9.3 Section 2 Background .1 Cosmology .11 Math Section 9.5

Book 9 Computational Mathematics Chapter 9 Ordinary Differential Equation Initial Value Problems

Section 4 Finite differences for second order non-linear D.E.s

Consider the non-linear differential equation

with the linear boundary conditions

Use equidistant points x0 = a, xk = x0 + kh, k =1, 2, . ., n - 1, h = (b - a)/n to replace Equation (4.1) and the boundarv conditions (4.2) by the non-linear system of n + 1 equations in the n + 1 unknowns yk , k = 0, 1, . . . , n

(4.3)

Let

Use iteration with r denoting the order of the approximation:

(4.5)

One linear system must be solved for each value of interval h. Exploiting the special form of the system, obtain its explicit solution (cf. [2])

(4.6)

where

are known and

(.4.7)

(4.8)

Note that on the right hand side of (4.6) only the fi(r) depend on r. Thus, the solution of (4.3) has been reduced to a rather simple scheme. The convergence of this process has been studied in [2].

Example 1: Using finite clifferences, find to an accuracy of 0.5?10-3 the solution of the boundary value problecn

Solution: Letting n = 10, h = 0.1, one finds

Equations (4.6) - (4.8) yield

1. Computation of the coefficients gik. By (4.11), for i = 1,

Hence

Again (4.11) yields for i = 2

whence

, etc.

All the results are displayed in the table:

2. Choice of the initial approximation. Select the function y0 = x(x - 1), the solution of the differential equation y" = 2 which satisfies the boundarv conditions y(0) = y(1) =. 0. Enter into the next table the values of xk = 0.1?k and the corresponding values of yk(0) = xk(xk.- 1), k =1, 2, . . . , 9, and compute fk(0)" = 2 + [ yk(0]2.

3. Computation of the first approximation The values yk(1) are given by (4.10) for r = 0:

For k = 1,

etc.

4. Computing the second approximation The yk(2) are found from (4.10) for r = 1:

In particular, for k = 1:

, etc.

A comparison of the values of yk(1) and yk(2) show a difference in the last digit, so that one can write yk ? yk(2).