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

**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 *a*_{i}(*x*) and *b*_{i}(*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 *c*_{i}. 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 d**egenerate kerne**l ([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: