelation Between the Attack on Pearl Harbor and the Attack September 111231 Words   |  5 PagesThe events of the terrorist attack on September 11, 2001 changed not just the American way of life, but the way in which terrorism will forever be viewed. Moments after the first plane changed our way of life by crashing into the first tower, it also changed the way people will live their lives for many generations from that point forward. For the first time on Continental United States soil, America was under siege. Months after the attack many political figures and citizens wondered aboutRead MoreThe Attack Of The United States2417 Words   |  10 PagesSeptember 11, 2001 is a day that will forever live in every American’s mind. The terrorist attack that took nearly 3,000 lives was highly publicized, many channels played live footage of New York City as the Twin Towers were attacked. 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Introduction Mobile Ad-hoc Network (MANET) attached in a dynamic manner and it is an assembly of wireless mobile nodes. Without any fixed infrastructure nodes making a temporary network where all nodes are arbitrarily free to move. In the network nodes are act as routers, which take part in finding and maintenance of routes to other nodes [1]. Wireless connection in MANET is highly misplay and due to mobility of nodes it goes downRead MoreCyber Attacks And Physical Damage1279 Words   |  6 Pages Cyber-Attacks and Physical Damage Name Course details Supervisor Institution Date â€Æ' Abstract The internet is considered to be one of the best things to have been invented in the world. It allows for the fastest transfer of information from one place to another with the least hassle. Furthermore, it has made communication and information easier to find due to the vast and c0neected nature of the world. However, in the recent years, not only have the positive outcomes of the internet become thingsRead MoreCyber Attack On The United States1674 Words   |  7 PagesAbstract Cyber attack has been a huge problem for so many years and there have been a lot of attempts to stop it but there have not been enough resources for this to happen. This paper offers more top to bottom clarification of Cyber attack, reasons, dangers, and defenselessness. It talks about the impact on individual; gives situations of Cyber attacks lastly clarifies ways that people can keep themselves from being casualties of Cyber attack. This paper will give insights on how Cyber attack impacted

The Big 10 Schools A Comparison of the Universities

The Big Ten Athletic Conference includes some of the countrys top public universities as well as one of the countrys top private universities. All are large research universities with significant masters and doctoral degree programs in addition to undergraduate programs.  On the athletic front, these Division I schools also have many strengths. Acceptance and graduation rates, however, vary widely. The table below puts the 14 Big Ten schools side-by-side for easy comparison. Fast Facts: The Big Ten Conference Northwestern University is the only private university in the conference, and it also the most selective.The Ohio State University has the largest undergraduate enrollment in the Big Ten. Northwestern is the smallest.The University of Nebraska has the lowest 4-year and 6-year graduation rates in the conference.The University of Iowa provides the highest percentage of students with grant aid. In the table below, you can click on a universitys name to get more admissions information including SAT score, ACT score, and GPA data for admitted students.   A Comparison of the Big Ten Universities University Undergrad Enrollment Acceptance Rate Grant Aid Recipients 4-Year Graduation Rate 6-Year Graduation Rate Illinois 33,955 62% 49% 70% 84% Indiana 33,429 77% 63% 64% 78% Iowa 24,503 83% 84% 53% 73% Maryland 29,868 47% 61% 70% 86% Michigan 29,821 23% 50% 79% 92% Michigan State 38,996 78% 48% 53% 80% Minnesota 35,433 52% 62% 65% 80% Nebraska 20,954 80% 75% 41% 69% Northwestern 8,700 8% 60% 84% 94% Ohio State 45,946 52% 74% 59% 84% Penn State 40,835 56% 34% 66% 85% Purdue 32,132 58% 50% 55% 81% Rutgers 35,641 60% 49% 61% 80% Wisconsin 31,358 52% 50% 61% 87% Undergraduate Enrollment: Northwestern University is obviously the smallest of the schools in the Big Ten while The Ohio State University is the largest. Even Northwestern, however, is a big school with over 22,000 students when graduate students are taken into consideration. Students looking for a more intimate college environment in which they will get to know their peers and professors well would do better at a liberal arts college than one of the members of the Big Ten. But for students looking for a large, bustling campus with lots of school spirit, the conference is certainly worth serious consideration. Acceptance Rate:  Northwestern is not just the smallest school in the Big Ten—it is also by far the most selective. Youre going to need high grades and standardized test scores to get in. Michigan is also highly selective, especially for a public institution. To get a sense of your chances of admission, check out these articles: SAT Score Comparison for the Big Ten | ACT Score Comparison for the Big Ten. Grant Aid:  The percentage of students receiving grant aid has been on the decline in recent years among most of the Big Ten schools. Iowa and Ohio State award grant aid to a sizable majority of students, but other schools dont do nearly as well. This can be a significant factor in choosing a school when Northwesterns price tag is over $74,000, and even a public university such as Michigan costs over $64,000 for out-of-state applicants. 4-Year Graduation Rate:  We typically think of college as a four-year investment, but the reality is that a significant percentage of students do  not  graduate in four years. Northwestern clearly does the best at getting students out the door in four years, in a large part because the school is so selective that it enrolls students who are entering well prepared for college, often with lots of AP credits. Graduation rates should be a factor when you consider a school, for a five- or six-year investment is clearly a very different equation than a four-year investment. Thats one or two more years of paying tuition, and fewer years of earning an income. Nebraskas 36% four-year graduation rate really stands out as a problem. 6-Year Graduation Rate:  There are lots of reasons why students dont graduate in four years--work, family obligations, co-op or certification requirements, and so on. For this reason, six-year graduation rates are a common measure of a schools success. The members of the Big Ten do pretty well on this front. All schools graduate at least two-thirds of students in six years, and most are above 80%. Here again Northwestern outperforms all the public universities--high cost and highly selective admissions has its benefits.   Data Source: The National Center for Education Statistics

The Adventures Of Huckleberry Finn By Mark Twain - 1649 Words

â€Å"I was thinking about how everybody can’t be everything to each other, but some people can be something to each other†(Winik). In The Adventures of Huckleberry Finn the main character, Huck, creates many relationships during his adventure on the river, and these relationships impacts what kind of person Huck becomes. Mark Twain in his novel The Adventures of Huckleberry Finn expresses how every single relationship, whether it lasts one week, several years, or a lifetime, shapes how a person lives his/her life. The Widow in The Adventures of Huckleberry Finn fills the role as Huck’s mother figure. The relationship that Huck has with the Widow can be seen as strained because of how the widow tries to instill proper morals and civilize Huck.†¦show more content†¦The Widow and Huck’s relationship progresses so much, from the beginning with Huck running away to him deciding to behave and act civilized. The relationship the Widow has made with Huck has impacted Huck to the point of him truly putting an effort towards being civilized. Without this relationship Huck would still be that wild uncivilized boy he had been before living with the widow. A negative relationship that Huck has is with his father who is referred to as â€Å"Pap†. Pap is an abusive, alcoholic racist who beats Huck and tries to take his money. Pap has only a negative impact on Huck by instilling terrible morals in Huck. The reader can tell how Huck feels about his father when he thinks, â€Å"Pap he hadn t been seen for more than a year, and that was comfortable for me; I didn t want to see him no more. He used to always whale me when he was sober and could get his hands on me; though I used to take to the woods most of the time when he was around†(Twain 11). It is obvious that Huck does not like the relationship he has with his father by him saying that he was â€Å"comfortable† with Pap being away for over a year. Even though Pap abuses Huck and is a drunk, Huck still likes the free and uncivilized aspect

Greatest Haitian Leader (Toussaint L’ouverture) - 1129 Words

Toussaint L’ouverture was born on May 20, 1743 in Haiti. Toussaint was one of the many who was born into slavery. Toussaint belonged to a very small class of slaves who had some privileges. Toussaint’s owner encouraged him to read and write; later on Toussaint became interested in books. This had a large impact on his political life. Toussaint was some of the few black males to be freed from slavery when he was in his thirties. After being freed from slavery Toussaint became a slave and landowner. An agent of change is a person, event or time period, which had significant impact on the society, which alters the way we do things today. Toussaint L’ouverture deserves to be an agent of change not only because he was an intelligent slave who overcame and defeated slavery for his colony, he also turn the society of slaves into independent state with there own government. He fought for the rights that the slaves were not receiving. Some of the rights that Toussaint Là ¢â‚¬â„¢ouverture was fighting for was things like equality and labor opportunities, and basic rights of a human. He made the stronger more powerful countries acknowledge the slaves as equals. Toussaint L’ouverture drove out the French and made them sell the land, which are now about 15 of our states in America. His hard work, effort, and the will to fight gave him a edge to achieve these accomplishments which goes down in history to be one of the greatest rebellions to have occurred. This has significant impact becauseShow MoreRelatedWhat Did The Haitian Revolution Do For End Racial Slavery And How Successful Was It?1381 Words   |  6 PagesCarrascoso 3/2/17 What did the Haitian Revolution do to end racial slavery and how successful was it in doing so? The Haitian Revolution was the first ever ‘successful’ slave rebellion, which took place from 1791-1804. This was followed by the Independence of Haiti, Racial Slavery is the forced labor of individuals and discrimination based on race, in which racism and hatred are the core. Racial slavery leads to the dehumanization of the slave. One of the greatest instances of racial slavery wasRead MoreHistory SBA1710 Words   |  7 PagesFrench and British colonies and from Europe. Cuba with seven documented significant insurrections in the 19th Century is second to Jamaica, which had 14 verified slave rebellions from the mid 18th Century to the mid 19th. The greatest slave revolution in Jamaica was the Baptist War of 1831-1832. It began simply as a general strike during the Christmas season. The slaves, led by one Samuel Sharp, wanted liberation and decent paid. It is not clear why it turned into a fully fledgedRead MoreRevolutionary Changes in the Atlantic World, 1750–185010951 Words   |  44 Pagesthe same time, these monarchs suppressed or banned radical ideas that promoted republicanism or attacked religion. 5. Many of the major intellectuals of the Enlightenment communicated with each other and with political leaders. Women were instrumental in the dissemination of their ideas, purchasing and discussing the writings of the Enlightenment thinkers and, in the case of wealthy Parisian women, making their homes available for salons

Historical Journeys free essay sample

Historical Journeys As I prepare to graduate from high school, I am excited about the opportunities that I will have as a college student to study history more in depth. I have been lucky enough in my life to have made several fascinating trips. Three trips in particular stand out as times when my knowledge base grew. When I returned home I searched for books on the areas and historical events where I had visited. When I was only ten years old, I traveled from Alabama to Washington state learning about the western expansion following the Louisiana Purchase in 1803. When I was thirteen, my family and I spent several weeks in Puerto Vallarta, Mexico. Finally, at age sixteen, my family visited England to trace both my mother’s and father’s family backgrounds. My love of history will be an important part of my life no matter what career field I may choose. We will write a custom essay sample on Historical Journeys or any similar topic specifically for you Do Not WasteYour Time HIRE WRITER Only 13.90 / page My grandfather was very excited about the chance to teach me to love the west as he did. From St. Louis, Missouri, we followed the Oregon Trail. I learned about Indians like Crazy Horse, Red Cloud, and Sitting Bull. Lewis and Clark, Jesse James, Kit Carson, George Armstrong Custer, and Buffalo Bill became as familiar as any of my friends. I remember walking the Little Big Horn battle field; I remember climbing Independence Rock; I remember seeing Mount Rushmore; I remember feeling the indentions in solid rock from the wagons on the Oregon Trail. These are only a few of the many memories that come to mind when I read or study any of the History of the West. My vacation in Mexico was culture shock. I knew Mexico was not a wealthy nation and I knew its history was impacted by the Mexican War. However, the widespread poverty was almost unimaginable. The lack of education and modern facilities; the poorly constructed houses and inferior transportation were obvious. I learned to appreciate the United States and was pleased when we headed home. I was interested in learning more about their history to try to understand why our two cultures were so different. All I need to say about England is The British Museum. We had time to spend four hours there. My family says they saw everything; I only made it to the Byzantine Empire. I expect to return this summer, with the hopes of spending at least another entire day at the museum. I realize I will need to make many trips to England if I am to visit the major historical locations. I look forward to those opportunities. We did visit the industrial area of England where my mother’s ancestors originated. We also visited York, where my father’s ancestors pillaged (Vikings). These are only three examples of all of the history I have learned by traveling and studying. However, I still feel my knowledge is only superficial and I will happily spend many hours learning from those who have done much a much more in depth study than I.

