Page 454 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 28: Integral Equations

reduced to an algebraic system of linear equations. If the integral of the given integral Notes
equation is replaced by a suitable sum then instead of dividing the basic integral into sub-
intervals of the same size, it may be useful to divide it according to the zeros of a certain
polynomial of Legendre. This method is developed in most books on numerical methods.

28.4 Description of Some Methods used in the solution of Fredholm

Integral Equation

In the next few units we are interested in studying the Fredholm integral equations. In the unit
29, we study the Fredholm equations by the method of successive approximation. In this iteration
method either the unknown function or the Kernel is iterated into a series known as Neumann’s
series. The convergence of the series depends upon the iterative parameter and the nature of the
Kernel as well as the function in the domain a x b, a t b when the Kernel K(x, t) and the
function f(x) are square integrable. For this purpose the function as well as Kernel has continuous
derivations. Then in the unit 31 we will study the solution of Fredholm equations with a special
type of Kernels known as Poincere Goursat Kernels (P.G.). In the light of P.G. Kernels the
existence and uniqueness of the solution of Fredholm equations of both kinds. In the unit 32 the
final unit the famous Fredholm theorem on the existence and uniqueness of the solutions is
described along with the conditions put on the functions.

Example: Express the differential equation
2
d y 9y = 18x y (0) y 0 ...(1)
dx 2 2

as an integral equation.
Solution: Integrating (1) from 0 to x,

x 2 x x
d y dx 9 y ( )du = 18xdx
u
dx 2
0 0 0
x
dy dy
u
or 9 y ( )du = 9x 2
dx dx x 0
0
x
u
or y ( ) y (0) 9 y ( )du = 9x 2
x
0
Again integrating
x t 3
x 9x
u
x
y ( ) y (0)x 9 dt y ( )du =
0 3
0 0
or
x x
y ( ) y (0) y (0)x 9 y ( )du dt = 3x 3
x
u
0 0

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