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

Section 2 **Finite differences for second order linear differential
equations**

Let *x*_{0} = *a*, *x*_{n} = *b, x*_{i}
= *x*_{0} + *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 *x*_{i }by
*y*_{i}, *y'*_{i}, *y"*_{i}. Next, replace the
derivatives *y*_{i}'(x), *y*_{i}"(*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*(*x*_{i}) 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 *x*_{i} = 0.1? *i* +
1, *i* = 1, 2, 3. Expansion of (2.5) yields the system

(2.6)

At the boundary point *x*_{0} = 1 and *x*_{4 }= 1.4:

Hence (2.6) has the solution

The exact solution *y *= (ln*x)*^{2}*/2 *of (2.4) at these points has the values