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.

Feminism Womens Status in Higher Education - 1448 Words

Throughout human history, customs derived from mythical and religious social traditions have provided cultural explanations regarding the evolution of the roles of men and women and the variations between the structures of the lives of the sexes (Sapiro 28). Although many see America as a country of equal opportunities for all, including women, most forget the struggle and hardship thousands faced to earn this God-given right. Professor Barry Bull of Indiana University speculates on the current infrastructure of American society: â€Å"If we do not expect this institution to treat citizens fairly in these various roles, the hope that we will be able to achieve anything like a recognizably just society is significantly undermined† (Bull 1).†¦show more content†¦Mary Wollstonecraft (1759-1797), eighteenth-century English writer, philosopher and advocate of women’s rights shared Blackwell’s beliefs that society as a whole would greatly benefit if women rece ived equal treatment and respect. Wollstonecraft concentrated her efforts in the fight for equality of women in the field of politics (Feminist Interpretations and Political Theory 37). Wollstonecraft concerned herself with refining women’s existence socially and politically through further distinguishing the distinctions between men and women in both the private and public sphere (37). Wollstonecraft saw the power and authority held by men in private spheres as well as in the sphere of politics as socially developed entirely (38). Similar to Wollstonecraft’s position, philosophers during the time of Enlightenment argued the artificiality of political authority among women, based on the premise that the natural authority between the sexes within the family structure blighted the authenticity of political authority among women (38). The governing ethic of conjugality dictates the treatment of women as wives, mothers and sexual partners (38). Obstacles such as these cont radict any progression made by political reform (40). As higher education became a prominent aspect of life in America, women’s education faced restricting ideologies which interpreted women’s mindsShow MoreRelatedWhen Thinking About Women’S Movements, One Is Likely To1115 Words   |  5 Pagesthinking about women’s movements, one is likely to associate them with an event that has taken place in either the United States or Canada. Nevertheless, it is evident that women s movements have historically occurred across the globe. Often, however, individuals do not take this into consideration when thinking about how these types of movements have had an influence on society in the 21st century. While reading the article: Political Culture, Catalan Nationalism, and the Women’s Movement in EarlyRead MoreGender Education, Work, And Society1394 Words   |  6 Pages Gender in Education, Work, and Society GMGT 1010 Siwei Cai 7790633 â€Æ' Inequality between male and female existed from the beginning of human civilization and still not be solved in today’s modern society. Discrimination is coming from social classes, and people always look down upon others in lower level. Although women are doing well in education and schooling, they are still facing inequality in work and society. So women will be in lower classes and be discriminated. The governmentRead MoreFeminist Student Culture And The Issues Of Postsecondary Education946 Words   |  4 Pagesstudent culture and the issues in postsecondary education. First, the paper reviews the various definitions and movements surrounding feminism. In addition, the paper analyzes the perspectives of women and men and whether they support or are against feminist student culture. Lastly, the paper discusses recommendations and future research for student affairs professions to maintain feminist student culture in higher education. Understanding feminism Feminism in the United States is often separated intoRead MoreAnalysis Of The Play Trifles By Susan Gillespe Essay1628 Words   |  7 Pagescreative source to human life† (Women’s History in America Presented by Women s International Center). In the play, Trifles, by Susan Gillespe illustrates how women pay more attention to the little aspects of a situation then men do. In the film, Thelma and Louis, they both live two different worlds but realize nobody would believe women base on the circumstances. â€Å"Women were long considered weaker than men and unable to perform work requiring muscular development†. (Women’s History in America PresentedRead MoreIn The Ancient World, Little Efforts Were Made To Protect1592 Words   |  7 Pagesvote, access to higher education and suffrage. These are some of the fundamental rights enjoyed by men in the society. In a bid to advocate and campaign for these rights, feminism was born. By definition, feminism is social and political movements which unified women around a number of issues which affected them. Spearheaded by feminists, these movements campaigned for women’s rights highlighted above among others. They have worked towards promoting historical changes for women’s rights and theirRead MoreFeminism And Women s Suffrage Essay1360 Words   |  6 PagesFeminism is a critique of patriarchy, on the one hand, and an ideology committed to women’s emancipation on the other. The feminist movement can be see mainly in 3 stages in the UK. It can be argued that feminism had its day due to lack of momentum behind a third and fourth stage of its movement resulting in a decline of feminism in this generation. In this essay, I will be focusing more attention on the first and second waves of feminism compared to third stage of feminism. The first wave of feminismRead MoreWomen During The Imperial Period Essay1638 Words   |  7 Pagesexamine how these changes in the political, economic and familial structures simultaneously gave rise to feminism in China and stifled it. I will first give a brief overview of the history of the role of women over the imperial period to the communist revolution to the modern day Chinese society. I will also break down the political, economic, and social factors that both shaped and stifled