Page 445 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes or
x
d h ( )
u
x
x
g
g 1 ( )g 2 ( ) h 1 ( )h 2 ( ) Z ( ) h 1 ( ) ( ) 2 Z ( )du = f ( )g 2 ( ).
x
x
x
x
x
x
u
x
2
u
dx g ( )
0 2
Simplifying equation (6) we have
x f ( )g ( )
x
x
x
u
h 1 ( )g 2 ( ) d h ( ) 2
x
Z ( ) 2 Z ( )du = ...(7)
u
x
x
x
x
x
x
u
x
x
x
g ( )g ( ) h ( )h ( ) du g ( ) g 1 ( )g 2 ( ) h 1 ( )h 2 ( )
0 1 2 1 2 2
So equation (7) is Volterra equation of the second kind. Putting
x
x
h ( )g ( ) d h ( )
u
K N ( , ) = 1 2 2
x
u
[ ( )g 2 ( ) h 1 ( )h 2 ( )] du g 2 ( )
u
a
x
x
g
x
1
x
x
f ( )g ( )
and f N ( ) = 2
x
x
g ( )g ( ) h ( )h ( )
x
x
x
1 2 1 2
equation (7) can be put into the form
x
x
Z ( ) K N ( , ) ( )du = f N ( ) ...(8)
x
x
u
u
Z
0
x
x
x
Knowing g 1 ( ), g 2 ( ), ( ),h 2 ( ) and f(x) we can then solve equation (8) by the methods of the
h
x
1
last unit.
Self Assessment
1. Solve the integral equation
x
x
x
y ( ) f ( ) e 2(x u ) y ( )du
u
0
26.3 Summary
 Volterra integral equation of the first kind may have a number of different kinds of
Kernels.
 It is sometimes useful to convert Volterra integral equation into Volterra integral equation
of the second kind.
 By converting Volterra integral equation into that of second order the method of solving
the Volterra integral equation of second kind may be employed.
26.4 Keyword
Volterra integral equation of the first kind is related to Volterra integral equation of the second
kind and the solution of Volterra integral equation of the first kind can be found by the methods
already used.
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