Page 453 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes Here K(x, t) the Kernel and f(x) the function are known and Q(x) is an unknown function on the
interval a x b.
Let (x) be a function which satisfies the Fredholm integral equation

b
t
Q
t
x
( ) = g ( ) K ( , ) ( )dx ...(3)
t
x
a
Here K ( , ) = K(x, t)
t
x
28.2 Types of Kernels
Just like in Volterra integral equation in the case of Fredholm integral equations are a variety of
Kernels as follows:
1. Symmetric Kernels: Kernels having properties
as K(x, t) = K(t, x)
are called symmetric Kernels.
2. Degenerate Kernels or Poincere Goursat type of Kernels. The Kernels of the type

n
K(x, t) = g i ( ) h i ( )
t
x
i 1
These Kernels play an important part in the development of Fredholm theory of integral
equation like the eigenvalue and eigenfunction problems.
3. Difference Kernels: The Kernels of the type

K(x, t) = K(x – t)
are known as difference Kernels. These types of Kernels do arise while converting a
differential equation with boundary conditions.
The conditions on Kernels are that they should be continuous and its partial derivatives should
be continuous. Also they should be square integrable.

28.3 Methods of Solving Fredholm Integral Equations

There are various methods of solving integral equations which can briefly summarized as follows:
(a) We can reduce integral equation to a differential equation which can be solved easily.

(b) The Fredholm integral equations can be solved by transform method. In this method the
Laplace transformation helps in writing an integral equation into an algebraic equation
and then by inverse Laplace transformation get the final solution.
(c) The Iteration Method: The most important method of solving the Fredholm integral
equation is the iterative method. In this method the unknown function is expanded in
powers of the iterated parameter. This series is known as Neumann series. There is an
other alternate approach in which the Kernels are iterated up to nth times and then solved
the integral equations. The famous iterative method are that of Picard’s methods or by
using the idea of L class Kernels in the iterative approaches.
2
(d) Numerical Methods: Sometimes the Kernel of the Fredholm equations is approximated by
a suitable Poincere Goursat Kernel on step functions, then the integral equations can be

446 LOVELY PROFESSIONAL UNIVERSITY