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

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 *x*_{0} = *a, x*_{k} = *x*_{0} + *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 *y*_{k} , *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 *f*_{i}^{(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 g_{ik}. 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 *y*_{0} = *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* x*_{k
}= 0.1?*k *and the corresponding values of *y*_{k}^{(0)} = *x*_{k}(*x*_{k}.-
1), *k* =1, 2, . . . , 9, and compute *f*_{k}^{(0)}" = 2 + [ *y*_{k}^{(0}]^{2}.

3. **Computation of the first approximation **The values *y*_{k}^{(1)}
are given by (4.10) for *r* = 0:

For *k* = 1,

etc.

4. **Computing the second approximation** The *y*_{k}^{(2)}
are found from (4.10) for *r* = 1:

In particular, for *k* = 1:

, etc.

A comparison of the values of *y*_{k}^{(1)} and *y*_{k}^{(2)}
show a difference in the last digit, so that one can write *y*_{k }?
*y*_{k}^{(2)}.