feminism. Finally, I will examine the challenges modern Chinese women still face. Throughout the imperial periodRead MoreEssay about The Facets of Womans Studies783 Words   |  4 Pagesinterdisciplinary academic field devoted to topics concerning women, gender and feminism; exploring our gender existence, how we perform femininity and masculinity and how this interacts with other aspects of our identities, such as race, ethnicity, socioeconomic status and sexuality. Women’s studies emerged in the late 1960’s and early 1970’s as a concerned women being misrepresentation and trivialization in the higher education curriculum and as well as being excluded from many positions of power authorityRead MoreFeminism Of Different Cultures : Women Without Liberation Essay1622 Words   |  7 PagesFeminism in Different Cultures: Women without Liberation Living in America, women’s rights are known as a sort of liberated revolution, whereas in other countries across the globe, the simplistic idea of self-importance in women is non-existent. For generations, harmful cultural practices have challenged feminist activists to conjure up a plan of action in the aid to help those women suffering in these countries. â€Å"In feminist legal studies, culture is often viewed as a deviation from the path ofRead MoreA Womans Journey Essay1336 Words   |  6 Pages The old definition for feminism was defined as working towards an overall goal as a group, to achieve economic and political power. Today, this new definition no longer holds true, because many women are misrepresented and confused by many new definitions of feminism. This confusion has created womens ability to take matters into her own hands, and follow her own goals and inspirations-whatever they may be. The first wave of the womens movement started when Abigail

Essay on The Reality of Human Cloning - 667 Words

The Reality of Human Cloning As aptly put by Rosa Beddington, the word â€Å"clone† has become one of the most emotive of all the terms coined by scientists which have entered popular vocabulary. I shall add another, and that will be the phrase â€Å"Dolly the sheep†. The conception of Dolly, the â€Å"baby† of scientist Ian Wilmut and his team has opened the possibility of cloning humans. The mention of Dolly brings to average the person, haunting connotations of â€Å"future replicas of living megalomaniacs and the resurrected dead†. Indeed, Dolly has provided misconceptions about cloning, which, to a certain extent is skewed. Deliberate cloning is as old as horticulture itself and is nothing new. It has yielded numerous benefits, particularly in†¦show more content†¦The questions are endless. Predicting the future of human cloning at present is indeed futile as it will only yield more needless speculations. As much as I recognize the dangers of human cloning, I see the hoo- ha of human cloning a matter of perspective and time frame. All the attention given it is a passing phase. Ever since the advent of the industrial revolution, mankind has demonstrated his unquenchable thirst for discovery through science and technology. The fields of medicine, computing, electronics were founded because of research done by numerous groups of people who wanted to find more answers to unexplained phenomena and to cure, solve and discover. A few centuries back, no one thought that tuberculosis could be cured but it subsequently could. The uproar of test tube babies by ethical and religious groups when the first of such babies was born was deafening. Through the years, such ethical groups had other concerns to contend with, examples being the euthanasia and organ transplants. All these scientific breakthroughs involved human lives. Despite the obvious differences, the possibility of human cloning can be said to be just another â€Å"concern† in the ever progressing field of science, with many other concerns to come. When science produces another breakthrough which proves more shocking than human cloning, people will shift their attention and fear to that something else and human cloning will be passà ©. Do not get me wrong, I am a stronglyShow MoreRelatedEssay on Human Cloning Can Make Immortality a Reality4036 Words   |  17 Pagesdebate on cloning. The March edition of the Life Extension Foundation’s (LEF) magazine vocally calls for American citizens to write to their Senators and stop an anti-cloning bill from passage through both Houses (See Figure 1.) While the public argues over short-term questions – such as what is the definition of cloning, at what point does life begin, and is cloning bad – we must examine the hidden future potential and consequences of therapeutic cloning. Scientists’ direct contact with cloning technologyRead MoreThe Debate Over Human Cloning Essay1526 Words   |  7 Pages Human cloning, an unknown wonder to the universe the world has yet to unfold. The idea of cloning can be a dream to some and others not so much, there’s just endless possibilities when it comes to cloning. With the thought of cloning humans comes reality and other things that can cause issues or keep the experiment stagnant. â€Å"Reality† includes the process of cloning humans, types of cloning, products of cloning, cost, and also ethical issues. Besides the issues and reality part of cloning, thisRead MoreShould Cloning Be Allowed?1440 Words   |  6 Pagesunmatched by any other. Though science fiction tends to bend reality to fit a story, human cloning is becoming a feasible practice. Recently there has been a successful cloning of a sheep, so scientists start to speculate the different uses of cloning human embryos. The three forms of cloning that stand out are reproductive cloning, therapeutic cloning, and cloning for scientific research. Cloning should be permitted, but only reproductive cloning should be permitted with a limit on the number of babiesRead MoreThe Invention Of An Achievement And Left Alone983 Words   |  4 Pagesprogressively been making steps forward in cloning whether it be humans, animals or most recently stem cells. With all these advancements in cloning it leaves people with many uncertainties regarding the morality, results and ethics behind all of it. Regardless of the uncertainties science continues to proceed with studies and experiments; what used to be something only heard of in science fiction books or movies is now a reality. Even though it’s now reality, is it something that should be pursuedRead MoreShould Cloning Be Researched?1135 Words   |  5 Pagesdebated by scientists was whether or not cloning should be researched. When anyone even mentions the word â€Å"cloning†, they think of all the benefits and cures that the media has portrayed the research for cloning will bring. Although, this belief is nowhere near the reality of the situation. The research for cloning not only pushes the ethical boundaries, but they also threaten the circle of life by trying to be god. Before even considering whether or not cloning should be researched or not, peopleRead MoreThe Morality Of Genetically Modified Animals For The Purpose Of Pharming And Xenotransplantation879 Words   |  4 Pagesgenetically modifying animals for the purpose of pharming and xenotransplantation: the potential benefits and the reality. Although genetic modification of mammals carries promises of being able to use said modified mammals as a source of materials used in pharmaceuticals or creating mammalian species with organs that can be transplanted in humans, the reality is that reproductive cloning in mammals has a low success rate. 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New inventions and discoveries play an important part in order for a breakthrough in science. Scientists are eager to study and dig deeper into the mysteries of life, to them experimenting with cloning is a major step in fully understanding the human body and itsRead MoreCloning Is Morally Wrong?1400 Words   |  6 PagesCloning has been identified in two ways – Reproductive cloning and Therapeutic cloning. In reproductive cloning, â€Å"the created embryo is impla nted into a human womb, leading to the birth of a human being† . Alternatively, therapeutic cloning creates human beings in embryo form, but the embryos are used for experimentation and eventually destroyed. Both ‘therapeutic’ and ‘reproductive’ cloning techniques create human life. I believe that life is sacred from conception, and must be valued. The embryoRead More America Needs More Cloning Essay examples1350 Words   |  6 PagesNeeds More Cloning Each day there are numerous automobile accidents on highways. Many people are critically injured, but sadly, it is often easier to repair the automobile. It can be fixed using spare parts. Human drivers and passengers do not have that luxury (Mooney and Mikos, 1999). In this situation, cloning could help. There has been much debate on the issue of cloning, ever since the famous sheep, Dolly, was cloned from a mammary cell. Since that first development of cloning, there

Economics Sports economics

Introduction Every organization and every event needs to be structured properly for successful completion and evaluation. Whether it is business or sports, the appropriate structure and the framework is to be adopted properly with the correct attainment of features. MLS is Major League Soccer which is a professional league of Soccer in United States and Canada. The league is one of the sports league in USA and it comprises of 19 teams among which, 16 teams are from US and 3 teams are from Canada. The League is founded in the year 1993 and this league is affiliated and sanctioned by US Soccer Federation. The teams can play in other major cups also like US Open Cup, Canadian Champion ship, CONCACAF Champions league. Instead of the initial financial and operational problems in the first few years and there were some major transformations at that period. After the changes, MLS has expanded in terms of financial and structural attributes and with the specific objective by the owners, the evolution has been incurred perfectly (Cairns, et al, 1990). Instead of the different owners for the different teams, MLS works as a single entity and each team is owned and controlled by the leagues investors. These investors or operators control their teams just like the owners control their teams in other major leagues and thus t he organizational structure and the different business and economical attributes are also different for the MLS. Here the different business proposition and attributes with the economic and management perspective are discussed and analysed and those will help to understand the restructure the future strategies or different future framework for MLS (Cairns, et al, 1990). Organizational structure of MLS MLS follows a unique structure among all the leagues present in North America. MLS is operated by single body structure. All the clubs who have joined with MLS are considered as the part of business entity. There is only a single ownership present for the existing clubs within MLS. All the existing clubs under the league are operated and controlled by the league itself. The existing clubs under the league can operate independently and they can organize their operation independently. The activities which are done by the front office are considered as the accountability of the club. The Board of Directors are also considered as the accountability of the league. The major shareholder group or the owners belong to the league itself. The company is recently adopting the expansion strategy by adding more clubs under the league. There is a culture among the league to share the revenue and profits among the clubs of the league. Presently, the league consists of 19 clubs out of them 16 clubs belong to USA and the remaining 3 belong to Canada. Though the clubs get the opportunity to run the daily operation independently but the control lies in the hand of the league itself. In case, the contracts between the players are required to be reviewed and even in case of establishing new contracts the league will take every decisions. The players contracts are made centrally and the clubs cannot take any decisions regarding this. The Major League Soccer maintains a franchise model in which the clubs under the Major league Soccer are accountable to provide franchise fees to the league. Generally, the North American clubs enjoy more liberty than the European clubs but, Major Soccer League is known