A Fatal Utopia Essay Example For Students

A Fatal Utopia Essay Two ways of exercising power over men, of controlling their relations, of separating out their dangerous mixtures. The plague stricken town, transversed throughout with hierarchy, surveillance, observation, writing; the town immobilized by the functioning of an extensive power that bears in a distinct way over all individual bodies-this is the utopia of the perfectly governed city Page 6 Michel Foucault This quote extracted from the Essay Panopticism written by Michel Foucault perfectly describes in detail the controls put on the citizens of Big Brothers Oceania in George Orwells 1984. Through control of relation, surveillance, and separating out their dangerous mixtures Big Brother obtains a government system, which is described by Foucault as a Utopia. The Utopia relies upon more than one method of control as noted by Foucault, but the far most significant method is surveillance. The telescreen received and transmittes simultaneously. Any sound that Winston made, about the level of a very low whisper, would be picked up by it; moreover, as long he remained within the field of vision which the metal plaque was commanded, he could be seen as well as heard Page 6 Orwell. In Orwells 1984 the surveillance is achieved by the distressing, constant presence of the telescreen. No individual was free of the thought that the vigilant telescreen would catch you in some undefiable act, which would lead to unquestionable vaporization. This enclosed, segmented space, observed at every point, in which the individuals are inserted in a fixed place, in which the slightest movements are supervised, in which all events are recordedà ¢Ã¢â€š ¬Ã‚ ¦ Page 5 Foucault The panopticon in Foucault essay is stressing exactly the same purpose as that of the telescreen in Orwells 1984. Simply that constant threat of punishment , but only if caught and in order to remain safe you must constantly behave as the ideal you are desired to be. To have the threat of being watched present, but not knowing whether or not in actuality you are being watched is what leads you to become that ideal member of the Utopia which in essence is exactly what the government wants. The effect of the telescreen was simply complete control over Winstons every move, thought, and word that came from his mouth. You watch as he lived in life in fear of the Party, in fear of the Thought Police stripping him of the little that he does have. His mind was racing full of thoughts he couldnt speak. His heart burned with hatred and anger for the party, which he couldnt act upon. You can see by the party watching him they suppressed what he said, and did all by posing that tiny threat of being under watch. The very same threat of being watched constantly can make ones stomach turn but to know that there is an outside force slowly integrating its way into your life can be just as nerve racking. Thus the Christian school must not simply train docile children; it must also make it possible to supervise the parents, to gain information as to their way of life, their resources, their piety, and their morals. Page 18 Foucault To truly understand someone you must not only know the individual, but those associated with that individual. By knowing and understanding the lifestyle that this person chooses to live you have gained more control than you can ever imagine. You have identified every thing that has and will take place within that persons life. You are now a part of their life due to your relations with those in their own lives. Foucault stresses the role of the parent in the lives of the child, and in 1984 you find it ironically twisted into the role of the children in the lives of the parent You watch as children are fed thoughts, absorb them, and with a young mind those thoughts are clenched tight. To them its simple there is one way, and it is the right way and anything other than that is wrong and you deserved the proper punishment. .u4c765d4286112113f24347aa09c72d0c , .u4c765d4286112113f24347aa09c72d0c .postImageUrl , .u4c765d4286112113f24347aa09c72d0c .centered-text-area { min-height: 80px; position: relative; } .u4c765d4286112113f24347aa09c72d0c , .u4c765d4286112113f24347aa09c72d0c:hover , .u4c765d4286112113f24347aa09c72d0c:visited , .u4c765d4286112113f24347aa09c72d0c:active { border:0!important; } .u4c765d4286112113f24347aa09c72d0c .clearfix:after { content: ""; display: table; clear: both; } .u4c765d4286112113f24347aa09c72d0c { display: block; transition: background-color 250ms; webkit-transition: background-color 250ms; width: 100%; opacity: 1; transition: opacity 250ms; webkit-transition: opacity 250ms; background-color: #95A5A6; } .u4c765d4286112113f24347aa09c72d0c:active , .u4c765d4286112113f24347aa09c72d0c:hover { opacity: 1; transition: opacity 250ms; webkit-transition: opacity 250ms; background-color: #2C3E50; } .u4c765d4286112113f24347aa09c72d0c .centered-text-area { width: 100%; position: relative ; } .u4c765d4286112113f24347aa09c72d0c .ctaText { border-bottom: 0 solid #fff; color: #2980B9; font-size: 16px; font-weight: bold; margin: 0; padding: 0; text-decoration: underline; } .u4c765d4286112113f24347aa09c72d0c .postTitle { color: #FFFFFF; font-size: 16px; font-weight: 600; margin: 0; padding: 0; width: 100%; } .u4c765d4286112113f24347aa09c72d0c .ctaButton { background-color: #7F8C8D!important; color: #2980B9; border: none; border-radius: 3px; box-shadow: none; font-size: 14px; font-weight: bold; line-height: 26px; moz-border-radius: 3px; text-align: center; text-decoration: none; text-shadow: none; width: 80px; min-height: 80px; background: url(https://artscolumbia.org/wp-content/plugins/intelly-related-posts/assets/images/simple-arrow.png)no-repeat; position: absolute; right: 0; top: 0; } .u4c765d4286112113f24347aa09c72d0c:hover .ctaButton { background-color: #34495E!important; } .u4c765d4286112113f24347aa09c72d0c .centered-text { display: table; height: 80px; padding-left : 18px; top: 0; } .u4c765d4286112113f24347aa09c72d0c .u4c765d4286112113f24347aa09c72d0c-content { display: table-cell; margin: 0; padding: 0; padding-right: 108px; position: relative; vertical-align: middle; width: 100%; } .u4c765d4286112113f24347aa09c72d0c:after { content: ""; display: block; clear: both; } READ: Biotechnology EssayThey kept on his tail for two hours, right through the woods, and then, when they got into Amersham, handed him over to the patrols. Page 50 Orwell Children punished the adults in the world of 1984, adults punish children in Foucault but either way by controlling family relations you gain inclusive control. That inclusive control is also gained by having the children punishing their elders which makes them some of the most dangerous members of society. Like the watchful eyes of the telescreen the children spy upon your every movement, looking for one slight transgression so they can remove one more defiant member of society. As the little girl followed around the man through the woods you saw the great lengths that these children go to in order to keep the land free of the supposed filth, and to Winston you know it isnt to surprising. He listened to Parsons tell the story and didnt feel anger or animosity towards the children but got that feeling of uneasiness which is what their purpose is. They are simply another threat in Big Brothers society and watch little threat is what gives him power. Of these threats comes one, which is the most evident and understood. The investigation as an authorization search for truth observed or attested was this opposed to the old procedures of the oath, the ordeal, the judicial duel, the judgment of God, or even of the transaction between private individuals. Page 29 Foucault This quote by Foucault is describing the fact that although certain acts are out of compassion, or whether it is moral obligation there is still a crime committed. A crime committed is punishment no matter what the incentive was which led to the action. In 1984 there is the Brotherhood. These are members of the population on Oceania who fight for their freedom. These people want to be heard, want to have equality, want to fight to vanquish the dictatorship imposed upon them by the powerful Big Brother. Morally what there are doing is admirable but in Oceania it is nothing but pure crime and defiance of the power inflicted upon them by Big Brother. These were people who wanted to help the world as a whole ready at moments notice to do anything to abolish Big Brother and what he stood for ready to , cheat, to forge, to blackmail, to corrupt the minds of children, to distribute habit forming drugs,à ¢Ã¢â€š ¬Ã‚ ¦.. Page 142 Orwell To have punishment for crimes no matter what the motive behind it was basically controlled Winstons life in Oceania. He wanted to speak out and tell the world his thoughts. He wanted to tell people how corrupt Oceania indeed was, but he couldnt. He couldnt because he had punishment staring him the face. He knew no matter what he would say, confess, or do in defense he would still be another political rebel silenced by death. He knew that political rebellion was the utmost defiant act upon the government nd all his life held it within afraid of its consequences. However as he got older those consequences entered his life. He knew he was getting involved in an organization, which had no end but death and took that choice. Whether or not his peace of mind was satisfying enough to put an end to his burning hatred within you will not know, because he was silenced as he knew he would be before his ideas were heard. Big Brothers cleverly organized, and cynical government silenced Winstons ideas. He used the methods of control presented to you in Michel Foucaults essay Panopticism, which are: surveillance, control of relations, and separating out their dangerous mixtures. The telescreens presence altered Winstons way of life. Control of relations changed Winstons outlook on the people in his life. Separating the dangerous mixtures is what killed him. He joined a mixture which was separated, separated by killing him. The ironic similarities within the two works are mind boggling and intriguing, but the truth in it all is what is truly remarkable.

aspectos problematicos del espaol essays

aspectos problematicos del espaol essays Aspectos problemticos del espa Conocimientos previos: vocabulario sobre medios de transporte, relaciones entre parejas, verbos de movimiento, presente, pasado/copretrito, futuro. No se dar la regla gramatical, se pretende que el alumno infiera los diferentes usos de las preposiciones. No se presentarn todos los usos de por y para, durante el transcurso de la clase se tomarn slo 4 modalidades: Que el alumno sea capaz de utilizar expresiones con las preposiciones por y para. Dar y pedir informacin en situaciones causales, de finalidad, temporalidad, finalidad y medios. Interactà ºe con sus compan sobre Estados de la Repà ºblica Mexicana. Que el alumno identifique diferentes estados de la Repà ºblica Mexicana. Que hable sobre dichos estados. Que el alumno infiera la reglas para la utilizacin de las preposiciones por y para. 1. Sensibilizacin (3minutos). El maestro har preguntas del tipo  ¿Alguna vez has tenido que separarte de algà ºn ser querido?" Con el objeto de contextualizar al alumno con el tema de la actividad. Presentacin de los personajes. 2. Dilogo. Juego de roles. (1 minuto) Dos alumnos leern el dilogo propuesto en donde se presentar la situacin que da pie al uso de dichas preposiciones. Se le proporcionar al alumno una hoja con preguntas, las cuales contestar con la ayuda de su compan) 3.1 Revisin de las preguntas acentuando la importancia de las preposiciones (todos los usos juntos) (5 minutos). 4. Ejercicios de transformacin (10 minutos) El maestro presentar fotografà ­as de personas y dar informacin extra para asà ­, de esta manera, procurar que el alum...

