Abstract
A variation of the direct Taylor expansion algorithm is suggested and applied to several linear and nonlinear differential equations of interest in physics and engineering, and the results are compared with those obtained from other algorithms. It is shown that the suggested algorithm competes strongly with other existing algorithms, both in accuracy and ease of application, while demanding a shorter computation time.
Highlights
IntroductionWith the advent of high speed personal computers and workstations and the decrease of the cost of computer resources in general, numerical methods and computer simulation have become an integral part of the scientific method and a third approach to the study of physical problems, in addition to theoretical and experimental methods
Open AccessWith the advent of high speed personal computers and workstations and the decrease of the cost of computer resources in general, numerical methods and computer simulation have become an integral part of the scientific method and a third approach to the study of physical problems, in addition to theoretical and experimental methods.The problem of solving differential equations numerically had long been of interest to mathematicians and scientists alike, long before the appearance of modern computers
To increase the order of the algorithm from four to six, one only needs to retain two additional terms in the Taylor expansion. Such an increase of the order of algorithm is not a trivial task in the RK (Runge-Kutta), ABM (Adams-Bashforth-Moulton), or Milne method. While generalizations of the latter methods to higher-order differential equations are not trivial, in the former case, it can be accomplished by incorporating Taylor expansions for higher derivatives, as we shall demonstrate in the following paragraph
Summary
With the advent of high speed personal computers and workstations and the decrease of the cost of computer resources in general, numerical methods and computer simulation have become an integral part of the scientific method and a third approach to the study of physical problems, in addition to theoretical and experimental methods. An improved version of the Euler method is obtained by retaining three terms in the Taylor expansion of the function instead of two, yielding a second order algorithm [6]. One can advance the solution from xn to xn+1 by using the value of the derivative at the midpoint of the interval rather that at the beginning, yn+1 This algorithm is known as the modified Euler method, the midpoint method, or the second-order Runge-Kutta method [1] [2] [7]. One can extend the Taylor algorithm as follows: From Equation (10) and its differentiation, the second and higher derivatives of the function are obtained and evaluated at x0. The function and its derivatives are Taylor expanded and advanced from subinterval to subinterval until the function is evaluated at the required value of the variable x , resulting in a much greater accuracy
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