as the one vital exception. The player Union consists of the labours of five different leagues and those are The National Football League Players Association, The National Basketball and Hockey Association, The major league soccer association and finally the Major League Basketball playe rs Association forms the labour Union. The economic and financial integration are happened by the collective bargaining agreement (CBA) among the players of the clubs and the leagues. The CBA process lasts for five years and the process of re-negotiation among the players and the league owners can create a hindrance in the progress of the organization. The organization structure can be observed from the view point of productivity and the productivity of the league comes from two distinct bases and those are, the team base and the league base. First the club owners used to develop the skills and strengths for their players but any individual club is unable to produce long term profits. The second stage of organizational production function then consists of the league structure and performance (Demmert, 1973). Literature on sports economics in respect of MLS There are two dimensions present in the sports economics which signifies the level of competition in the sports market or the rules and regulations adopted by the sports market. The first dimension is related to the labour market where the movement of the players are highly regulated and the second dimension is the regulation implemented in the finance market. The finance market refers to that market which signifies the revenue collection either individually or collectively (Kesenne, 2010). The central body has the power to collect and distribute the revenues among the existing clubs. The two-dimensional model can provide four possible outcomes. Source: The sport business future (Aaron C.T. Smith) The revenue collection may be done by the competitor team and can be retained by them without distributing the revenue amount. The two dimensional framework which can raise four different types of possibilities can be shown by the above diagram. The 1st quadrant in the above diagram represents the free market where the finance market and the labour market are unregulated in nature (Cairns, et al, 1990). The home team is responsible for collecting the revenue but for example, the television promotions can be managed by the individual clubs. The quadrant two represents the unregulated type of labour market and the regulated type of finance market. Whatever the revenue will come can be used to redistribute those among the poorly structured clubs and teams but in this system the players would have the choice to choose the clubs where they want to play. The quadrant 3 represents the regulated labour market and the regulated finance market and this is considered as the mostly regulated and tight market scenario. Quadrant four finally, represents the unregulated finance and restricted labour market. The players contribution should be equally distributed and the talent of the players will also be equally distributed among the participating team (Rascher, 2012). The sources of revenues are generated in the North American context from different broad sources and those are selling of the tickets, second source is the facility based revenue which signifies that with the improved facilities the demand of the games would be increased and thus sports expenditure would also increase. The concept of transfer payment does not present in the context of North America and most of the leagues encourage prohibition of selling the players for cash (Cairns, et al, 1990). The ticket sales along with the fees for media rights, television and broadcasting rights etc are the major sources of the revenues of the company. The leagues have evaluated new form of earning revenues including the premium ticket selling, by introducing the luxury boxes and by introducing the club seats. The club seats refer to the special seating arrangement which is arranged in a special area of the club. The premium tickets are sold almost two to four times more than the normal ticket prices. Sponsorship is another source of obtaining revenue by the league (Sloane, 2006). In the sports economics it is argued about which objective a firm or a lunge should follow. One objective is the profit maximization objective which is considered as the basic objective of any economy. The second objective is the utility maximization objective which is required for the long term development of an organization. For determining the price of the tickets the league depends on the long term objectives of the business. The wages that are required to offer to the employees also depend on the broader objective of the organization. Most of the economists have observed that American leagues are dependent on the utility maximization approach. For satisfying the needs and requirements of the stakeholders of the leagues the leagues follow the profit maximization objectives as the profit maximization is the prime objective of any organization (Ferguson, et al, 2001). For the clubs under MLS winning the match will always signify a prestigious issue. Every league will prefer a wealt hy and prestigious franchise house. The clubs are therefore needs to apply a equilibrium market structure where the prices and the wage level would be fixed in accordance to both the profit maximizing objective and the utility maximizing objective. The economic effect of the sports lies in the broader sense and many economists have argued that, the introduction of new stadiums are not able to generate the economic welfare to the local areas but in the broader sense the soccer can introduce economic welfare on the country basis. Business Management within MLS The Major League Soccer is one of the major games or sport activity within US as stated before and this league is followed by a numerous number of people within the region. The average attendance is also higher than the attendance in NBA and other sports. The league has been profitable from the time of demonstrating single entity ownership structure within the league. The umbrella marketing and media and advertising scenario has also helped to