Graduate Residence Essay Sample

Graduate Residence Essay SampleWhen it comes to writing a sample APA style graduate reflection essay, the key is to think about each paragraph as a self-contained essay. Your task as a grader is to determine whether you need to edit and eliminate any of the statements within.To begin, the first paragraph should be focused on addressing the student's interest or question asked in the admission letter, especially if the student asks for more specific information. The author should then go on to state their name, and describe who they are. For example, if the applicant asked 'How did you come to be the graduate student in this graduate composition class? ', the author could say 'John Doe, PhD, is a professor at Westmont College, in Fort Collins, Colorado.'In the next paragraph, the writer should start out by first answering the author's statement about what he or she hopes to get out of the reader. This is a great opportunity to address all of the writer's thoughts and feelings regardin g the topic. This allows the reader to get a feel for who the writer is and what type of writer they are. After all, this is supposed to be an informal reflection on the topic that will hopefully help the writer to gain clarity.The third paragraph deals with setting the tone for the rest of the essay, starting off by stating that the author does not agree with the viewpoint of any of the other viewpoints discussed. For example, if the writer states that they do not agree with the author's statement that immigration is bad for the country, then the author could answer the statement by saying 'In my field, we rarely discuss that topic. However, I did meet some of the students that are discussing the issue at the local college here in Boulder. Their discussion caused me to reflect on that issue and I think the issue is important, even if I have many strong opinions about the issue.'The fourth paragraph sets up the direction of the essay to follow by providing a place where the writer c an start by starting out with the reader's interest. For example, if the writer states that they wrote the article to help them find a college, they should then start the letter by stating that the student is looking for the best graduate school to attend. The last paragraph then continues to address the author's personal beliefs, calling attention to the writer's personal story and past experiences.Following the fifth paragraph, the next paragraph begins the author's personal experience, concluding by thanking the student for allowing them to share their opinion with the reader. It is important that the student acknowledge that the student is trying to help and educate the reader, and that the author should also encourage the student to continue to write about what the author finds interesting or needs to know.Following the sixth paragraph, the author might then ask the reader to consider what the person does not agree with the author. For example, if the author writes that people should learn Spanish, the author could then thank the student for allowing them to share their opinion and, at the same time, question the individual's opinion, stating that they do not like seeing immigrants in the United States.Graders are encouraged to closely scrutinize the writer's information, grammar, and writing style. Each time the grader asks the author to revise the essay, the grader should be questioning not only the opinions of the writer, but also their own perspective and actions. Graders are encouraged to continue to ask the author to revise the essay as long as they still believe that the piece still contains excellent information.

My Building-Block Approach to Writing a Novel

My Building-Block Approach to Writing a Novel My Building-Block Approach to Writing a Novel Dan Burns is the author of A Fine Line and Recalled to Life. He is also an award-winning writer for the screen and stage. In this article, Dan talks about overcoming the â€Å"where do I start?† question that plagues many writers at the onset of their career. His advice for overcoming that doubt? Start small, and see where your story takes you. Stop asking, â€Å"Where do I start?† and just startMany people talk about being a writer but have difficulty actually getting started. Writing is hard, and often the ideas are there but the approach is elusive. Where do I start?  I get that: it’s scary to put yourself out there and not see your ideas pan out the way you want them to. That’s why I recommend starting small and building on an idea over time. This approach helps me maintain my status as a working writer and overcome roadblocks. If putting your ideas out there scares you, start small and build your story slowly. As for Sebastian Drake, he continues to bug me to write as his story still has so much room for growth. I can only imagine where he’ll take me next.A Fine Line  is available in hard cover  and on  Amazon Kindle.For more information, visit Dan Burns' website.Please  share your thoughts, experiences, or any questions for Dan Burns, in the comments below!

UDP protocol issues Research Paper Example | Topics and Well Written Essays - 1500 words

UDP protocol issues - Research Paper Example Reliable and unreliable do not denote whether certain IT tools will perform properly or not, rather it denotes whether the IT tools can deliver guaranteed services. A reliable IT system usually sets up connections and accordingly confirms the accurateness of data transmission and integrity of data. Reliable protocols perform effectively in such a medium which has high tendency to errors. In contrast, an unreliable IT system makes no attempt to develop a connection and do not verify the data. It also does not make any endowment for data recovery or loss of data. Unreliable protocols perform effectively on such kind of medium which has low possibility of data loss and low tendency of error rates. In this context, it can be stated that Transmission Control Protocol (TCP) is considered as a usual reliable protocol and User Datagram Protocol (UDP) is an example of unreliable protocol (Zandy & Miller, 2002). Unreliable Communication and Its Impact on Application Complexity Communication ch annels usually use one of the two types of protocols namely TCP or UDP. In case of TCP i.e. reliable communication, there exists the acknowledgement of received data, while in UDP, no response exists on communication. Thus, in UDP applications, there can be no guarantee for delivery of information or duplication of information. Unreliable communication of UDP minimizes the complexity of applications which help to reduce overhead. UDP applications are usually simpler than TCP applications, as the key objective is to develop a boundary between networking application procedures, operating in higher layers. UDP takes the data, packages it in simplified format, and sends it to the IP address for transmission. This low level of complexity helps to ascertain rapid transmission of information without any delay. Thus, in short, it can be stated that unreliable communication can reduce complexity and increase rapidity of applications (Sinopoli et al., 2004). Issues Related To the Reliability of an Application That Uses UDP There are several applications where UDP is used and one of the most popular applications is Voice over IP (VoIP). Detailed Analysis of the Application VoIP has become one of the newest applications in the internet. Traditionally, voice and signaling data was directed only through dedicated circuit by using standard TCP protocol on physical links such as telephony channel. This exchange of information occurs from both directions at the similar time. In telephony channel, the voice is perceived through microphone and transmitted to the communication channel. However, in VoIP, the voice is electronically encoded through Pulse Code Modulation (PCM) and then directed through internet channel by data packets. Whereas in typical telephony system, voice data are transmitted at a continuous rate, in VoIP, comparatively small data packets are transmitted at a continuous rate. However, the advantage of VoIP over telephony system is that a high-speed network can transport the data packets for different voice channels and also possibly share with other types of data simultaneously (Bu et al., 2006). UDP protocol is used for transmitting voice data on VoIP network. UDP is considered as ‘send and forget’ protocol with no prerequisite for transmitter to recall sent data packets, because it can lead to reaction error (Bu et al., 2006).

Albert Hourani and Ibn Khaldoun Essay Example | Topics and Well Written Essays - 1750 words

Albert Hourani and Ibn Khaldoun - Essay Example Hourani, in his book, presents his political views while he traces the emergence of Islam, which was born into a pagan area influenced by Christianity and Judaism, where the great powers exercised no direct control that is a breeding ground in both the political and religious. A new political order, headed by a new elite, following the Arab tribes of Mecca, went up between the two empires. The precise role that Mohammed has played in these events is uncertain because the history of his time was reconstituted in writing nearly a century after his death. These biographies tell of events concerning his life, his revelation and his action may not have been fabricated. Muhammad began to communicate his message, announcing the end of the world, the arrival of Reckoning, asking to submit to the Will of Allah, the Arabic word for the God of monotheism. His relations with other tribes deteriorated so quickly that he left Mecca for Medina in the Hijra, the Hijra, which marks the beginning of t he Muslim era. In Medina, Muhammad gathered an increased number of supporters and fought to Quraysh. For Ibn Khaldoun, morality matters in the relationship between the individual and the state to the extent that any resolutions or acts taken by the ruler should be evaluated by the moral principles of the society. As one of the great political thinkers and anthropologists, he was greatly affected by the Islamic traditions that values morality above one’s own interests.

Individual Solve a Problem Paper Essay Example for Free

Individual Solve a Problem Paper Essay This document PHL 458 Week 2 Individual Solve a Problem Paper has solutions of the following question: Write a 1,050- to 1,400-word paper in which you use the creative process to solve a problem with which you have experience. Include the following: †¢ Describe each stage in the creative process using Ch. 5 of your text. †¢ Search for a personal challenge. Explain how you used the techniques to develop curiosity in your search (see Ch. 6). †¢ Express the problem created by this personal challenge (see Ch. 7). Refine your expression of the problem by considering multiple perspectives and deciding which expression is best. Include both the initial and the refined version. Justify your revision. †¢ Investigate the problem by obtaining necessary information (see Ch. 8). List the questions must answer to understand the problem. Identify a source of information to answer each question. †¢ Conclude your investigation with a thorough explanation of the problem, including answers to each of the questions you identified. †¢ Produce ideas toward solution of this problem (see Ch. 9). List all of your ideas—whether they are ridiculous or serious is not important. †¢ Discuss the two ideas that you think are the most imaginative, original, and positive potential solutions to the problem. †¢ Format your paper consistent with APA guidelines. General Questions General General Questions Write a 1,050- to 1,400-word paper in which you use the creative process to solve a problem with which you have experience. Include the following: Describe each stage in the creative process using Ch. 5 of your text. Search for a personal challenge. Explain how you used the techniques to develop curiosity in your search (see Ch. 6). Express the problem created by this personal challenge (see Ch. 7). Refine your expression To get this material Click this link https://bitly.com/12CdUy1 Avoid distractions when you are in class. Sure, it may be tempting to send a text or write a note in class, but by becoming distracted, you are not going to learn. You can prevent distractions by limiting what you bring to class with you; for instance, leave your cell phone in your dorm. General Questions General General Questions Write a 1,050- to 1,400-word paper in which you use the creative process to solve a problem with which you have experience. Include the following: Describe each stage in the creative process using Ch. 5 of your text. Search for a personal challenge. Explain how you used the techniques to develop curiosity in your search (see Ch. 6). Express the problem created by this personal challenge (see Ch. 7). Refine your expression of the problem by considering multiple perspectives and deciding which expression is best. Include both the initial and the refined version. Justify your revision. Investigate the problem by obtaining necessary information (see Ch. 8). List the questions must answer to understand the problem. Identify a source of information to answer each question. Conclude your investigation with a thorough explanation of the problem,mincluding answers to each of the questions you identified. Produce ideas toward solution of this problem (see Ch. 9). List all of your ideas—whether they are ridiculous or serious is not important. Discuss the two ideas that you think are the most imaginative, original, and positive potential solutions to the problem. Format your paper consistent with APA guidelines.

The Dehumanization Process in the Narrative of the Life of Frederick Do

The Dehumanization Process in the Narrative of the Life of Frederick Douglass: An American Slave Throughout American history, minority groups were victims of American governmental policies, and these policies made them vulnerable to barbaric and inhumane treatment at the hands of white Americans. American slavery is a telling example of a government sanctioned institution that victimized and oppressed a race of people by indoctrinating and encouraging enslavement, racism and abuse. This institution is injurious to slaves and slave holders alike because American society, especially in the south, underwent a dehumanization process in order to implement the harsh and inhumane doctrine. In the episodic autobiography Narrative of the Life of Frederick Douglass: An American Slave, Frederick Douglass illustrates, through personal experience, the brutality and violence of slave life. As a young boy, Douglass is sent to Baltimore, an event that gives him temporary relief from the harsh conditions on the plantation. In Baltimore, he teaches himself to read and write and begins a jou rney to mental freedom that eventually paves the way to his successful escape to the North. Despite the hardships he endures, â€Å"most amazing is the indestructible total humanity of [Douglass] whom society called a thing, a chattel to be bought and sold† (Rexroth 134). Amazingly, Douglass realizes at a young age that the institution of slavery poisons people who might otherwise act in good and decent ways. His autobiography focuses primarily on the ill effects slavery has on slaves; however, he also acknowledges the damage that enforcing the laws of slavery has on slave holders. Through the use of imagery, Douglass masterfully illustrates the dehuman... ...oppression due to the lack of liberty, education, humanity, or justice. Works Cited Douglass, Frederick. Narrative of the Life of Frederick Douglass: An American Slave. The Heath Anthology of American Literature. Ed. Paul Lauter. Vol 1. 4th ed. Houghton Mifflin Company: New York, 2002. 1817-1880. Franklin, Bruce. â€Å"Animal Farm Unbound.† Frederick Douglass’s Narrative of the Life of Frederick Douglass. Ed. Harold Bloom. New York: Chelsea House, 1988. 29-43. Houston PBS Online. Resource Book: People and Events Frederick Douglass 1818-1895. http://www.pbs.org/wgbh/aia/part4/4p1539.html Rexroth, Kenneth. â€Å"Frederick Douglass.† The Elastic Retort: Essays in Literature and Ideas. The Continuum, 1973: 108-11. Rpt. in Nineteenth Century Literature Criticism. Eds. Laura Lanzen Harris and Sheila Fitzgerald. Vol. 7. Detroit: Gale, 1984. 133-34.

