Page 431 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes b
x
u
y
or K ( , ) ( ) f ( ) ...(2)
u
x
a
In these cases the limits of integrations are fixed by some constants and the unknown variable
appears inside the integral sign. These equations are known as Fredholm integral equations of
the second kind (1) and the first kind (2) respectively.
We can also have integral equations of the following type.
x
u
u
y
y ( ) f ( ) K ( , ) ( )du ...(3)
x
x
x
a
x
x
x
u
or K ( , ) ( )du f ( ) ...(4)
y
u
a
In the equations (3) and (4) the limits of integration depends on the independent variable.
Equations (3) and (4) are known as Volterra integral equations of the second kind and the first
kind respectively.
We can take up the various types of integral equations and study them and devise methods of
solving them. The solution of the integral equation is based on the properties of the Kernels
K (u, x) as well as the function f(x).
In this unit we concentrate on the Volterra integral equations and in particular see how the
solution of the Volterra integral equations are carried out along with the discussion of the
L -Kernel.
2
25.2 Volterra Integral Equations

In the previous unit we had seen some difficulties in the solutions of the integral equation by
converting them into an algebraic system of equations. It is seem there that when dealing with
integral equation of the first kind we find the mean values of the function in the successive
1 1 2
intervals 0, , , .... and so therefore the equation (2) of that section will possess infinite
n n n
many solutions.
To avoid these difficulties, Vito Volterra investigated the solution of the integral equations in
which the Kernel satisfies the conditions
K(x, y) = 0 if u > x ...(1)
This corresponds (in the sense of the previous unit) to the simple case of a system of algebraic
linear equations where the elements of the determinant above the main diagonal are all zero.
We rewrite the integral equations of Volterra type of the second kind and first kind as follows:
t
x
y
u
x
u
x
y ( ) K ( , ) ( )du f ( ) ...(2)
0
t
y
and K ( , ) ( )du f ( ) ...(3)
x
u
x
u
0
In this section we shall study the Volterra integral equation of the second kind (2) that we can
readily solve by Picard’s process of successive approximation as discussed in unit 6. We state by
setting y (x) = f(x) and then determine y (x):
0 1
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