Page 427 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes Thus y(x) is given by the integral equation
1
u
x
y
u
x
y(x) = K ( , ) ( )du ( ) ...(7)
0
Self Assessment
2. Express
2y ( ) 3 ( ) 2 ( ) 4e t 2cos t with initial conditions y (0) 4, (0) 1, into
y
x
y
x
y
x
integral equation.
(Hint: Integrate the differential equation twice and use initial conditions.)
24.4 Relation between Integral Equations and Algebraic System of
Linear Equations
Consider the general linear Fredholm integral of the second kind for a function (x) of the type

1
x
( ) K ( , )dy = f ( ) (0 x 1) ...(1)
x
y
x
0
and the linear Fredholm equation of the first kind is given by
1
x
y
y
x
K ( , ) ( )dy = f ( ) (0 x 1) ...(2)
0
The problem of solving (1) and (2) can be considered as a generalization of the problem of
solving a set of n linear algebraic equations in n unknown:
n
a x = b . (r = 1, 2, ...n) ...(3)
rs s
r
s 1
For this purpose we divide the interval (0 x 1) into n segments and define
K(x, y) = K (r, s = 1, 2, ... n)
rs
and f(x) = f
r
Here, x, y are divided into strips as ...(4)

r 1 r
< x (r = 1, 2, ... n)
n n
s 1 s
< y (s = 1, 2, 3, ... n)
n n
Then equation (1) becomes

n s /n
y
x
( ) = f r K rs ( )dy r 1 x r ...(5)
s 1 (s 1)/n n n
Equation (5) shows that if a function ( )x exists it must be a step function, i.e.

420 LOVELY PROFESSIONAL UNIVERSITY