Inc Narrative Report Essay

For our group, the INCredibles, we are assigned to organize an event which has something to do with a group and its cooperation. The two most important thing when we say â€Å"LEADERSHIP†. As described, â€Å"LEADERSHIP† is the activity of leading a group of people or an organization or the ability to achieve a specific goal. Leader steps up in times of crisis, and is able to think and act creatively in difficult situations. Unlike management, leadership cannot be taught, although it may be learned and enhanced through coaching or mentoring. Through this as a student, we are after a good and responsible leader that could really lead us for the best. Through teambuilding activities, different strategies and different leadership is being standout. Teambuilding is defined as the ability to identify and motivate individual employees to form a team that stays together, works together, and achieves together. For our event, entitled â€Å"Many Paths, One Spirit, A legacy of Leadership† we are after for a good leader and especially this event is after for a true and a strong bond. Our event took place at Ouans worth farm, Kanlurang Mayao, Lucena City. The call time of the organizers is at 7:00 in the morning. And for our participants which are the DSC Officers, their call time is at 9:00 in the morning at Manuel S. Enverga CTHRM Study Area. When we got at the venue, everyone is busy designating and planning the last minute petty things to be solved. Then after that, we started putting thing into place. We started cutting papers for the souvenirs and started blowing up the balloons, while other members of the group is still at Zharina’s House and busy preparing the food for our event. We also decorated and started arranging the tables at the conference room and putting some finishing touches. Then after that everyone is settled and other members already know their designated stations for the Amazing Race, and we are only waiting for the arrival of our participants but the DSC Chairman arrived as early as 8:00 in the morning. Then our participants arrived at 10:00 in the morning and it also started the registration, the assigned for the registration is Lindsey Hidalgo, Vera Javier and Karen Tagon. After the participants finished the registration, the program officially begins. The event started with a prayer hoping the event would be successful and also at the same time to attain our main goal that is to have unity, cooperation and a strong bond among the DSC Officers. As the participants log their names at our registration forms, they were at the same time given their IDs and they are also to choose a balloon that indicates which group they would belong. After that Czarina Miranda the emcee of that day’s event, she welcomes whole heartedly the participants and calls on immpediately Nadine Almozara for her to start the day with a prayer. After the Opening Prayer, the Opening Remarks was given by Nina Lumenario. Then the emcee immediately instructed the participants on how the Amazing Race would work and what are the rules. The stations includes the following: Calamansi relay, bilangan ng monggo, spell me DSC, kadena, obstacle, pulot ng basura, harina, and eat all you can (includes ampalaya, banana, calamansi and tomsto). Then after that, the emcee instructed the participants that its time for them to know their respective groups. The first group includes Alexis, Meco and Nino. The second group includes Crio, Hazel, Tyrone, and Chesca. And the last group includes Paul, Alvin and Janica. After that the groups where formed and they need to make a name for their group and to present a yell. Paul’s group is named as the â€Å"TAMARights† inspired by the FEU Tamaraws, they performed their yell first. Alexis’ group is named as â€Å"Orange and Lemons† inspired by a Filipino Local Band. And the last group, Crio’s group is named as â€Å"BLUEtooth† inspired by a cellphone specs. Crio, Paul and Meco added spice whenever they throw jokes to each opther that makes also the event fun. And its now time for the groups to start the Amazing Race. The process is that they where given a clue that would took them to their destination and in that way they would be starting in dif ferent stations. In every stations that they’ll be accomplishing the Station Manager would give them a letter that is a key for a keyword â€Å"CTHRM DSC† and the first to complete this letters, they are the winner. So the game started it’s a thrill. We could really see the cooperation and the hardworks of our participants and we are really thankful because they really supported our event. The first group to arrived at the conference room and completed the letters is Paul’s Group and the are the TAMARights. Second who arrived is Alexis’ Group, the Orange and Lemons. And last is Crio’s Group, the BLUEtooth. So because Crio’s group finish last Meco started throwing jokes at Crio. And once4 again everyone laugh and enjoyed. Then after the picture taking. We started distributing lunch for our participants, our lunch is extraordinary because it’s in a banana leaf and we need to eat together but its really fun. We enjoyed eating like it’s all back to basic. After we finish our lunch, the second game begins and it is to solve a jigsaw puzzle. Alexis’ Group won the challenge, next is Crio’s group and lastly it’s Paul’s group. The next activity is to make their flags that would be symbolizing their group and everyone really made a good job. The event was tiring but it was really worth it, it’s fun and it really bonded the relationships of the participants and also the organizers. Alvin, the DSC Chairman, gave some message that really made us realize and also touches us. Immediately after that Donna, our Event Manager gave a Closing Remarks and after the program everyone was having a free time to sit and jam with the participants and to distribute their snacks and at the same time Paul gave us some intermission number and he does some impersonations. That ends our event and we really think that our event achieves our goals. And as we finish this event we are hoping that this would give our participants a more clearer and more deeper understanding about leadership.

