10.9 Section 2 Background .1 Cosmology .11 Math Math 2.1.11

Book 9 Computational Mathematics Chapter 10 Partial Differential Equations

Section 10 Replacement of a kernel by a degenerate kernel

Consider the Fredolm equation of the second kind

(10.1)

The kernel K(x,s) is said to be degenerate if it can be represented in the form

(10.2)

where the functions ai(x) and bi(s), i = 1, 2, . . , n are linearly independent on the interval [a,b].

The proposed method ([2], [13], [17]) is based on the fact that then (10.1) with a degenerate kernel can be solved exactly. Replace the degenerate kernel K(x,s) by the degenerate kernel

(10.3)

and look for an approximate solution of (10.1) in the form

(10.4)

where

(10.5)

Substitution of (10.4) into (10.5) yields

In the notation

(10.6)

one arrives at

(10.7)

a system of linear equations in ci. Write the approximate solution of (10.1) in the form (10.4). Use a section of the Taylor or Fourier series for the function K(x,s) as a degenerate kernel ([2], [17]).

Example 1: Solve approximately the equation

(10.8)

Solution: Replace the kernel K(x, s) = sinh (x, s) by the sum of the first three terms of its Taylor expansion

and seek the solution of (10.8) in the form

Letting

find the coefficients of the system (10.7) from (10.6):

Thus,

Applying iteration to this system, one finds

Hence, an approximate solution of (10.8) is

Exercises

Find approximate solutions of the following integral equations be replacement of their kernels by the first three terms of their Taylor's series: