Numerical differential equation analysis package - Free Essay 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! 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.

Macroeconomic Implications Education As Foundation Of Nations Economy - 825 Words

Macroeconomic Implications: Education As A Foundation Of Nations Economy (Essay Sample) Content: Macroeconomics Name Instructor’s name Institution Education as A Foundation of Nations Economy According to me, education is simply a continuous learning process that equips one with the knowledge to face all aspects of life challenges. Education illustrates its significance to the growth of the economy as a promoter of production quality, technology advancement and human development. Education helps individuals to attain success in various fields and also helps in setting higher objectives which helps them surpass the ordinary persons, especially in the job market. In this article, am going to discuss the educational system in both Ireland and Taiwan and its macroeconomic implications. Irish Education System In Ireland, education is mandatory for all children aged 6-16 until completion of the three-year second level education. The education system comprises of primary, secondary, vocational and tertiary education. Under primary or first-level education entirely covers basic education including science education and environment, social, mathematics, language, health education, physical integration, and education arts (drama and music). Secondary education offers several kinds of post-primary schools; comprehensive, community, secondary and vocational schools. After primary education, the majority of pupils go to vocational/secondary schools till they attain a leaving certificate after the 12th grade. Second level education constitutes a three-year junior series accompanied by a two or three-year senior series. In vocational education, 30% of secondary school students are offered with practical skills that can help them earn a living. Third-level education. It comprises of a technological sector, university sector and colleges of education. It also includes independent private institutions. There are universities which govern themselves and the offer degree programs at bachelor, masters and doctorate level. The whole population of Ireland as per 2018 statistics is approximated to be 4,803,748 people. The population that attend primary, secondary and higher education institution is approximated to be 1,091,632 people. This implies that at least 50% of this statistical population have attained primary education, 34% have acquired secondary education and at 16% have acquired post-secondary education. Taiwan Education System In Taiwan, the ministry of education has the responsibility of setting and maintaining policies governing education and also managing all public education. The education system is made up of primary education -elementary schools. Secondary education-junior, senior and vocational schools. Higher education includes institutions such as universities, colleges and technical institutes ADDIN ZOTERO_ITEM CSL_CITATION {"citationID":"OI1kfnQG","properties":{"formattedCitation":"(Woo, 1991)","plainCitation":"(Woo, 1991)","noteIndex":0},"citationItems":[{"id":498,"uris":["http://zotero.org/users/local/MqS91Xa3/items/855BMJQK"],"uri":["http://zotero.org/users/local/MqS91Xa3/items/855BMJQK"],"itemData":{"id":498,"type":"article-journal","title":"Education and economic growth in Taiwan: A case of successful planning","container-title":"World Development","page":"1029–1044","volume":"19","issue":"8","source":"Google Scholar","shortTitle":"Education and economic growth in Taiwan","author":[ {"family":"Woo","given":"Jennie Hay"}],"issued":{"date-parts":[["1991"]]}}}],"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"} (Woo, 1991). Taiwan as a province roughly homes a population of 23 million people out of which 5,384,926 people are formally literate at various levels. Total of 40% of this population attends primary education, 31% secondary school level and 29% post-secondary education. Macroeconomic Implications of Education In as much as education seems to benefit an individual, it is one of the foundations of economic growth in many countries and in this case, Ireland and Taiwan. some of the positive macroeconomic implications of education include employment. From statistics employment figures in both countries have illustrated a considerable increase, because the educated young graduates have occupied several net professional job opportunities offered by the government. Employed people are capable of meeting their basic needs and thus contributing to the overall income of the nation. One’s income is another one’s expenditure. Education is an investment. In the long run edu...

A Guide to the Origins and Celebration of Kwanzaa

Unlike Christmas, Ramadan, or Hanukkah, Kwanzaa is unaffiliated with a major religion. One of the newer American holidays, Kwanzaa originated in the turbulent 1960s to instill racial pride and unity in the black community. Now, fully recognized in mainstream America, Kwanzaa is widely celebrated. The U.S. Postal Service debuted its first Kwanzaa stamp in 1997, releasing a second commemorative stamp in 2004. In addition, former U.S. presidents Bill Clinton and George W. Bush recognized the day while in office. But Kwanzaa has its share of critics, despite its mainstream status. Are you considering celebrating Kwanzaa this year? Discover the arguments for and against it, whether all blacks (and any non-blacks) celebrate it and the impact of Kwanzaa on American culture. What Is Kwanzaa? Established in 1966 by the African-American professor, activist and author Ron Karenga (or Maulana Karenga), Kwanzaa aims to reconnect black Americans to their African roots and recognize their struggles as a people by building community. It is observed every year between Dec. 26 and Jan. 1. Derived from the Swahili term, â€Å"matunda ya kwanza,† which means â€Å"first-fruits,† Kwanzaa is based on African harvest celebrations such as the seven-day Umkhost of Zululand. According to the official Kwanzaa website, â€Å"Kwanzaa was created out of the philosophy of Kawaida, which is a cultural nationalist philosophy that argues that the key challenge in black people’s [lives] is the challenge of culture, and that what Africans must do is to discover and bring forth the best of their culture, both ancient and current, and use it as a foundation to bring into being models of human excellence and possibilities to enrich and expand our lives.† Just as many African harvest celebrations run for seven days, Kwanzaa has seven principles known as the Nguzo Saba. They are: umoja (unity); kujichagulia (self-determination); ujima (collective work and responsibility); ujamaa (cooperative economics); nia (purpose); kuumba (creativity); and imani (faith). Celebrating Kwanzaa During Kwanzaa celebrations, a mkeka (straw mat) rests on a table covered by kente cloth, or another African fabric. On top of the mkeka sits a kinara (candleholder) in which the mishumaa saba (seven candles) go. The colors of Kwanzaa are black for the people, red for their struggle, and green for the future and hope that comes from their struggle, according to the official Kwanzaa website. Mazao (crops) and the kikombe cha umoja (the unity cup) also sit on the mkeka. The unity cup is used to pour tambiko (libation) in remembrance of ancestors. Lastly, African art objects and books about the life and culture of African people sit on the mat to symbolize commitment to heritage and learning. Do All Blacks Observe Kwanzaa? Although Kwanzaa celebrates African roots and culture, the National Retail Foundation found that just 13 percent of African Americans observe the holiday, or approximately 4.7 million. Some blacks have made a conscious decision to avoid the day because of religious beliefs, the origins of the day and the history of Kwanzaa’s founder (all of which will be covered later). If you’re curious about whether a black person in your life observes Kwanzaa because you want to get him or her a related card, gift, or another item, simply ask. Don’t make assumptions. Can Non-Blacks Celebrate Kwanzaa? While Kwanzaa focuses on the black community and African Diaspora, people from other racial groups may join in the celebration. Just as people from a range of backgrounds partake in cultural celebrations such as Cinco de Mayo, Chinese New Year or Native American powwows, those who aren’t of African descent may celebrate Kwanzaa. As the Kwanzaa Web site explains, â€Å"The principles of Kwanzaa and the message of Kwanzaa has a universal message for all people of good will. It is rooted in African culture, and we speak as Africans must speak, not just to ourselves, but to the world.† New York Times  reporter Sewell Chan grew up celebrating the day. â€Å"As a child growing up in Queens, I remember attending Kwanzaa celebrations at the American Museum of Natural History with relatives and friends who, like me, were Chinese-American,† he  said. â€Å"The holiday seemed fun and inclusive (and, I admit, a bit exotic), and I eagerly committed to memory the Nguzo Saba, or seven principles†¦Ã¢â‚¬  Check local newspaper listings, black churches, cultural centers or museums to find out where to celebrate Kwanzaa in your community. If an acquaintance of yours celebrates Kwanzaa, ask for permission to attend a celebration with her. However, it would be offensive to go as a voyeur who doesn’t care about the day itself but is curious to see what it’s about. Go because you agree with the principles of the day and are committed to implementing them in your own life and community. After all, Kwanzaa is a day of tremendous significance for millions of people. Objections to Kwanzaa Who opposes Kwanzaa? Certain Christian groups who regard the holiday as pagan, individuals who question its authenticity and those who object to founder Ron Karenga’s personal history. A group called the Brotherhood Organization of a New Destiny (BOND), for one, labeled the holiday as racist and anti-Christian. In an article in the self-avowed right-wing anti-muslim magazine FrontPage, BOND founder the Rev. Jesse Lee Peterson takes issue with the trend of preachers incorporating Kwanzaa into their messages, calling the move â€Å"a horrible mistake† which distances blacks from Christmas. â€Å"First of all, as we’ve seen, the whole holiday is made up,† Peterson argues. â€Å"Christians who celebrate or incorporate Kwanzaa are moving their attention away from Christmas, the birth of our Savior, and the simple message of salvation: love for God through his Son.† The Kwanzaa Web site explains that Kwanzaa isn’t religious or designed to replace religious holidays. â€Å"Africans of all faiths can and do celebrate Kwanzaa, i.e., Muslims, Christians, Jews, Buddhists†¦,† the site says. â€Å"For what Kwanzaa offers is not an alternative to their religion or faith but a common ground of African culture which they all share and cherish.† African Roots? and a Troubled Founder Even those who don’t oppose Kwanzaa on religious grounds may take issue with it because Kwanzaa is not an actual holiday in Africa and, furthermore, the customs founder Ron Karenga based the holiday on roots in Eastern Africa. During the  transatlantic slave trade, however, blacks were taken from Western Africa, meaning that Kwanzaa and its  Swahili  terminology aren’t part of most African Americans’ heritage. Another reason people choose not to observe Kwanzaa is the background of Ron Karenga. In the 1970s, Karenga was  convicted  of felony assault and false imprisonment. Two black women from the Organization Us, a black nationalist group with which he’s still affiliated, were reportedly victimized during the attack. Critics question how Karenga can be an advocate for unity within the black community when he himself was allegedly involved in an attack on black women. Wrapping Up While Kwanzaa and its founder are sometimes subject to criticism, journalists such as Afi-Odelia E. Scruggs celebrate the holiday because they believe in the principles it espouses. In particular, the values Kwanzaa gives to children and to the black community at large are why Scruggs observes the day. Initially, Scruggs thought Kwanzaa was contrived, but seeing its principles at work changed her mind. In a  Washington Post  column, she wrote, â€Å"I’ve seen Kwanzaa’s ethical principles work in many little ways. When I remind the fifth-graders I teach that they aren’t practicing ‘umoja’ when they disturb their friends, they quiet down. †¦When I see neighbors turning vacant lots into community gardens, I’m watching a practical application of both ‘nia’ and ‘kuumba.’† In short, while Kwanzaa has inconsistencies and its founder a troubled history, the holiday aims to unify and uplift those who observe it. Like other holidays, Kwanzaa can be used as a positive force in the community. Some believe this outweighs any concerns about authenticity.

William Faulkners A Rose for Emily and Barn Burning Essay

Symbolism in William Faulkners A Rose for Emily and Barn Burning If we compare William Faulkners two short stories, A Rose for Emily and Barn Burning, he structures the plots of these two stories differently. However, both of the stories note the effect of a father ¡Ã‚ ¦s teaching, and in both the protagonists Miss Emily and Sarty make their own decisions about their lives. The stories present major idea through symbolism that includes strong metaphorical meaning. Both stories affect my thinking of life. Both  ¡Ã‚ §A Rose for Emily ¡Ã‚ ¨ and  ¡Ã‚ §Barn Burning ¡Ã‚ ¨ address the influence of a father, and the protagonists of both stories make their own†¦show more content†¦He notifies the landlord of the fire, and runs away from his family.  ¡Ã‚ §He [Sarty] did not look back ¡Ã‚ ¨ ( ¡Ã‚ §Barn Burning ¡Ã‚ ¨, 25). He does not want to let his father controlling him anymore. He wants to start his own life. Both the stories present major ideas through symbolism. Faulkner uses particular objects to link the tales with his metaphorical meaning.  ¡Ã‚ §A Rose for Emily ¡Ã‚ ¨ does not explicitly involve a rose. Faulkner notes the rose only twice, in the title and the third paragraph from the last,  ¡Ã‚ §Ã‚ ¡Kthis room decked and furnished as for a bridal: upon the valance curtains of faded rose color, upon the rose-shaded lights ¡K ¡Ã‚ ¨ ( ¡Ã‚ §A Rose for Emily, 129). But the significant symbolic meaning of the rose strongly affects the readers ¡Ã‚ ¦ perception of Miss Emily. It stirs the readers to sympathize with Miss Emily. Rose stands for true love, expectation and the most resplendent period of life. Miss Emily adorns her room as a bridal chamber in rose color, representing a woman who yearns for true love and dreams of a fairyland where she and her beloved can stay together forever. For years, Miss Emily ¡Ã‚ ¦s father drove away all the young men who want to date with h er. Her father thwarted her to experiencing love. In her dreary existence, Homer Barron is the only bright spot, one  ¡Ã‚ §rose ¡Ã‚ ¨. Like a wilted rose, she keeps his body, forever. It reminds her of the joy she once had in her otherwise emptyShow MoreRelatedWilliam Faulkner’s Barn Burning and A Rose for Emily Essay1157 Words   |  5 Pages â€Å"Barn Burning† is a story filled with myth. This coming of age story features a boy stuck in a family with a father who can be thought of as Satan, and can be easily seen as connected to myths of Zeus and Cronus. The connection to Zeus is further elaborated when William Faulkner’s â€Å"A Rose for Emily† is also considered. These two stories along with a few others provided an amazing view of the south. Many characters or families can be viewed as groups that lived in the south duringRead MoreEssay on A Rose for Emily and Barn Burning856 Words   |  4 PagesWilliam Faulkner some would say was one of the great writers of American literature during the twentieth century. His stories many times had a gothic plot and contained odd or supernatural ideas and characters. He had many notable works, two of which were â€Å"A Rose for Emily† and â€Å"Barn Burning†. â€Å"A Rose for Emily† and â€Å"Barn Burning† are similar in the way that William Faulkner portrays the characters and the tone he uses in both. Emily the main character in â€Å"A Rose for Emily† and Sartoris theRead MoreWilliam Faulkner s A Rose For Emily And Barn Burning796 Words   |  4 PagesWilliam Faulkner has said that when you are writing a novel, there is a lot of room to add some fluff and be a bit careless with your ideas, but when writing a short story there is no room for â€Å"trash†, as he calls it. However, this can be a tough thing to do when you have a lot to say. Even Faulkner could have difficulty following along with his beliefs. After having read and considered A Rose for Emily and Barn Burning, I do believe that William Faulkner has been true to his beliefs in both of theseRead MoreWilliam Faulkner s A Rose For Emily1801 Words   |  8 Pages William Faulkner is known for his many short stories, however, many has wondered what has influenced him in writing these stories. Like his well known, most famous short story â€Å"A Rose for Emily†, which has always been compared to â€Å"Barn Burning†, one of Faulkner’s other short story. It only make sense to compare them two together because these two stories has may similarities , whether it may be in setting , characters or style they favor each other . Nevertheless they also have many differencesRead MoreComparing the Setting of Barn Burning to that of A Rose for Emily1352 Words   |  6 PagesComparing the Setting of Barn Burning to that of A Rose for Emily William Faulkner has written some of the most unique novels and short stories of any author, and, to this day, his stories continue to be enjoyed by many. Both â€Å"Barn Burning† and â€Å"A Rose for Emily† tell about the life of southern people and their struggles with society, but Faulkner used the dramatic settings of these two stories to create a mood unlike any other and make the audience feel like they too were a part of these southernRead MoreWilliam Faulkner s A Rose For Emily1810 Words   |  8 Pages William Faulkner is one amazing writing ,who is known for his many short stories .However, many has wondered what has influenced him in writing these stories . Like his well known, most famous short story â€Å"A Rose for Emily†, which has always been compared to â€Å"Barn Burning†, one of Faulkner’s other short story. It only make sense to compare them two together because these two stories has may similarities , whether it may be in setting , characters or style they favor each other . NeverthelessRead MoreA Rose For Emily And Barn Burning By William C. Faulkner949 Words   |  4 Pagesshort stories were written by William C. Faulkner who embodied the Southern sensibility, and to this day his stories continue to be enjoyed by many. Faulkner was born from a rich family who had accumulated wealth before the Civil War, but like many families in the South they had lost all of it during the conflict. His family moved to Oxford, Mississippi which is the basis for the fictional town of Jefferson in most of his stories from Yoknapatawpha County. Faulkner’s stories create a mood to makeRead MoreComparing and Contrasting Barn Burning and A Rose for Emily1141 Words   |  5 PagesHunter Taylor Dr. William Bedford English 1102-011 10 September 2013 Comparing and Contrasting â€Å"A Rose for Emily† and â€Å"Barn Burning† In William Faulkner’s short stories â€Å"A Rose for Emily† and â€Å"Barn Burning† the characters are both guilty of committing terrible crimes. However, Miss Emily in â€Å"A Rose for Emily† and Abner Snopes in â€Å"Barn Burning† are both portrayed very differently from each other. A few things to consider while reading these short stories is how each of these characters is characterizedRead MoreA Rose For Emily By William Faulkner1138 Words   |  5 Pagespity and sacrifice which have been the glory of his past.† With these words, American author William Faulkner described the duty of an author in his Noble Prize acceptance speech. Under further examination of Faulkner’s works, one would expect to find that he followed his own job description. However, two of his most well-known short stories seem to be contradicting. Neither â€Å"A Rose for Emily† nor â€Å"Barn Burning† is uplifting because the protagonists strugg le with their communities, loyalty to their fathersRead More William Faulkner Essay1234 Words   |  5 Pagesthe beginning of the twentieth century (William Faulkner; 699). He was the son to Murray C. and Maud Butler Faulkner (Hoffman 13). Growing up in the South in the early 1900s meant being exposed to harsh racism. He watched the blacks endure unbelievable amounts of cruelty and was amazed at how the blacks conducted themselves with such dignity. He witnessed, first hand, what discrimination is and could not comprehend why this goes on. In many of Faulkners works I found that he portrayed blacks as

Stereotypes Of Stereotypes Are Everywhere - 2445 Words

Stereotypes are everywhere. No matter which part of the world you are in, no matter who you talk to, there will always be stereotypes. In Stuyvesant, for example, the main stereotype deals with Asians. Typically, when most people think of Asians, their first instinct is to imagine an extremely studious child with millions of extracurricular who stays upridiculouslylate doing homework. He or she sits in a dark room, illuminated by only a small desk lamp, bending over the twenty page math homework assignment due for weeks in advance. In addition, if an Asian isn t getting straight A s in all of his or her classes, then the only valid conclusion is that they must be doing drugs. I mean, there is absolutely no other possible reason for a student of Asian descent to not be a straight-A student, right? Another relatively popular stereotype in Stuy is about Russian people. There is a relatively large number of Russian kids in Stuyvesant, and all of them have to deal with a couple of false preconceptions about their culture. All of the students of Russian descent are, without a doubt, permanently-drunk alcoholics. They have cold personalities and are very straight to the point. They rarely smile, but when they do, it is not a smile that isfriendly,but more of a smirk. They also all have an IV of vodka under their clothes to keep themselves constantly intoxicated without getting caught, right? I fall into the large group of Russians that are forced to fall victim to theShow MoreRelatedEffects Of Stereotyping In Schools1122 Words   |  5 Pageson hate or fear. Stereotyping can cause behaviors that will be carried into ones adulthood. How we stereotype someone changes them? It changes the way we act towards them because of how we classify them. Stereotypes make people treat others differently, which begins to affect those who are being judged. The way people act towards a person can begin to shape that person. Stereotyping is everywhere, and that’s not good because people’s judgement of others changes how that person looks at himself.Read MoreStereotypes And The Athletic Snob1083 Words   |  5 PagesStereotype- a widely held but fixed and oversimplified image or idea of a particular type of person or thing. Stereotypes show up everywhere, and the stories The Outsiders, a realistic fiction novel by S.E. Hinton, and the short story â€Å"The Athletic Snob† by Sam Barnes are no exceptions. In The Outsiders, the town Ponyboy Curtis lives in is divided in two: the rich, wild Socials, or Socs, and the quiet, tough Greasers. Throughout the course of the novel, Ponyboy, his brothers, and his friend s startRead MoreStereotypes of Africa: How Much Do You Know?669 Words   |  3 PagesThe word stereotype can be defined as â€Å"a widely held but fixed and oversimplified image or idea of a particular type of person or thing. Stereotypes can be found everywhere, from schools to our views on the world. There are many stereotypes about Africans and Africa as a whole, and just like a majority of all stereotypes, they couldn’t be more wrong. One of the stereotypes about Africa is that it’s a country. It has often been argued that Africa isn’t a continent when, in actuality, it is the secondRead MoreRacial Stereotypes755 Words   |  4 PagesThere are many different stereotypes in the world today. They can be used for different categories like age, gender and race. Stereotypes are formed by the media, passed down from many generations and also just the populations need to understand the social world around us. Racial stereotypes make up large portion of stereotypes in todays society. Racial stereotypes can be used for comedic effect and our found to be funny by a majority of people, but they can also be depicted as hate to an ethnicRead MoreEssay on Anti-Gay Bullying Stereotypes and Suicides825 Words   |  4 PagesAnti-Gay Bullying 1 Anti-Gay Bullying Stereotypes and Suicides HU300: Art and Humanities: Twentieth Century and Beyond Anti-Gay Bullying 2 Anti-Gay Bullying Stereotypes and Suicides Anti-gay bulling has increased over the years. There are more gays and lesbians committing suicide as a result. Asher Brown, a 13-year-old Houston, Texas teen committed suicide because he could not take the daily ridiculing of being bullied at school for years. Asher wasRead MoreEthnic Stereotyping : Nereotyping, And Racial Stereotypes910 Words   |  4 PagesStereotypes Stereotypes refers to the features imposed upon individual groups which are conventional, formulaic and exaggerated regarding to their nationality, race and sexual alignment, among many others (Stuart Ewen Elizabeth Ewen; 2006). These features tend to be over simplications of the groups involved. For instance, somebody who meets some few people from a certain country and finds them to be old fashioned and quit may spread to all the people from the country in question are reserved andRead MoreStereotypes - A Hasty Generalization Essay example961 Words   |  4 PagesStereotypes are everywhere and can be about anyone. Generalized remarks about gender, sexual orientation, religion, ethnicity or age are common forms of stereotyping. Any time someone makes hasty groupings whether by race, gender or an individual and makes a blanket judgment about them is stereotyping. 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Research shows that stereotypes operates off of emotions and below

Employee Motivation Research Paper - 2403 Words

