Numerical Analysis Using MATLAB and Spreadsheets
This concise text provides complete, clear, and detailed explanations of the principal numerical analysis methods and well known functions used in science and engineering.
TABLE OF CONTENTS Numerical Analysis Using MATLAB and Spreadsheets
Chapter 9: Solution of Differential Equations by Numerical Methods
This chapter is an introduction to several methods that can be used to obtain approximate solutions of differential equations. Such approximations are necessary when no exact solution can be found. The Taylor, Runge-Kutta, Adams', and Milne's methods are discussed.
9.1 Taylor Series Method
We recall from Chapter 6 that the Taylor series expansion about point a is
Now, if x 1 > a is a value close to a, we can find the approximate value y 1 of f( x 1) by using the first k + 1 terms in the Taylor expansion of f( x 1) about x = a. Letting h 1 = x ? a in (9.1), we get:
Obviously, to minimize the error f( x 1) ? y 1 we need to keep h 1 sufficiently small.
For another value x 2 > x 1, close to x 1, we repeat the procedure with h 2 = x 2 ? x 1 ; then,
In general,
Example 9.1
Use the Taylor series method to obtain a solution of
correct to four decimal places for values x 0 = 0.0, x 1 = 0.1, x 2 = 0.2, x 3 = 0.3, x 4 = 0.4, and x 5 = 0.5 with the initial condition y( 0) = 1
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