Numerical differential equation analysis package - Free Essay Example

Sample details Pages: 31 Words: 9220 Downloads: 1 Date added: 2017/06/26 Category Statistics Essay Did you like this example? The Numerical Differential Equation Analysis package combines functionality for analyzing differential equations using Butcher trees, Gaussian quadrature, and Newton-Cotes quadrature. Butcher Runge-Kutta methods are useful for numerically solving certain types of ordinary differential equations. Deriving high-order Runge-Kutta methods is no easy task, however. There are several reasons for this. The first difficulty is in finding the so-called order conditions. These are nonlinear equations in the coefficients for the method that must be satisfied to make the error in the method of order O (hn) for some integer n where h is the step size. The second difficulty is in solving these equations. Besides being nonlinear, there is generally no unique solution, and many heuristics and simplifying assumptions are usually made. Finally, there is the problem of combinatorial explosion. For a twelfth-order method there are 7813 order conditions! Don’t waste time! Our writers will create an original "Numerical differential equation analysis package" essay for you Create order This package performs the first task: finding the order conditions that must be satisfied. The result is expressed in terms of unknown coefficients aij, bj, and ci. The s-stage Runge-Kutta method to advance from x to x+h is then where Sums of the elements in the rows of the matrix [aij] occur repeatedly in the conditions imposed on aij and bj. In recognition of this and as a notational convenience it is usual to introduce the coefficients ci and the definition This definition is referred to as the row-sum condition and is the first in a sequence of row-simplifying conditions. If aij=0 for all ij the method is explicit; that is, each of the Yi (x+h) is defined in terms of previously computed values. If the matrix [aij] is not strictly lower triangular, the method is implicit and requires the solution of a (generally nonlinear) system of equations for each timestep. A diagonally implicit method has aij=0 for all ij. There are several ways to express the order conditions. If the number of stages s is specified as a positive integer, the order conditions are expressed in terms of sums of explicit terms. If the number of stages is specified as a symbol, the order conditions will involve symbolic sums. If the number of stages is not specified at all, the order conditions will be expressed in stage-independent tensor notation. In addition to the matrix a and the vectors b and c, this notation involves the vector e, which is composed of all ones. This notation has two distinct advantages: it is independent of the number of stages s and it is independent of the particular Runge-Kutta method. For further details of the theory see the references. ai,j the coefficient of f(Yj(x)) in the formula for Yi(x) of the method bj the coefficient of f(Yj(x)) in the formula for Y(x) of the method ci a notational convenience for aij e a notational convenience for the vector (1, 1, 1, ) Notation used by functions for Butcher. RungeKuttaOrderConditions[p,s] give a list of the order conditions that any s-stage Runge-Kutta method of order p must satisfy ButcherPrincipalError[p,s] give a list of the order p+1 terms appearing in the Taylor series expansion of the error for an order-p, s-stage Runge-Kutta method RungeKuttaOrderConditions[p], ButcherPrincipalError[p] give the result in stage-independent tensor notation Functions associated with the order conditions of Runge-Kutta methods. ButcherRowSum specify whether the row-sum conditions for the ci should be explicitly included in the list of order conditions ButcherSimplify specify whether to apply Butchers row and column simplifying assumptions Some options for RungeKuttaOrderConditions. This gives the number of order conditions for each order up through order 10. Notice the combinatorial explosion. In[2]:= Out[2]= This gives the order conditions that must be satisfied by any first-order, 3-stage Runge-Kutta method, explicitly including the row-sum conditions. In[3]:= Out[3]= These are the order conditions that must be satisfied by any second-order, 3-stage Runge-Kutta method. Here the row-sum conditions are not included. In[4]:= Out[4]= It should be noted that the sums involved on the left-hand sides of the order conditions will be left in symbolic form and not expanded if the number of stages is left as a symbolic argument. This will greatly simplify the results for high-order, many-stage methods. An even more compact form results if you do not specify the number of stages at all and the answer is given in tensor form. These are the order conditions that must be satisfied by any second-order, s-stage method. In[5]:= Out[5]= Replacing s by 3 gives the same result asRungeKuttaOrderConditions. In[6]:= Out[6]= These are the order conditions that must be satisfied by any second-order method. This uses tensor notation. The vector e is a vector of ones whose length is the number of stages. In[7]:= Out[7]= The tensor notation can likewise be expanded to give the conditions in full. In[8]:= Out[8]= These are the principal error coefficients for any third-order method. In[9]:= Out[9]= This is a bound on the local error of any third-order method in the limit as h approaches 0, normalized to eliminate the effects of the ODE. In[10]:= Out[10]= Here are the order conditions that must be satisfied by any fourth-order, 1-stage Runge-Kutta method. Note that there is no possible way for these order conditions to be satisfied; there need to be more stages (the second argument must be larger) for there to be sufficiently many unknowns to satisfy all of the conditions. In[11]:= Out[11]= RungeKuttaMethod specify the type of Runge-Kutta method for which order conditions are being sought Explicit a setting for the option RungeKuttaMethod specifying that the order conditions are to be for an explicit Runge-Kutta method DiagonallyImplicit a setting for the option RungeKuttaMethod specifying that the order conditions are to be for a diagonally implicit Runge-Kutta method Implicit a setting for the option RungeKuttaMethod specifying that the order conditions are to be for an implicit Runge-Kutta method $RungeKuttaMethod a global variable whose value can be set to Explicit, DiagonallyImplicit, or Implicit Controlling the type of Runge-Kutta method in RungeKuttaOrderConditions and related functions. RungeKuttaOrderConditions and certain related functions have the option RungeKuttaMethod with default setting $RungeKuttaMethod. Normally you will want to determine the Runge-Kutta method being considered by setting $RungeKuttaMethod to one of Implicit, DiagonallyImplicit, and Explicit, but you can specify an option setting or even change the default for an individual function. These are the order conditions that must be satisfied by any second-order, 3-stage diagonally implicit Runge-Kutta method. In[12]:= Out[12]= An alternative (but less efficient) way to get a diagonally implicit method is to force a to be lower triangular by replacing upper-triangular elements with 0. In[13]:= Out[13]= These are the order conditions that must be satisfied by any third-order, 2-stage explicit Runge-Kutta method. The contradiction in the order conditions indicates that no such method is possible, a result which holds for any explicit Runge-Kutta method when the number of stages is less than the order. In[14]:= Out[14]= ButcherColumnConditions[p,s] give the column simplifying conditions up to and including order p for s stages ButcherRowConditions[p,s] give the row simplifying conditions up to and including order p for s stages ButcherQuadratureConditions[p,s] give the quadrature conditions up to and including order p for s stages ButcherColumnConditions[p], ButcherRowConditions[p], etc. give the result in stage-independent tensor notation More functions associated with the order conditions of Runge-Kutta methods. Butcher showed that the number and complexity of the order conditions can be reduced considerably at high orders by the adoption of so-called simplifying assumptions. For example, this reduction can be accomplished by adopting sufficient row and column simplifying assumptions and quadrature-type order conditions. The option ButcherSimplify in RungeKuttaOrderConditions can be used to determine these automatically. These are the column simplifying conditions up to order 4. In[15]:= Out[15]= These are the row simplifying conditions up to order 4. In[16]:= Out[16]= These are the quadrature conditions up to order 4. In[17]:= Out[17]= Trees are fundamental objects in Butchers formalism. They yield both the derivative in a power series expansion of a Runge-Kutta method and the related order constraint on the coefficients. This package provides a number of functions related to Butcher trees. f the elementary symbol used in the representation of Butcher trees ButcherTrees[p] give a list, partitioned by order, of the trees for any Runge-Kutta method of order p ButcherTreeSimplify[p,,] give the set of trees through order p that are not reduced by Butchers simplifying assumptions, assuming that the quadrature conditions through order p, the row simplifying conditions through order , and the column simplifying conditions through order all hold. The result is grouped by order, starting with the first nonvanishing trees ButcherTreeCount[p] give a list of the number of trees through order p ButcherTreeQ[tree] give True if the tree or list of trees tree is valid functional syntax, and False otherwise Constructing and enumerating Butcher trees. This gives the trees that are needed for any third-order method. The trees are represented in a functional form in terms of the elementary symbol f. In[18]:= Out[18]= This tests the validity of the syntax of two trees. Butcher trees must be constructed using multiplication, exponentiation or application of the function f. In[19]:= Out[19]= This evaluates the number of trees at each order through order 10. The result is equivalent to Out[2] but the calculation is much more efficient since it does not actually involve constructing order conditions or trees. In[20]:= Out[20]= The previous result can be used to calculate the total number of trees required at each order through order10. In[21]:= Out[21]= The number of constraints for a method using row and column simplifying assumptions depends upon the number of stages. ButcherTreeSimplify gives the Butcher trees that are not reduced assuming that these assumptions hold. This gives the additional trees that are necessary for a fourth-order method assuming that the quadrature conditions through order 4 and the row and column simplifying assumptions of order 1 hold. The result is a single tree of order 4 (which corresponds to a single fourth-order condition). In[22]:= Out[22]= It is often useful to be able to visualize a tree or forest of trees graphically. For example, depicting trees yields insight, which can in turn be used to aid in the construction of Runge-Kutta methods. ButcherPlot[tree] give a plot of the tree tree ButcherPlot[{tree1,tree2,}] give an array of plots of the trees in the forest {tree1, tree2,} Drawing Butcher trees. ButcherPlotColumns specify the number of columns in the GraphicsGrid plot of a list of trees ButcherPlotLabel specify a list of plot labels to be used to label the nodes of the plot ButcherPlotNodeSize specify a scaling factor for the nodes of the trees in the plot ButcherPlotRootSize specify a scaling factor for the highlighting of the root of each tree in the plot; a zero value does not highlight roots Options to ButcherPlot. This plots and labels the trees through order 4. In[23]:= Out[23]= In addition to generating and drawing Butcher trees, many functions are provided for measuring and manipulating them. For a complete description of the importance of these functions, see Butcher. ButcherHeight[tree] give the height of the tree tree ButcherWidth[tree] give the width of the tree tree ButcherOrder[tree] give the order, or number of vertices, of the tree tree ButcherAlpha[tree] give the number of ways of labeling the vertices of the tree tree with a totally ordered set of labels such that if (m, n) is an edge, then mn ButcherBeta[tree] give the number of ways of labeling the tree tree with ButcherOrder[tree]-1 distinct labels such that the root is not labeled, but every other vertex is labeled ButcherBeta[n,tree] give the number of ways of labeling n of the vertices of the tree with n distinct labels such that every leaf is labeled and the root is not labeled ButcherBetaBar[tree] give the number of ways of labeling the tree tree with ButcherOrder[tree] distinct labels such that every node, including the root, is labeled ButcherBetaBar[n,tree] give the number of ways of labeling n of the vertices of the tree with n distinct labels such that every leaf is labeled ButcherGamma[tree] give the density of the tree tree; the reciprocal of the density is the right-hand side of the order condition imposed by tree ButcherPhi[tree,s] give the weight of the tree tree; the weight (tree) is the left-hand side of the order condition