demonstrating profits for the company. The league has developed the infrastructure for the stadiums and the play grounds and with them, more advertisement or ownerships has expanded. The league is now being shown in ESPN network with a long term seal signed and MP and Silva is also appointed with a partnership right for the television broadcasting. The league is also being sponsored by Adidas and many more company and also most every team has a separate sponsor for them. These also help to increase the profitability and the business structure f or the Major League Soccer. The teams are operated separately and for each team, there is an operational or business partner present within the league. Few teams have acquired also breakeven in the year 2012 and according to Forbes, the teams will be also more profitable due to the increase in soccer and the league within the people. The league also control costs by introducing the share revenues among the franchises and holds player contracts despite of players being contracted by the individual teams. The business strategy of the league is more concentrated into holistic marketing rather than individualistic marketing. The Major League Soccer is a private business entity and it does not publish the revenue or the share details to the public (Demmert,1973). So the public disclosure and the business strategy are more concentrated with the single entity ownership business approach and this will include the future consideration and policies for profitable league operations. Assessment of organization performance and structure in MLS IN the other major leagues like La Liga, BPL, Bundesliga etc. the organizational structure is more into perfect competition status and the owners of the clubs are also into individualistic approach. The leagues are bounded by the owners and their funding as the owners also find high risks attached with the operation of the clubs (Dietl and Lang, 2008). If an owner wants to direct the risk or the liquidity to other options, they are limited. No individual club owner is economically viable to approach or invest in other forms of business with the up growing risks attached to the club in those leagues other than selling the club. Thus specific league status and to safeguard the investors into the leagues, a different framework or process ne implied and applied with the leagues. The hold u risks and the cost to the owner are sometimes great and thus the leagues are also affected by the owners and stakeholders decisions (Szymanski and Ross, 2007). Therefore, classical form of the leagues can be more implemented and addressed as the value chain would be more under unified ownership and the owners can distribute the risks with associated teams for getting better performance and better stability within the leagues. An alternative business organization and organizational structure of vertical integration can be implemented within perception. The value chain can be more admired with the forward integration for the activities like team development into cup run or championship production. The back ward integration can be also seen as the introduction of the league owners into the leagues (Dietl and Lang, 2008). In the Major League Soccer of US, there is a single entity for commencing the league strategies and objectives and for doing that, voting is taken. The democratic issues within the League are more into accessible and adoptable rather than the other leagues. The participants are the club owners and they are more into owning the league rather than owning the single club. In this organizational structure, the risk is less and the clubs can be more prosperous with the perfect competition and endeavours attained by the owners (Downward and Dawson, 2000). This business consideration is unique and the owners do have the risks distributed within the clubs equally. There are also proper policies and rules to distribute the wealth or salary among the players. Apart from the designated player, core players or key player policies, all the players are in the same league or in the same salary caps. The team owners are also responsible for structuring the league with the financial and management pr ocesses and thus these improvements are to be observed within the Major League Soccer in US. There are also proper training and development activities which are integrated within the Major League Soccer and these helps to improve or emerge new young players within the soccer world. There is additional advertisers or fundraisers present in the league to introduce funds for the development and achievement of the league (Vrooman, 1995). Discussion and conclusion The above discussion of the events is examined by different frameworks and different techniques to understand the impact of the Major League Soccer in US with the appropriate relevance to the structure of the league. As seen from the discussion, the structure and organizational framework is different from the other major leagues and the league organization and proposition is also different from the others. The Major League Soccer is organized on the single entity and the owners own the league rather than the clubs individually and this activity is better in some points of organizational strategies (Dietl and Lang, 2008). The major change or difference is in the light of structure and strategies with the business orientation. The clubs are also been equally judged and synchronised by the owners and the league organization committee is responsible as the central authority for the teams and the players (Downward and Dawson, 2000). This structure implies the appropriate distribution of p ower and risks. But it is also responsible for the delaying of decisions or strategic implementation of the policies. As per the voting or decision taking rights, the single entity league structure is more adoptable and suitable to justify fair chance and structure of the leagues. But the salary caps or the decisions are to be more lenient or at par with other major leagues and the players and the clubs are to be more adjustable with the requirement. Other than that, no permanent restructuring is needed to be emphasised within the Major League Soccer.