Motivating Employees Introduction Employees are motivated by both intrinsic and extrinsic rewards. In order for the reward system to be effective, it must encompass both sources of motivation. Studies have found that among employees surveyed, money was not the most important motivator, and in some instances managers have found money to have a de-motivating or negative effect on employees. This research paper addresses the definition of rewards in the work environment context, the importance of rewarding employees for their job performance, motivators to employee performance such as extrinsic and intrinsic rewards, Herzberg’s two-factor theory in relation to rewarding employees, Hackman and Oldman model of job enrichment that†¦show more content†¦Cicerone et al (2007) suggests that â€Å"Rewarding employees for their job performance that meets or exceeds customer expectations is important because: âž ¢ It tells employees what standards their job performance must meet. âž ¢ When employees kn ow that customers expect a particular level of performance, they’ll be more cooperative about performing at that level than if a performance standard seems to be based on a manager’s impulse. âž ¢ Rewards improve employee job performance. âž ¢ The need to discipline employees is reduced because employee job performance meets customer expectations more often. This creates a more pleasant work environment for managers and their employees. âž ¢ Customers’ expectations are more likely to be met by employee job performance even when a manager isn’t present. ( ¶. 8) Motivators According to Bateman Snell (2009), Motivators to employee job performance are centered on extrinsic and intrinsic rewards. Extrinsic rewards are characteristics of the workplace that attract and retain people. They revolve around organization and management policies, working conditions, pay, benefits, and other so-called â€Å"hygiene† factors. Intrinsic rewards are motivators that provide employees personal satisfaction in the performance of their jobs such as opportunities for personal and career growth, recognition and the feeling of achievement in the successful completion of a task. (p. 486). Herzberg’s two-factor theory suggestsShow MoreRelatedMotivation To Improve Performance Through Employee Involvement.1571 Words   |  7 Pages Motivation to Improve Performance through Employee Involvement Charlese Mason Leadership and Organizational Behavior; 520 Dr. Laura Jones Strayer University February 13, 2017 Content 1 Introduction (Motivation through Involvement) 2 The Important Road Ahead (Optimizing Value and Performance) A. What is Motivation B. What is Employee Involvement 3 Leading Characteristics (Management Styles/Organizational Behavior) A. Understanding the Sticks and Carrots Read MoreHow Leaders Motivate Employees And Organizations831 Words   |  4 Pageslike to address the problem of how leaders motivate employees in organizations. This is a general topic which can have a wide range of applicability (and that’s part of its appeal to me). But the problem of employee motivation, and how leaders can stimulate and/or make use of that motivation, is critical to the success of any organization. A leader can have the perfect plan of action and yet fall short of meeting their objective simply because they did not know how to generate momentum among theirRead MoreFrederick Herzberg1492 Words   |  6 Pagesarticle written by Frederick Herzberg himself are covered in this paper. Mr. Herzberg’s theory of management focuses on one area mainly. The area of focus deals with job satisfaction and everything that leads to job satisfaction. Unlike my previous papers, this paper will focus on one main subject. I will try to explain in detail the Herzberg theory. â€Å"Introduction† The Herzberg theory is the subject of this paper. The purpose of this paper is to explain Mr. Herzberg’s management theory. 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The results indicate that skill-enhancing practices w ere more positively related to human capital and less positively related to employee motivation than motivation-enhancing practices and opportunity-enhancing practices. Moreover, the three dimensions of HR systems were related to financial outcomes both directly and indirectly by influencing human capital and employee motivation as well as voluntaryRead MoreOrganisational Culture and Motivation1496 Words   |  6 PagesReading INTRODUCTION AND RATIONALE As Desson and Clouthier (2010) state, culture is an important factor in both attracting and retaining desirable employees. The extent to which an employee’s needs and expectations are fulfilled will determine the motivation, job satisfaction and performance levels (Mullins, 2005, p. 499) which would be influenced by culture. 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Eleven Case and the Industrial Relations System