imposed by tree ButcherPhi[tree] give (tree) using tensor notation ButcherSigma[tree] give the order of the symmetry group of isomorphisms of the tree tree with itself Other functions associated with Butcher trees. This gives the order of the tree f[f[f[f] f^2]]. In[24]:= Out[24]= This gives the density of the tree f[f[f[f] f^2]]. In[25]:= Out[25]= This gives the elementary weight function imposed by f[f[f[f] f^2]] for an s-stage method. In[26]:= Out[26]= The subscript notation is a formatting device and the subscripts are really just the indexed variable NumericalDifferentialEquationAnalysis`Private`$i. In[27]:= Out[27]//FullForm= It is also possible to obtain solutions to the order conditions using Solve and related functions. Many issues related to the construction Runge-Kutta methods using this package can be found in Sofroniou. The article also contains details concerning algorithms used in Butcher.m and discusses applications. Gaussian Quadrature As one of its methods, the Mathematica function NIntegrate uses a fairly sophisticated Gauss-Kronrod-based algorithm. The Gaussian quadrature functionality provided in Numerical Differential Equation Analysis allows you to easily study some of the theory behind ordinary Gaussian quadrature which is a little less sophisticated. The basic idea behind Gaussian quadrature is to approximate the value if an integral as a linear combination of values of the integrand evaluated at specific points: Since there are 2n free parameters to be chosen (both the abscissas xi and the weights wi) and since both integration and the sum are linear operations, you can expect to be able to make the formula correct for all polynomials of degree less than about 2n. In addition to knowing what the optimal abscissas and weights are, it is often desirable to know how large the error in the approximation will be. This package allows you to answer both of these questions. GaussianQuadratureWeights[n,a,b] give a list of the pairs (xi, wi) to machine precision for quadrature on the interval a to b GaussianQuadratureError[n,f,a,b] give the error to machine precision GaussianQuadratureWeights[n,a,b,prec] give a list of the pairs (xi, wi) to precision prec GaussianQuadratureError[n,f,a,b,prec] give the error to precision prec Finding formulas for Gaussian quadrature. This gives the abscissas and weights for the five-point Gaussian quadrature formula on the interval (-3, 7). In[2]:= Out[2]= Here is the error in that formula. Unfortunately it involves the tenth derivative of f at an unknown point so you dont really know what the error itself is. In[3]:= Out[3]= You can see that the error decreases rapidly with the length of the interval. In[4]:= Out[4]= Newton-Cotes As one of its methods, the Mathematica function NIntegrate uses a fairly sophisticated Gauss-Kronrod based algorithm. Other types of quadrature formulas exist, each with their own advantages. For example, Gaussian quadrature uses values of the integrand at oddly spaced abscissas. If you want to integrate a function presented in tabular form at equally spaced abscissas, it wont work very well. An alternative is to use Newton-Cotes quadrature. The basic idea behind Newton-Cotes quadrature is to approximate the value of an integral as a linear combination of values of the integrand evaluated at equally spaced points: In addition, there is the question of whether or not to include the end points in the sum. If they are included, the quadrature formula is referred to as a closed formula. If not, it is an open formula. If the formula is open there is some ambiguity as to where the first abscissa is to be placed. The open formulas given in this package have the first abscissa one half step from the lower end point. Since there are n free parameters to be chosen (the weights) and since both integration and the sum are linear operations, you can expect to be able to make the formula correct for all polynomials of degree less than about n. In addition to knowing what the weights are, it is often desirable to know how large the error in the approximation will be. This package allows you to answer both of these questions. NewtonCotesWeights[n,a,b] give a list of the n pairs (xi, wi) for quadrature on the interval a to b NewtonCotesError[n,f,a,b] give the error in the formula Finding formulas for Newton-Cotes quadrature. option name default value QuadratureType Closed the type of quadrature, Open or Closed Option for NewtonCotesWeights and NewtonCotesError. Here are the abscissas and weights for the five-point closed Newton-Cotes quadrature formula on the interval (-3, 7). In[2]:= Out[2]= Here is the error in that formula. Unfortunately it involves the sixth derivative of f at an unknown point so you dont really know what the error itself is. In[3]:= Out[3]= You can see that the error decreases rapidly with the length of the interval. In[4]:= Out[4]= This gives the abscissas and weights for the five-point open Newton-Cotes quadrature formula on the interval (-3, 7). In[5]:= Out[5]= Here is the error in that formula. In[6]:= Out[6]= Runge-Kutta Methods From Wikipedia, The Free Encyclopedia Jump to: navigation, search In numerical analysis, the Runge-Kutta methods (German pronunciation:[kta]) are an important family of implicit and explicit iterative methods for the approximation of solutions of ordinary differential equations. These techniques were developed around 1900 by the German mathematicians C. Runge and M.W. Kutta. See the article on numerical ordinary differential equations for more background and other methods. See also List of Runge-Kutta methods. Contents 1 The common fourth-order Runge-Kutta method 2 Explicit Runge-Kutta methods o 2.1 Examples 3 Usage 4 Adaptive Runge-Kutta methods 5 Implicit Runge-Kutta methods 6 References 7 External links The Common Fourth-Order Runge-Kutta Method One member of the family of Runge-Kutta methods is so commonly used that it is often referred to as RK4, classical Runge-Kutta method or simply as the Runge-Kutta method. Let an initial value problem be specified as follows. Then, the RK4 method for this problem is given by the following equations: where yn + 1 is the RK4 approximation of y(tn + 1), and Thus, the next value (yn + 1) is determined by the present value (yn) plus the product of the size of the interval (h) and an estimated slope. The slope is a weighted average of slopes: k1 is the slope at the beginning of the interval; k2 is the slope at the midpoint of the interval, using slope k1 to determine the value of y at the point tn + h / 2 using Eulers method; k3 is again the slope at the midpoint, but now using the slope k2 to determine the y-value; k4 is the slope at the end of the interval, with its y-value determined using k3. In averaging the four slopes, greater weight is given to the slopes at the midpoint: The RK4 method is a fourth-order method[needs reference], meaning that the error per step is on the order of h5, while the total accumulated error has order h4. Note that the above formulae are valid for both scalar- and vector-valued functions (i.e., y can be a vector and f an operator). For example one can integrate Schrdingers equation using the Hamiltonian operator as function f. Explicit Runge-Kutta Methods The family of explicit Runge-Kutta methods is a generalization of the RK4 method mentioned above. It is given by where (Note: the above equations have different but equivalent definitions in different texts). To specify a particular method, one needs to provide the integer s (the number of stages), and the coefficients aij (for 1 j i s), bi (for i = 1, 2, , s) and ci (for i = 2, 3, , s). These data are usually arranged in a mnemonic device, known as a Butcher tableau (after John C. Butcher): 0 c2 a21 c3 a31 a32 cs as1 as2 as,s 1 b1 b2 bs 1 bs The Runge-Kutta method is consistent if There are also accompanying requirements if we require the method to have a certain order p, meaning that the truncation error is O(hp+1). These can be derived from the definition of the truncation error itself. For example, a 2-stage method has order 2 if b1 + b2 = 1, b2c2 = 1/2, and b2a21 = 1/2. Examples The RK4 method falls in this framework. Its tableau is: 0 1/2 1/2 1/2 0 1/2 1 0 0 1 1/6 1/3 1/3 1/6 However, the simplest Runge-Kutta method is the (forward) Euler method, given by the formula yn + 1 = yn + hf(tn,yn). This is the only consistent explicit Runge-Kutta method with one stage. The corresponding tableau is: 0 1 An example of a second-order method with two stages is provided by the midpoint method The corresponding tableau is: 0 1/2 1/2 0 1 Note that this midpoint method is not the optimal RK2 method. An alternative is provided by Heuns method, where the 1/2s in the tableau above are replaced by 1s and the bs row is [1/2, 1/2]. If one wants to minimize the truncation error, the method below should be used (Atkinson p.423). Other important methods are Fehlberg, Cash-Karp and Dormand-Prince. Also, read the article on Adaptive Stepsize. Usage The following is an example usage of a two-stage explicit Runge-Kutta method: 0 2/3 2/3 1/4 3/4 to solve the initial-value problem with step size h=0.025. The tableau above yields the equivalent corresponding equations below defining the method: k1 = yn t0 = 1 y0 = 1 t1 = 1.025 k1 = y0 = 1 f(t0,k1) = 2.557407725 k2 = y0 + 2 / 3hf(t0,k1) = 1.042623462 y1 = y0 + h(1 / 4 f(t0,k1) + 3 / 4 f(t0 + 2 / 3h,k2)) = 1.066869388 t2 = 1.05 k1 = y1 = 1.066869388 f(t1,k1) = 2.813524695 k2 = y1 + 2 / 3hf(t1,k1) = 1.113761467 y2 = y1 + h(1 / 4 f(t1,k1) + 3 / 4 f(t1 + 2 / 3h,k2)) = 1.141332181 t3 = 1.075 k1 = y2 = 1.141332181 f(t2,k1) = 3.183536647 k2 = y2 + 2 / 3hf(t2,k1) = 1.194391125 y3 = y2 + h(1 / 4 f(t2,k1) + 3 / 4 f(t2 + 2 / 3h,k2)) = 1.227417567 t4 = 1.1 k1 = y3 = 1.227417567 f(t3,k1) = 3.796866512 k2 = y3 + 2 / 3hf(t3,k1) = 1.290698676 y4 = y3 + h(1 / 4 f(t3,k1) + 3 / 4 f(t3 + 2 / 3h,k2)) = 1.335079087 The numerical solutions correspond to the underlined values. Note that f(ti,k1) has been calculated to avoid recalculation in the yis. Adaptive Runge-Kutta Methods The adaptive methods are designed to produce an estimate of the local truncation error of a single Runge-Kutta step. This is done by having two methods in the tableau, one with order p and one with order p 1. The lower-order step is given by where the ki are the same as for the higher order method. Then the error is which is O(hp). The Butcher Tableau for this kind of method is extended to give the values of : 0 c2 a21 c3 a31 a32 cs as1 as2 as,s 1 b1 b2 bs 1 bs The Runge-Kutta-Fehlberg method has two methods of orders 5 and 4. Its extended Butcher Tableau is: 0 1/4 1/4 3/8 3/32 9/32 12/13 1932/2197 7200/2197 7296/2197 1 439/216 8 3680/513 -845/4104 1/2 8/27 2 3544/2565 1859/4104 11/40 16/135 0 6656/12825 28561/56430 9/50 2/55 25/216 0 1408/2565 2197/4104 1/5 0 However, the simplest adaptive Runge-Kutta method involves combining the Heun method, which is order 2, with the Euler method, which is order 1. Its extended Butcher Tableau is: 0 1 1 1/2 1/2 1 0 The error estimate is used to control the stepsize. Other adaptive Runge-Kutta methods are the Bogacki-Shampine method (orders 3 and 2), the Cash-Karp method and the Dormand-Prince method (both with orders 5 and 4). Implicit Runge-Kutta Methods The implicit methods are more general than the explicit ones. The distinction shows up in the Butcher Tableau: for an implicit method, the coefficient matrix aij is not necessarily lower triangular: The approximate solution to the initial value problem reflects the greater number of coefficients: Due to the fullness of the matrix aij, the evaluation of each ki is now considerably involved and dependent on the specific function f(t,y). Despite the difficulties, implicit methods are of great importance due to their high (possibly unconditional) stability, which is especially important in the solution of partial differential equations. The simplest example of an implicit Runge-Kutta method is the backward Euler method: The Butcher Tableau for this is simply: It can be difficult to make sense of even this simple implicit method, as seen from the expression for k1: In this case, the awkward expression above can be simplified by noting that so that from which follows. Though simpler then the raw representation before manipulation, this is an implicit relation so that the actual solution is problem dependent. Multistep implicit methods have been used with success by some researchers. The combination of stability, higher order accuracy with fewer steps, and stepping that depends only on the previous value makes them attractive; however the complicated problem-specific implementation and the fact that ki must often be approximated iteratively means that they are not common. References J. C. Butcher, Numerical methods for ordinary differential equations, ISBN 0471967580 George E. Forsythe, Michael A. Malcolm, and Cleve B. Moler. Computer Methods for Mathematical Computations. Englewood Cliffs, NJ: Prentice-Hall, 1977. (See Chapter 6.) Ernst Hairer, Syvert Paul Nrsett, and Gerhard Wanner. Solving ordinary differential equations I: Nonstiff problems, second edition. Berlin: Springer Verlag, 1993. ISBN 3-540-56670-8. William H. Press, Brian P. Flannery, Saul A. Teukolsky, William T. Vetterling. Numerical Recipes in C. Cambridge, UK: Cambridge University Press, 1988. (See Sections 16.1 and 16.2.) Kaw, Autar; Kalu, Egwu (2008), Numerical Methods with Applications (1st ed.), www.autarkaw.com. Kendall E. Atkinson. An Introduction to Numerical Analysis. John Wiley Sons 1989 F. Cellier, E. Kofman. Continuous System Simulation. Springer Verlag, 2006. ISBN 0-387-26102-8. External links Runge-Kutta Runge-Kutta 4th Order Method Runge Kutta Method for O.D.E.s Numerical integration First order methods Euler method Backward Euler Semi-implicit Euler Exponential Euler Second order methods Verlet integration Velocity Verlet Crank-Nicolson method Beemans algorithm Midpoint method Heuns method Newmark-beta method Leapfrog integration Higher order methods Runge-Kutta methods List of Runge-Kutta methods Linear multistep method Retrieved from https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods Categories: Numerical differential equations | Runge-Kutta methods This page was last modified on 28 November 2009 at 11:21. Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. See Terms of Use for details. Wikipedia is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. Contact us Privacy policy About Wikipedia Disclaimers Higher Order Taylor Methods Marcelo Julio Alvisio Lisa Marie Danz May 16, 2007 Introduction Differential equations are one of the building blocks in science or engineering. Scientists aim to obtain numerical solutions to differential equations whenever explicit solutions do not exist or when they are too hard to find. These numerical solutions are approximated though a variety of methods, some of which we set out to explore in this project. We require two conditions when computing differential equations numerically. First, we require that the solution is continuous with initial value. Otherwise, numerical error introduced in the representation of the number in computer systems would produce results very far from the actual solution. Second, we require that the solution changes continuously with respect to the differential equation itself. Otherwise, we cannot expect the method that approximates the differential equation to give accurate results. The most common methods for computing differential equations numerically include Eulers method, Higher Order Taylor method and Runge-Kutta methods. In this project, we concentrate on the Higher Order Taylor Method. This method employs the Taylor polynomial of the solution to the equation. It approximates the zeroth order term by using the previous steps value (which is the initial condition for the first step), and the subsequent terms of the Taylor expansion by using the differential equation. We call it Higher Order Taylor Method, the lower order method being Eulers Method. Under certain conditions, the Higher Order Taylor Method limits the error to O(hn), where n is the order used. We will present several examples to test this idea. We will look into two main parameters as a measure of the effectiveness of the method, namely accuracy and efficiency. Theory of the Higher Order Taylor Method Definition 2.1 Consider the differential equation given by y0(t)= f(t,y), y(a)= c. Then for ba, the nth order Taylor approximation to y(b) with K steps is given by yK, where {yi} is defined recursively as: t0 = a y0 = y(a)= c ti+1 = ti + h h2 f hn n1f yi+1 = yi + hf(ti,yi)+ (ti,yi)+ +(ti,yi) 2 t n! tn1 with h =(b a)/K. It makes sense to formulate such a definition in view of the Taylor series expansion that is used when y(t) is known explicitly. All we have done is use f(t,y) for y0(t), ft(t,y) for y00(t), and so forth. The next task is to estimate the error that this approximation introduces. We know by Taylors Theorem that, for any solution that admits a Taylor expansion at the point ti, we have h2 hn h(n+1) y(ti+1)= y(ti)+ hy0(ti)+ y00(ti)+ + y(n)(ti)+ y(n+1)() 2 n!(n + 1)! where is between ti and ti+1 Using y0 = f(t,y), this translates to h2 f hn (n1)fh(n+1) (n)f y(ti+1)= y(ti)+hf(ti,yi)+ (ti,yi)++(ti,yi)+ (,y()) 2 t n! t(n1) (n + 1)! t(n) Therefore, the local error, that is to say, the error introduced at each step if the values calculated previously were exact, is given by: 1 (n)f Ei =(hn+1)(,y()) (n + 1)! tn which means that 1 (n)f max (hn+1)(,y()) Ei [a,b] (n + 1)! tn 23 We can say Ei = O(hn+1). Now, since the number of steps from a to b is proportional to 1/h, we multiply the error per step by the number of steps to find a total error E = O(hn). In Practice: Examples We will consider differential equations that we can solve explicitly to obtain an equation for y(t) such that y0(t)= f(t,y). This way, we can calculate the actual error by subtracting the exact value for y(b) from the value that the Higher Order Taylor method predicts for it. To approximate values in the following examples, the derivatives of f(t,y) were computed by hand. MATLAB then performed the iteration and arrived at the approximation. Notice that the definitions given in the previous section could also have been adapted for varying step size h. However, for ease of computation we have kept the step size constant. In our computations, we have chosen step size of (b a)/2k, which resulted in K =2k evenly spaced points in the interval. Example 3.1 We consider the differential equation 1+ t y0(t)= f(t,y)= 1+ y with initial condition y(1) = 2. It is clear that y(t)= t2 +2t +6 1 solves this equation. Thus we calculate the error for y(2) by subtracting the approximation of y(2) from y(2), which is the exact value. Recall that we are using h =2k because (b a)=1. The following table displays the errors calculated. k = 1 k = 2 k = 3 k = 4 order = 1 .0333 .0158 .0077 .0038 order = 2 .0038 .0009 .0002 .0001 order = 3 .0003269 .0000383 .0000046 .0000006 Runge-Kutta Methods The Taylor methods in the preceding section have the desirable feature that the F.G.E. is of order O(hN ), and N can be chosen large so that this error is small. However, the shortcomings of the Taylor methods are the a priori determination of N and the computation of the higher derivatives, which can be very complicated. Each Runge-Kutta method is derived from an appropriate Taylor method in such a way that the F.G.E. is of order O(hN ). A trade-off is made to perform several function evaluations at each step and eliminate the necessity to compute the higher derivatives. These methods can be constructed for any order N. The Runge-Kutta method of order N = 4 is most popular. It is a good choice for common purposes because it is quite accurate, stable, and easy to program. Most authorities proclaim that it is not necessary to go to a higher-order method because the increased accuracy is offset by additional computational effort. If more accuracy is required, then either a smaller step size or an adaptive method should be used. The fourth-order Runge-Kutta method (RK4) simulates the accuracy of the Taylor series method of order N = 4. The method is based on computing yk+1 as follows: (1) yk+1 = yk + w1k1 + w2k2 + w3k3 + w4k4, where k1, k2, k3, and k4 have the form (2) k1 = h f (tk , yk ), k2 = h f (tk + a1h, yk + b1k1), k3 = h f (tk + a2h, yk + b2k1 + b3k2), k4 = h f (tk + a3h, yk + b4k1 + b5k2 + b6k3). By matching coefficients with those of the Taylor series method of order N = 4 so that the local truncation error is of order O(h5), Runge and Kutta were able to obtain the 490 CHAP. 9 SOLUTION OF DIFFERENTIAL EQUATIONS following system of equations: (3) b1 = a1, b2 + b3 = a2, b4 + b5 + b6 = a3, w1 + w2 + w3 + w4 = 1, w2a1 + w3a2 + w4a3 = 1 2, w2a2 1 + w3a2 2 + w4a2 3 = 1 3 , w2a3 1 + w3a3 2 + w4a3 3 = 1 4 , w3a1b3 + w4(a1b5 + a2b6) = 1 6 , w3a1a2b3 + w4a3(a1b5 + a2b6) = 1 8 , w3a2 1b3 + w4(a2 1b5 + a2 2b6) = 1 12 , w4a1b3b6 = 1 24 The system involves 11 equations in 13 unknowns. Two additional conditions must be supplied to solve the system. The most useful choice is (4) a1 = 1 2 and b2 = 0. Then the solution for the remaining variables is (5) a2 = 1 2 , a3 = 1, b1 = 1 2 , b3 = 1 2 , b4 = 0, b5 = 0, b6 = 1, w1 = 1 6 , w2 = 1 3 , w3 = 1 3 , w4 = 1 6 The values in (4) and (5) are substituted into (2) and (1) to obtain the formula for the standard Runge-Kutta method of order N = 4, which is stated as follows. Start with the initial point (t0, y0) and generate the sequence of approximations using (6) yk+1 = yk + h( f1 + 2 f2 + 2 f3 + f4) 6 , SEC. 9.5 RUNGE-KUTTA METHODS 491 where (7) f1 = f (tk , yk ), f2 = f tk + h 2 , yk + h 2 f1 , f3 = f tk + h 2 , yk + h 2 f2 , f4 = f (tk + h, yk + h f3). Discussion about the Method The complete development of the equations in (7) is beyond the scope of this book and can be found in advanced texts, but we can get some insights. Consider the graph of the solution curve y = y(t) over the first subinterval [t0, t1]. The function values in (7) are approximations for slopes to this curve. Here f1 is the slope at the left, f2 and f3 are two estimates for the slope in the middle, and f4 is the slope at the right (a)). The next point (t1, y1) is obtained by integrating the slope function (8) y(t1) y(t0) = _ t1 t0 f (t, y(t)) dt. If Simpsons rule is applied with step size h/2, the approximation to the integral in (8) is (9) _ t1 t0 f (t, y(t)) dt h 6 ( f (t0, y(t0)) + 4 f (t1/2, y(t1/2)) + f (t1, y(t1))), where t1/2 is the midpoint of the interval. Three function values are needed; hence we make the obvious choice f (t0, y (t0)) = f1 and f (t1, y(t1)) f4. For the value in the middle we chose the average of f2 and f3: f (t1/2, y(t1/2)) f2 + f3 2 . These values are substituted into (9), which is used in equation (8) to get y1: (10) y1 = y0 + h 6 f1 + 4( f2 + f3) 2 + f4 . When this formula is simplified, it is seen to be equation (6) with k = 0. The graph for the integral in (9) is shown in Figure 9.9(b). 492 CHAP. 9 SOLUTION OF DIFFERENTIAL EQUATIONS y t m1 = f1 m2 = f3 m3 = f4 m4 = f4 (t0, y0) y = y(t) (t1, y(t1)) t0 t1/2 t1 (a) Predicted slopes mj to the solution curve y = y(t) z t (t0, f1) (t1/2, f2) (t1/2, f3) (t1, f4) t0 t1/2 t1 (b) Integral approximation: h 6 y(t1) y0 = ( f1 + 2f2 + 2f3 + f4) Figure 9.9 The graphs y = y(t) and z = f (t, y(t)) in the discussion of the Runge-Kutta method of order N = 4. Step Size versus Error The error term for Simpsons rule with step size h/2 is (11) y(4)(c1) h5 2880 . If the only error at each step is that given in (11), after M steps the accumulated error for the RK4 method would be (12) _M k=1 y(4)(ck) h5 2880 b a 5760 y(4)(c)h4 O(h4). The next theorem states the relationship between F.G.E. and step size. It is used to give us an idea of how much computing effort must be done when using the RK4 method. Theorem 9.7 (Precision of the Runge-Kutta Method). Assume that y(t) is the solution to the I.V.P. If y(t) C5[t0, b] and {(tk , yk)}M k=0 is the sequence of approximations generated by the Runge-Kutta method of order 4, then (13) |ek| = |y(tk ) yk| = O(h4), |_k+1| = |y(tk+1) yk hTN (tk , yk)| = O(h5). SEC. 9.5 RUNGE-KUTTA METHODS 493 In particular, the F.G.E. at the end of the interval will satisfy (14) E(y(b), h) = |y(b) yM| = O(h4). Examples 9.10 and 9.11 illustrate Theorem 9.7. If approximations are computed using the step sizes h and h/2, we should have (15) E(y(b), h) Ch4 for the larger step size, and (16) E y(b), h 2 C h4 16 = 1 16 Ch4 1 16 E(y(b), h). Hence the idea in Theorem 9.7 is that if the step size in the RK4 method is reduced by a factor of 12 we can expect that the overall F.G.E. will be reduced by a factor of 1. Example 9.10. Use the RK4 method to solve the I.V.P. y_ = (t y)/2 on [0, 3] with y(0) = 1. Compare solutions for h = 1, 12 , 14 , and 18 . Table 9.8 gives the solution values at selected abscissas. For the step size h = 0.25, a sample calculation is f1 = 0.0 1.0 2 = 0.5, f2 = 0.125 (1 + 0.25(0.5)(0.5)) 2 = 0.40625, f3 = 0.125 (1 + 0.25(0.5)(0.40625)) 2 = 0.4121094, f4 = 0.25 (1 + 0.25(0.4121094)) 2 = 0.3234863, y1 = 1.0 + 0.25 0.5 + 2(0.40625) + 2(0.4121094) 0.3234863 6 = 0.8974915. _ Example 9.11. Compare the F.G.E. when the RK4 method is used to solve y_ = (ty)/2 over [0, 3] with y(0) = 1 using step sizes 1, 12 , 14 , and 18 Table 9.9 gives the F.G.E. for the various step sizes and shows that the error in the approximation to y(3) decreases by about 1 16 when the step size is reduced by a factor of 1/2. E(y(3), h) = y(3) yM = O(h4) Ch4 where C = 0.000614. _ A comparison of Examples 9.10 and 9.11 and Examples 9.8 and 9.9 shows what is meant by the statement The RK4 method simulates the Taylor series method of order N = 4. For these examples, the two methods generate identical solution sets {(tk , yk)} 494 CHAP. 9 SOLUTION OF DIFFERENTIAL EQUATIONS Table 9.8 Comparison of the RK4 Solutions with Different Step Sizes for y_ = (t y)/2 over [0, 3] with y(0) = 1 yk tk h = 1 h = 12 h = 14 h = 18 y(tk ) Exact 0 1.0 1.0 1.0 1.0 1.0 0.125 0.9432392 0.9432392 0.25 0.8974915 0.8974908 0.8974917 0.375 0.8620874 0.8620874 0.50 0.8364258 0.8364037 0.8364024 0.8364023 0.75 0.8118696 0.8118679 0.8118678 1.00 0.8203125 0.8196285 0.8195940 0.8195921 0.8195920 1.50 0.9171423 0.9171021 0.9170998 0.9170997 2.00 1.1045125 1.1036826 1.1036408 1.1036385 1.1036383 2.50 1.3595575 1.3595168 1.3595145 1.3595144 3.00 1.6701860 1.6694308 1.6693928 1.6693906 1.6693905 Table 9.9 Relation between Step Size and