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

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

Section 2 Finite differences for second order linear differential equations

Let x0 = a, xn = b, xi = x0 + ih. i = l, 2, . . ., n - 1, h = (b - a)/n be a system of equidistant points and

Denote the approximate values of y(x) and its derivatives at the point xi by yi, y'i, y"i. Next, replace the derivatives yi'(x), yi"(x) by the divided finite differences

(2.1)

in order to arrive at the system of simultaneous equations

(2.2)

If n is large, a direct solution of the system (2.2) is rather difficult; Section III presents a simple method, developed especially for systems of this kind. An error estimate of the method of finite differences for Problem (1.3) and (1.4) has the form

where y(xi) is the exact solution and

The accuracy of the difference method can be increased considerably by employment of multi-point difference schemes for the derivatives ([39].)

Practical problems frequently encounter equations, in which the functions p(x), q(x) and f(x) are given by tables with spacing h. It is only natural to solve such equations by the difference method with the given spacing).

Example 1: Use finite differences to solve the boundary value problem

(2.4)

Solution: Apply (2.1) to (2.4) to arrive at the finite difference equation

(2.4)

An obvious rearrangement yields the system

(2.5)

With h = 0.1, one obtains the three interval points xi = 0.1? i + 1, i = 1, 2, 3. Expansion of (2.5) yields the system

(2.6)

At the boundary point x0 = 1 and x4 = 1.4:

Hence (2.6) has the solution

The exact solution y = (lnx)2/2 of (2.4) at these points has the values