Question: Discuss about the Eleven Case and the Industrial Relations System. Answer: Introduction: 7-Eleven, the largest convenience store chain of Australia has been under the scanner for over seven years. The reason for this being the alleged systematic underpayment of wages to its employees mostly international students. Investigations have also given indications of rosters and time sheets being doctored, store financials having wage bills that are understated and explosive documents in relation to payroll compliance (Hobday, 2016). The store reviews have provided further evidence of deep rooted rot within the Australian empire of 7-Eleven (The Sydney Morning Herald, 2015). This essay will discuss a brief summary of the report that was released by Fair Work Ombudsman (FWO) in April, 2016 on the findings of the inquiry that it carried out on the franchisees of 7-Eleven in view of the above allegations of systematic non-compliance with federal workplace laws. Besides this, it will also explain the failings or gaps that are present in the industrial relations system which enabled the convenience store to underpay its employees. Finally, recommendations will be provided which will help in protecting the rights of the employees so that they are prevented from being exploited on the same lines in future. Brief Summary of the 7 Eleven Inquiry The Fair Work Ombudsman carried out an inquiry into the franchise network of 7 Eleven. This was prompted by allegations from the employees of 7-Eleven regarding the systematic non-compliance with the federal laws in relation to the workplace. It had also come to light that several franchisees were falsifying the records in a deliberate move for disguising the underpayment of wages (Australian Government, 2016). The FW Act permits inquiry into any practice or act which is in contradiction to the Act, an instrument of fair work or safety net contractual entitlement (Australian Government, 2015). Thus, the purpose of such an inquiry by FWO was identifying if the allegations had a basis within the store network of 7-Eleven, what factors were the driving forces for non-compliant behaviour, were these factors being disguised and in case yes, then how, how was it possible to expose them and who was responsible for it. The inquiry looked to access the role as well as the involvement of the companys head office and if the franchise operating model was in itself a contributing factor to the unlawful behaviour that certain franchisees demonstrated. The aim of the inquiry was also to find out if the workers themselves turned out to be in participants in the non-compliance either inadvertently or reluctantly. The inquiry for structured in a way that it included several major activity components particul arly investigations on the basis of the assistance requests from the individual employees, testing allegations that were anonymous, engaging with 7-Eleven and forensic auditing of a sample of the stores in an in-depth manner (Australian Government, 2016). Based on the continuous concerns of the employees as well as intelligence the strategic inquiry by FWO started in a sample of 20 stores of 7-Eleven throughout the provinces of Queensland, NSW and Victoria in June, 2014. The aim of the inquiry was to test the allegations by means of site inspections in a coordinated manner followed by an analysis of record keeping of a sample of 20 stores of the company. Several in-depth investigations were also undertaken which revealed concerning levels of non-compliance with both the Fair Work Regulations 2009 and the Fair Work Act 2009 and many instances of records being deliberately manipulated for disguising underpayment of wages. Three of the five stores were found to be underpaying their staff (Australian Government, 2016). The investigations of sample stores and other stores which were investigated as a result of wider inquiry led to several enforcement action comprising filing of seven matters in the Federal Circuit Court, issuance of 20 caution letters, one enforceable undertaking, issuance of 3 notices of compliance and 14 notices of infringement along with the recovery of more than $293, 500 for the employees (Australian Government, 2016). It was found out by the inquiry that the approach of 7-Eleven to the matters of the workplace promoted compliance ostensibly but it also did not detect in an adequate manner or address the issue of deliberate non-compliance. Consequently, it compounded the matter. In certain specific instances, misleading and false records were created by the franchisees for satisfying the auditing as well as payroll regime of 7-Eleven while they continued to underpay their workers (Briton, 2015). Despite the presence of such signs, 7-Eleven did not make any substantial alterations in either its store review processes or its payroll system for targeting the risks related to incorrect record-keeping. The inquiry also suggested that the records or store practices were not interrogated sufficiently by the payroll section for uncovering the signs of non-compliance where they were hidden by the franchisee even though a reasonable basis was present before 7-Eleven to conduct inquiry and take action (Austra lian Government, 2016). The Failings (Gaps) are in the Industrial Relations System that Enabled 7-Eleven to Underpay its Workers The formal allegations related to non-compliance led to the identification of certain themes as well as practices related to gaps in the system of industrial relations which led to the workers being underpaid. The workforce of 7-Eleven largely consisted of international students from the backgrounds that were non-English speaking. Although the international student visas permit them to work for 40 hours in a fortnight, several of the students were working in excess of this. They are paid correctly on paper but half their pay is taken back by blackmailing them, indicating the existence of a cash-back scheme (Cox, 2016). Thus, they were employed in breach of their visa conditions and threatened with deportation in case they complained. Their wage records are falsified to show that they were working as per the requirements of the visa. A common feature of these kinds of allegations was the training periods that were unpaid and ranged from one shift to 14 days of work(Nunweek, 2015). It thus came to light that the intersection between the visa framework as well as the workplace relations system was apparently multiplying the vulnerable situation in which the workers found themselves. The international students in Australia are permitted to work for only 20 hours in a week but they were forced to work for 40 hours by the franchisees and then paid only for 20 hours. Therefore, if their base wage per hour is $24, they effectively receive only $12 (Branley, 2015). As per the law in Australia, the minimum wages cannot be undercut by the employers, even though the employees might make an offer of accepting the low rates. The employees also have to maintain at all times, accurate records related to time-and-wages. There are minimum pay rates that apply to everyone in Australia and are not negotiable. The employer cannot take advantage of any worker especially an overseas employee who may have limited knowledge of English and their rights of the workplace but the student s were threatened that if they tried to complaint they will end up losing their visas or being deported due to the breach of the conditions in their own visas (Nunweek, 2015). This way the franchisees took advantage of the gaps and exploited the vulnerability of these students by catching them in a vicious trap. Recommendations Recommendations which will protect the rights of employees from being exploited along similar lines in the future should first of all recognise the need of balancing the range of regulatory as well as policy settings that every framework is looking to address. In light of this, it is recommended that companies implement arrangements for effective governance which ensure that all the federal workplace laws are complied with. The companies having franchisees can also establish a staff consultative forum which can have representatives of the employees from the different parts of the network and this should be separate from the franchisees. These companies should accept their ethical as well as moral responsibility to make sure that their stores meet the expectations of the society and the community for fair, safe and equal work opportunities for every employee. The corporates like 7-Eleven also need to review their operating models for ensuring periodic reviews of their financial viabil ity and also legal exposure with respect to their agreements of franchising. An independent external party may be engaged for self-auditing their compliance. Steps should be taken for improving the employment related practices of the franchisees by implementation of changes that are not only permanent but also sustainable for a franchise model for ensuring that workplace relations law like the Fair Work Act, 2009 and other instruments related to it are complied with in full for every employee in all the franchisees. Conclusion The FWO inquiry was conducted particularly for figuring out if 7-Eleven had a part to play in the alleged employment record falsification and wage underpayment by the franchisees. It also looked for identifying and addressing the non-compliance drivers, the motivations along with the actions of the workplace participants. Its aim was to understand in a better way the roles of 7-Eleven, its franchises as well as its employees respectively in the culture and operating model of the network. It found that 7-Eleven did not take steps for rectifying this incorrect act of the franchisees despite having the sufficient reason to do so. Advantages were taken of the gaps in law by employing vulnerable international students in breach of their visa conditions. They were underpaid, exploited and threatened with deportation in case they complained. For avoiding such kind of exploitation in future and protecting the employee rights, a sustained effort is needed for which resources have to be alloca ted for a longer time period. A sustainable behaviour change has to be driven top down and the culture of compliance has to be rebuilt by a strong leadership as false record-keeping has apparently been ingrained within each and every aspect of the network. References Australian Government. (2016). A Report of the Fair Work Ombudmans Inquiry into 7- Eleven: Identifying and addressing the drivers of non-compliance in the 7-Eleven network . Australia: Australian Government. Australian Government. (2015 ). Fair Work Handbook. Retrieved August 31, 2016, from Fairwork Ombudsman: https://www.fairwork.gov.au/ArticleDocuments/712/Fair-Work-Handbook.pdf.aspx Australian Government. (2016). Fair Work Obudsman: Statement of 7-Eleven. Retrieved August 31, 2016, from Australian Government: https://www.fairwork.gov.au/about-us/news-and-media-releases/2016-media-releases/april-2016/20160409-7-eleven-presser Branley, A. (2015 ). 7-Eleven staff work twice as long at half pay rate, investigation reveals. ABC News . Briton, B. (2015 ). 7-Eleven scandal: The tip of a low-wage iceberg. Guardian (Sydney) . Cox, D. (2016 ). 7-Eleven wage scam: Union says it has evidence cash-back scheme is 'still alive and kicking'. ABC News . Hobday, L. (2016 ). 7-Eleven wage underpayment claims taking too long: Allan Fels. ABC News . Nunweek, J. (2015 ). Lessons from 7-Elevens scam. Overland Literary Journal . The Sydney Morning Herald. (2015). How 7 Eleven is Ripping off its workers.. The Sydney Morning Herald .