F.G.E. for the RK4 Solutions to y_ = (t y)/2 over [0, 3] with y(0) = 1 Step size, h Number of steps, M Approximation to y(3), yM F.G.E. Error at t = 3, y(3) yM O(h4) Ch4 where C = 0.000614 1 3 1.6701860 0.0007955 0.0006140 12 6 1.6694308 0.0000403 0.0000384 14 12 1.6693928 0.0000023 0.0000024 18 24 1.6693906 0.0000001 0.0000001 over the given interval. The advantage of the RK4 method is obvious; no formulas for the higher derivatives need to be computed nor do they have to be in the program. It is not easy to determine the accuracy to which a Runge-Kutta solution has been computed. We could estimate the size of y(4)(c) and use formula (12). Another way is to repeat the algorithm using a smaller step size and compare results. A third way is to adaptively determine the step size, which is done in Program 9.5. In Section 9.6 we will see how to change the step size for a multistep method. SEC. 9.5 RUNGE-KUTTA METHODS 495 Runge-Kutta Methods of Order N = 2 The second-order Runge-Kutta method (denoted RK2) simulates the accuracy of the Taylor series method of order 2. Although this method is not as good to use as the RK4 method, its proof is easier to understand and illustrates the principles involved. To start, we write down the Taylor series formula for y(t + h): (17) y(t + h) = y(t) + hy_ (t) + 1 2 h2 y__ (t) + CT h3 + , where CT is a constant involving the third derivative of y(t) and the other terms in the series involve powers of h j for j 3. The derivatives y_ (t) and y__ (t) in equation (17) must be expressed in terms of f (t, y) and its partial derivatives. Recall that (18) y_ (t) = f (t, y). The chain rule for differentiating a function of two variables can be used to differentiate (18) with respect to t, and the result is y__ (t) = ft (t, y) + fy(t, y)y_ (t). Using (18), this can be written (19) y__ (t) = ft (t, y) + fy(t, y) f (t, y). The derivatives (18) and (19) are substituted in (17) to give the Taylor expression for y(t + h): y(t + h) = y(t) + h f (t, y) + 1 2 h2 ft (t, y) + 1 2 h2 fy(t, y) f (t, y) + CT h3 + . (20) Now consider the Runge-Kutta method of order N = 2, which uses a linear combination of two function values to express y(t + h): (21) y(t + h) = y(t) + Ah f0 + Bhf1, where (22) f0 = f (t, y), f1 = f (t + Ph, y + Qhf0). Next the Taylor polynomial approximation for a function of two independent variables is used to expand f (t, y) (see the Exercises). This gives the following representation for f1: (23) f1 = f (t, y) + Phft (t, y) + Qhfy(t, y) f (t, y) + CPh2 + , 496 CHAP. 9 SOLUTION OF DIFFERENTIAL EQUATIONS where CP involves the second-order partial derivatives of f (t, y). Then (23) is used in (21) to get the RK2 expression for y(t + h): y(t + h) = y(t) + (A + B)h f (t, y) + BPh2 ft (t, y) + BQh2 fy(t, y) f (t, y) + BCPh3 + . (24) A comparison of similar terms in equations (20) and (24) will produce the following conclusions: h f (t, y) = (A + B)h f (t, y) implies that 1 = A + B, 1 2 h2 ft (t, y) = BPh2 ft (t, y) implies that 1 2 = BP, 1 2 h2 fy(t, y) f (t, y) = BQh2 fy(t, y) f (t, y) implies that 1 2 = BQ. Hence, if we require that A, B, P, and Q satisfy the relations (25) A + B = 1 BP = 1 2 BQ = 1 2 , then the RK2 method in (24) will have the same order of accuracy as the Taylors method in (20). Since there are only three equations in four unknowns, the system of equations (25) is underdetermined, and we are permitted to choose one of the coefficients. There are several special choices that have been studied in the literature; we mention two of them. Case (i): Choose A = 12 . This choice leads to B = 12 , P = 1, and Q = 1. If equation (21) is written with these parameters, the formula is (26) y(t + h) = y(t) + h 2 ( f (t, y) + f (t + h, y + h f (t, y))). When this scheme is used to generate {(tk , yk)}, the result is Heuns method. Case (ii): Choose A = 0. This choice leads to B = 1, P = 12 , and Q = 12 . If equation (21) is written with these parameters, the formula is (27) y(t + h) = y(t) + h f t + h 2 , y + h 2 f (t, y) . When this scheme is used to generate {(tk , yk)}, it is called the modified Euler-Cauchy method. Numerical Methods Using Matlab, 4th Edition, 2004 John H. Mathews and Kurtis K. Fink ISBN: 0-13-065248-2 Prentice-Hall Inc. Upper Saddle River, New Jersey, USA https://vig.prenhall.com/ Deriving the Runge-Kutta Method Deriving the midpoint method The Taylor method is the gold standard for generating better numerical solutions to first order differential equations. A serious weakness in the Taylor method, however, is the need to compute a large number of partial derivatives and do other symbolic manipulation tasks. For example, the second order Taylor method for the equation y( t) = f(t,y(t)) is yi+1 = yi + h f(ti ,yi ) + h2 2 f t ( ti ,yi ) + f ( ti ,yi ) f y ( ti ,yi ) Higher order formulas get even uglier. The Midpoint method arises from an attempt to replace the second order Taylor method with a simpler Euler-like formula yi+1 = yi + h f(ti + ,yi + ) We can solve for the best values for and by applying a first order Taylor expansion to the term f(ti + ,yi + ): yi+1 = yi + h f ( ti ,yi ) + f t ( ti ,yi ) + f y ( ti ,yi ) + 2f t y ( ti ,yi ) The choices of and that make this look as close as possible to the second order Taylor formula above are = h2 = h2 f(ti ,yi ) leading to the so-called midpoint rule: 1 yi+1 = yi + h f(ti + h2 ,yi + h2 f(ti ,yi)) This formula has a simple interpretation. Essentially what we are doing here is driving an Euler estimate half way across the interval [ti , ti+1] and computing the slope f(ti + h2 ,yi + h2 f(ti ,yi)) at that midpoint. We then rewind back to the point ( ti ,yi ) and drive an Euler estimate all the way across the interval to ti+1 using this new midpoint slope in place of the old Euler slope. The Runge-Kutta Method The textbook points out that it is possible to derive similar methods by starting with more complex Euler-like formulas with more free parameters and then trying to match those Euler-like methods to higher order Taylor formulas. The Runge-Kutta method is essentially an attempt to match a more complex Euler-like formula to a fourth order Taylor method. The problem with this is that the Euler-like formula needed to match all the complexity of the fourth order Taylor method formula is quite complex. The textbook states in exercise 31 at the end of section 5.4 that the formula required is yi+1 = yi + h6 f(ti ,yi ) + h3 f(ti + 1 h,yi + 1 h f(ti ,yi)) + h3 f (ti + 2 h,yi + 2 h f (ti + 2 h, yi + 3 h f ( ti ,yi ))) + h6 f (ti + 3 h, yi + 3 h f (ti + 4 h, yi + 5 h f ( ti + 6 h,yi+ 7 h f ( ti ,yi)))) It is very messy to do so, but this form can expanded out and matched against the Taylor formula of order four. This allows us to solve for all the unknown coefficients. A somewhat cleaner alternative derivation is based on the following argument. Another way to solve for yi+1 is to compute this integral !t i+1 t i y( t) dt = y(ti+1) y(ti ) = yi+1 yi We can imagine beginning to compute the integral by noting that y( t) = f(t,y(t)) !t i+1 t i y( t) dt = !t i+1 t i f ( t,y( t)) dt 2 Unfortunately, we can not do the integral on the right hand side exactly, because we dont know what y(t) is. That is, after all, the unknown we are trying to solve for. Even though we cant compute the integral on the right exactly, we can estimate it. For example, applying Simpsons rule to the integral produces the estimate !t i+1 t i f ( t,y( t)) dt h3 f ( ti ,y( ti))+4 f ti+ti+1 2 ,y ti+ti+1 2 + f ( ti+1,y( ti+1)) The Runge-Kutta method takes this estimate as a starting point. The thing we need to do to make this estimate work is to find a way to estimate the unknown terms y((ti + ti+1) /2) and y(ti+1) . The first step is to rewrite the estimate as h3 f ( ti ,y( ti))+2 f ti+ti+1 2 ,y ti+ti+1 2 +2 f ti+ti+1 2 ,y ti+ti+1 2 + f ( ti+1,y( ti+1)) We write the middle term twice because we are going to develop two different estimates for y((ti + ti+1) /2). The thinking is that the mistakes we make in developing those two interior estimates may partly cancel each other out. Here is how we will develop our estimates. 1. y(ti ) is just yi . We estimate the first y((ti + ti+1) /2) by driving the original Euler slope k1 = f(ti ,yi ) half-way across the interval: 2. k1 = f(ti ,y(ti)) y ti + ti+1 2 yi + h/2 k1 As in the midpoint rule, we compute a second slope at that midpoint we just estimated. We then rewind to the start and drive that slope half-way across the interval again. 3. k2 = f(ti + h/2,yi + h/2 k1 ) y ti + ti+1 2 yi + h/2 k2 We use the second estimated midpoint to compute another slope and then drive that slope all the way across the interval. 4. 3 We use the second estimated midpoint to compute another slope and then drive that slope all the way across the interval. 4. k3 = f(ti + h/2,yi + h/2 k2 ) y(ti+1) = yi + h k3 k4 = f(ti + h,yi + h k3 ) Substituting all of these estimates into the Simpsons rule formula above gives yi+1 yi = !t i+1 t i f ( t,y( t)) dt h3 f ( ti ,y( ti))+ 2 f ti+ti+1 2 ,y ti+ti+1 2 +2 f ti+ti+1 2 ,y ti+ti+1 2 + f ( ti+1,y( ti+1)) or yi+1 = yi + h3 (k1 + 2 k2 + 2 k3 + k4 ) Summary Of The Method k1 = f(ti ,yi ) k2 = f(ti + h/2,yi + h/2 k1 ) k3 = f(ti + h/2,yi + h/2 k2 ) k4 = f(ti + h,yi + h k3 ) yi+1 = yi + h3(k1 + 2 k2 + 2 k3 + k4 ) 4 Taylor Series Methods: To derive these methods we start with a Taylor Expansion: y(t+_t) _ y(t) + _ty0(t) + 1 2 _t2y00(t) + + 1 r! y(r)(t)_tr. Lets say we want to truncate this at the second derivative and base a method on that. The scheme is, then: yn+1 = yn + fn_t + f0 tn 2 _t2. The Taylor series method can be written as yn+1 = yn +_tF (tn, yn,_t) where F = f + 1 2_tf0. If we take the LTE for this scheme, we get (as expected) LTE(t) = y(tn +_t) y(tn) _t f(tn, y(tn)) 1 2 _tf0(tn, y(tn)) = O(_t2). Of course, we designed this method to give us this order, so it shouldnt be a surprise! So the LTE is reasonable, but what about the global error? Just as in the Euler Forward case, we can show that the global error is of the same order as the LTE. How do we do this? We have two facts, y(tn+1) = y(tn) + _tF (tn, y(tn),_t), and yn+1 = yn +_tF (tn, yn,_t) where F = f + 1 2_tf0. Now we subtract these two |y(tn+1) yn+1| = |y(tn) yn +_t (F(tn, y(tn)) F(tn, yn)) + _tLTE| _ |y(tn) yn|+_t |F(tn, y(tn)) F(tn, yn)|+_t|LTE| . Now, if F is Lipschitz continuous, we can say en+1 _ (1 + _tL)en+_t|LTE|. Of course, this is the same proof as for Eulers method, except that now we are looking at F, not f, and the LTE is of higher order. We can do this no matter which Taylor series method we use, how many terms we go forward before we truncate. Advantages And Disadvantages Of The Taylor Series Method: advantages a) One step, explicit b) can be high order c) easy to show that global error is the same order as LTE disadvantages Needs the explicit form of derivatives of f. 4 Runge-Kutta Methods To avoid the disadvantage of the Taylor series method, we can use Runge-Kutta methods. These are still one step methods, but they depend on estimates of the solution at different points. They are written out so that they dont look messy: Second Order Runge-Kutta Methods: k1 = _tf(ti, yi) k2 = _tf(ti + __t, yi + _k1) yi+1 = yi + ak1 + bk2 lets see how we can chose the parameters a,b, _, _ so that this method has the highest order LTE possible. Take the Taylor expansions to express the LTE: k1(t) = _tf(t, y(t)) k2(t) = _tf(t + __t, y + _k1(t) = _t _ f(t, y(t) + ft(t, y(t))__t+ fy(t, y(t))_k1(t) + O(_t2) _ LTE(t) = y(t+_t) y(t) _t a _t f(t, y(t))_t b _t (ft(t, y(t))__t+ fy(t, y(t)_k1(t) + f(t, y(t))_t + O(_t2) = y(t+_t) y(t) _t af(t, y(t)) bf(t, y(t)) bft(t, y(t))_ bfy(t, y(t)_f(t, y(t))+ O(_t2) = y0(t) + 1 2 _ty00(t) (a + b)f(t, y(t)) _t(b_ft(t, y(t))+ b_f(t, y(t))fy(t, y(t)) + O(_t2) = (1 a b)f + ( 1 2 b_)_tft + ( 1 2 b_)_tfyf + O(_t2) So we want a = 1 b, _ = _ = 1 2b . Fourth Order Runge-Kutta Methods: k1 = _tf(ti, yi) (1.3) k2 = _tf(ti + 1 2 _t, yi + 1 2 k1) (1.4) k3 = _tf(ti + 1 2 _t, yi + 1 2 k2) (1.5) k4 = _tf(ti+_t, yi + k3) (1.6) yi+1 = yi + 1 6 (k1 + k2 + k3 + k4) (1.7) The second order method requires 2 evaluations of f at every timestep, the fourth order method requires 4 evaluations of f at every timestep. In general: For an rth order Runge- Kutta method we need S(r) evaluations of f for each timestep, where S(r) = 8 : r for r _ 4 r + 1 for r = 5 and r = 6 _ r + 2 for r _ 7 5 Practically speaking, people stop at r = 5. Advantages of Runge-Kutta Methods 1. One step method global error is of the same order as local error. 2. Dont need to know derivatives of f. 3. Easy for Automatic Error Control. Automatic Error Control Uniform grid spacing in this case, time steps are good for some cases but not always. Sometimes we deal with problems where varying the gridsize makes sense. How do you know when to change the stepsize? If we have an rth order scheme and and r + 1th order scheme, we can take the difference between these two to be the error in the scheme, and make the stepsize smaller if we prefer a smaller error, or larger if we can tolerate a larger error. For Automatic error control yo are computing a useless (r+1)th order shceme . . . what a waste! But with Runge Kutta we can take a fifth order method and a fourth order method, using the same ks. only a little extra work at each step.