Foundations of Couples - Marriage - and Family

Question: Discuss about the Foundations of Couples, Marriage, and Family. Answer: Introduction: face a range of ethical conflict during their daily work. No matter where they function in their roles, they are faced with ethical decisions that can affect them as well as their patients. Ethical conflict in healthcare and nursing is ranked as the top ethical challenge facing health care today. Ethical dilemma occurs when a person is not able to act on ones ethical choice. Therefore, constraints interfere with acting in a way that an individual believes to be right. According to Canadian Nurse Association, ethical conflict is a situation where nurses cannot fulfill their ethical duty thus failing to pursue their right course of action (In Lundy, In Janes, Lundy, 2016). Ethics involves doing well and not causing harm. Nurses encounter many ethical issues in their workplace. The ethical conflict that happens in their day-to-day work entails them making a decision (Harvey Land, 2016). The decisions they make are affected by many factors including principles, values, as well as exper iences. Harry Nelson is a patient who is ailing from end-stage cancer. He is receiving palliative care at his home where he has brought up his family. Harry has not prepared an advanced care directive, and his condition is worsening, and he is aware that his death is imminent. Harry wishes to die in his home, but his daughter Stacey is stressed because of his fathers condition. Therefore, she calls an ambulance because she believes his father can live longer if he receives further treatment. When the palliative care crew gets at Harry Nelson home, they find considerable conflict, and this becomes complex on how to solve it. Harry has the desire to stay at home, and on the other hand, Stacey wants him to go to the hospital for further treatment. The ambulance transports Harry because according to its guidelines, Harry meets the criteria of medical treatment and transport. Therefore, Harry is taken to hospital despite his objections. According to the case study above, it is clear that there is a considerable conflict. Critical medical issues like Harry Nelson condition involve far-reaching decisions about the life and the death of a severely sick patient, and this is very challenging for the healthcare staffs. Furthermore discussing such serious decisions with the patient becomes impossible due to their stubbornness thus resulting in an increased need to communicate with the relatives or legal representative. Trust and confidence, therefore, become challenged thus inviting conflict. According to the case study, the considerable conflict occurred between the palliative care and Harry Nelsons daughter Stacey in the matter of whether or not to limit the life-sustaining treatment and the patients death due to Harry Nelson wish to stay at home. Principles of healthcare ethics According to the actions of the palliative care crew, they acted in accordance to the relevant principles of healthcare, code of ethics as well as the code of conduct in healthcare professional. According to research, critical medical issues present the major ethical theories and application in healthcare as part of the foundation for the study of ethics. The foundation shows how principles are used in healthcare and how they are applied in care of ethical conflict. The principles that are commonly used in healthcare ethics are non-maleficence, beneficence, and autonomy. The nurses use these principles to make decisions in case of serious illness in patients like that of Harry Nelson. This principle relates to the first ethical requirement in healthcare that procedure done by any healthcare professional should not harm the patient or others in the community. In healthcare ethics, there is no clear evidence whether a health care professional want to do harm or avoid doing harm. Nevertheless, the debate comes in when we consider the word harm. Different ethical theories give the various meaning of harm. For example, a deontologist would define harm as that which prevents people from performing their duty or that, which is opposed, to the formal condition of moral law. On the other hand, a consequentiality would define harm as that which prevents the good thus leading to less good. However, in healthcare, harm is that which worsens the condition of a patient (Evans, 2017). In many fields, they may say harm include pain, inconvenience, expense or even discomfort. Nevertheless, in healthcare settings, harm is considered physical harm. According to the case study, the palliative care crew acted in non-maleficence. The crew did not mean to harm Harry Nelson despite his wish to stay at home. First, his health was deteriorating, and therefore, it is the duty of the healthcare crew to look after him without neglecting him. Even though in clinical setting non-maleficence can occur due to negligence and as a violation of autonomy, a patient needs to have a legal document that specifies what actions should be taken for their health (Maville Huerta, 2013). In harm as a violation of autonomy, a patient may elect not to receive treatment because of their condition as they are aware they are about to die. Therefore, imposing treatment on a person who desires to lose life quality then it is wrong. According to the case study, Harry is very weak and fully aware that his death is nearing thus he wishes to remain at home with no further treatment and die in his home. However, the palliative care crew transports him against his w ill upon his daughters insistence. This, however, is not wrong because the crew was taking Harry to the hospital for further treatment they did not mean any harm. Second, Harry had not prepared an advanced care directive. Therefore, their actions are not questionable as they acted in best of interests and reasonable decision (Smith, 2016). Beneficence principle advocates that any performance should be done with the intention of doing well for the patient. It demands that healthcare staffs develop and maintain skills and knowledge, considers the circumstances of all patients and strive for overall benefit. Beneficence involves more than just avoiding harm (Dossey, Keegan, In Barrere, 2016). It involves a level of altruism that is not present from just refraining from harm. The healthcare ethical guideline of having to involve in beneficent acts means that the healthcare professionals are ethically required to take optimistic steps to assist the patients. Concerning the ethical theories, the principle advocates for the maximum good for an utmost number just as the principle of consequentialism (Shafer-Landau, 2012). The beneficence principle is important in healthcare ethics and therefore, ethical egoism, which is the belief that our basic obligation is a virtue, is not connected from healthcare (Maville Huerta, 2013). Therefore, in the case study, the healthcare professionals transported Harry to the hospital because they wanted to help him. They did not hold the negligent or deficient action to perform a beneficent act. Healthcare workers are expected to portray the act of kindness and courtesy as it is part of the common morality of healthcare. Nevertheless, the principles of beneficence together with non-maleficence are not sufficient in healthcare actions due to the people who defined harm and good. Many patients are ignorant of why physicians are doing certain treatments n them. This is evident in the case study, as Harry does not want any further treatment thus having self-interest of just staying at his place and having a wish of dying at his own home. Therefore, the palliative care crew who is avoiding harm has no say in Harry Nelson wish. However, according to (Tobin, 2012) competent patients have the assertion of making decisions for themselves, which now brings the conflict of Harry being taken to the hospital and his desire to stay at home. Thus, due to the inadequacy of the two principle health professional can include the principle of Autonomy in their decisions concerning a patient who has a critical condition. This principle necessitates that the patient has the sovereignty of intent, deed, as well as thought when making verdict concerning healthcare procedures (Fletcher, 2015). Therefore, the decision taken must be free of coaxing for the patient to makes fully choice. In addition, the patient must be aware of the risks and likelihood of success in the procedures. However, in most times when a patient is in a critical condition, it is always difficult t expect the patient to operate under fully informed consent. In the case study, Harry has the right to make any decision he wants without anyone going against him. On the other hand, the palliative care crew has the obligation of acting a way that benefits Harry without harming him. Therefore, due to the conflict of his desire and the best interest of the palliative crew, the healthcare professionals should be considerable in extending Harry Nelson quality life by doing further treatment. However, Harry is competent and understands that his condition is critical and he is aware that his death is about to happen and that is why he does not want any further treatment. This makes it difficult for the palliative crew to decide whether to act upon Harry or upon his daughter insistence. It is therefore at this point the principles of healthcare ethics non-maleficence, beneficence and autonomy are important for the healthcare professionals in making decisions in any healthcare conflict. Professional Code of Ethics Since 1953, the code in healthcare has served as the standard for nurses globally (In Capuzzi In Stauffer, 2015). The code of ethics for nurses is a channel for action based on social standards and morals. The code of ethics makes it clear that inherent in nursing is respect for human rights which include, right to life, dignity and to be treated with respect (In Friberg In Creasia, 2016). The code of ethics directs nurses in daily choices thus supporting their denial to perform in activities that conflict with caring and healing. In the case study, the palliative care crew strived to guard the health and the safety of the patient. The nurses do not only focus on giving care to a patient but also to promote safety and protection as well as the rights of the patient (Butts Rich, 2016). In the case, Harry, being taken to the hospital against his will was to protect his health. Even though he had wished to remain at his home, the lack of an advanced care directive makes the palliative to focus on their actions and encounters with Harry Nelson. Therefore, their act drives protocols and promotes the wellbeing, protects his life (In Gerrish In Lathlean, 2015). Also in the code of ethics in the case study, the healthcare professionals conducted themselves professionally thus practicing care regardless of the unrestricted consideration, nature of health problem and Harry Nelson wish. Besides, the palliative crew owed Harry Nelson commitment of care. The ethical codes in nursing say that the nurses responsibility is foremost to their patients. Even though the crew was overcome by the considerable conflict, in the code of ethics, the nurse must always recall that their primary responsibility is to be an advocate to and for their patient and not any influence like in the insistence of Stacey Harry Nelson daughter. Professional Code of Conduct According to the International Patient Charter, codes sets out the customary of behavior that nurses are expected to maintain in their professional practice (Fowler American Nurses Association, 2015). The code of conduct advises nurses on what is expected of them. The codes are in line with the statutory to maintain the health and safety of patient (Sorkin, August, Fischer, 2017). According to the case study, the healthcare professionals acted in agreement with the code of conducts for nurses which include integrity, improving health and accountability (Waddill-Goad Sigma Theta Tau International, 2016). In the case study, the crew acted in full honor to safeguard the life of Harry. Also as the healthcare services are in agreement with promoting health. Therefore, the palliative care crew did just to take Harry in the hospital as the code of conduct acts nurses to support and empower those with chronic conditions. Besides accountability comes in when Stacey asks the crew to take her father for further treatment. The crew agrees since it is their duty to address peoples concerns. Other national and international Charters Other national healthcare charters like "you and Your Health Service" is created through efforts from many concerned individuals. The parties include advocacy groups, individual supporters who assist in the enhancement of the National policy for health services. The charter is an element of the program of quality enhancement initiatives developed to involve service users in influencing optimum healthcare (In Lundy, In Janes Lundy, 2016). The International Charter for Human Values in Healthcare is another healthcare charter. It is an inter-professional collaborative involving people globally working together to restore core human ethics to health care (Tobin, 2012). The human values consist of respect for people, integrity, justice, as well as a commitment to excellence. Ethical Theory Different ethical theories exist, and they are applied in different situations to help in healthcare professional thinking as well as making decisions. The theory of ethical principals is mostly used in approaching ethics in healthcare. The ethical principal is viewed as a tool of making a decision as well as solving problems, which is relinquished by appealing to good moral principles (Brixey et al., 2015). The theory emphasizes the four ethical healthcare principles, which are autonomy, justice, beneficence, and non-maleficence. These principles are accepted in healthcare as they assist in providing sound moral reason while taking actions. This theory is increasingly acceptable as a reliable and practical framework for solving moral problems in health care. The principal ethical theory has therefore gained contemporary discussions on ethical issues in healthcare. Therefore, it is essential for nurses to have knowledge and understanding of this model. Conclusion In conclusion, it is essential for the healthcare professionals to have the full competence of the health care principles, code of ethics, as well as the code of conducts to help them in their professional practice. The principles of healthcare ethics can assist health care at the time of handing critically ill patients, as mostly they have to make decisions that are in agreement with promoting health and preventing harm in the patient. The commitments of the nurses in their care include providing safety, competent and ethical care. However, the healthcare professionals get frustrated in their efforts to enact their professional in the way that is acceptable to them. Due to that then ethical conflict is created. If however, palliative care crew like in the case study were not able to identify the ethical conflict and dilemma then they would not have been able to make any right decision and resolve the situation while as well building up of moral residue. References Brixey, J. J., Brixey, J. E., Saba, V., McCormick, K. (2015).Essentials of Nursing Informatics Study Guide. Butts, J. B., Rich, K. L. (2016).Nursing ethics: Across the curriculum and into practice. Burlington, MA : Jones Bartlett Learning In Friberg, E. E., In Creasia, J. L. (2016).Conceptual foundations: The bridge to professional nursing practice. Dossey, B. M., Keegan, L., In Barrere, C. (2016).Holistic nursing: A handbook for practice. Evans, D. R., Hearn, M. T., Uhlemann, M. R., Ivey, A. E. 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