Page 462 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Richa Nandra, Lovely Professional University Unit 30: Neumann s Series

Unit 30: Neumann s Series Notes

CONTENTS
Objectives

Introduction
30.1 Fredholm Integral Equations, Successive Approximation Neumann s Series
30.2 Successive Approximation for the Resolvent Kernel

30.3 Summary
30.4 Keywords

30.5 Review Question
30.6 Further Readings

Objectives

After studying this unit, you should be able to:

 Find lots of similarities of the description of the successive approximation approach in
regard to getting Neumann Series.
 Observe that the unknown function can either be expanded in power series of or the
resolvent Kernel is expanded in power series in .
 Understand the convergence of the Neumann Series as given in unit 29.

Introduction

For small values of the solution of the Fredholm equation can be determined as power series
known as Neumann s Series.
The resolvent kernel is an analytic function of the parameter but it is not an entire function of
the whole complex plane.

30.1 Fredholm Integral Equations, Successive Approximation
Neumann s Series

Consider the Fredholm integral equations of the first kind and second kind:

b
t
y
x
x
f ( ) K ( , ) ( )dt ...(1)
t
a
and
b
t
x
y
t
x
y ( ) f ( ) K ( , ) ( )dt ...(2)
x
a
In these equations y(x) is an unknown function that has to be found and f(x) and K(x, t) are given
as function and the Kernel of the integral equations. Unless in the case of Volterra integral
equation, here the limits of the integral are fixed as constants a and b. The range of x and t are
given as a x b and a t b. Depending upon the nature of Kernel K(x, t) a suitable method of
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