Page 452 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Sachin Kaushal, Lovely Professional University Unit 28: Integral Equations

Unit 28: Integral Equations Notes

CONTENTS
Objectives
Introduction

28.1 Fredholm Equations
28.2 Types of Kernels
28.3 Methods of Solving Fredholm Integral Equations

28.4 Description of Some Methods used in the solution of Fredholm Integral Equation
28.5 Summary
28.6 Keywords
28.7 Review Question
28.8 Further Readings

Objectives

After studying this unit, you should be able to:

 Classify the type of Fredholm integral equations.
 Classify the Kernel of any integral equation i.e. is it symmetric or Poincere Goursat type
or of different type?
 Choose the right method of solving the integral equation.

Introduction

You have learnt in the previous few units the Volterra integral equation of the second and first
kind.
You will find similarities and differences in approach between the two types of integral equations.

28.1 Fredholm Equations

In the last three units we studied one type of integral equation known as Volterra integral
equation. In the next few units we are interested in studying an other integral equation known
as Fredholm integral equation.

In the case of Volterra integral equation we saw that linear differential equations with initial
condition lead us to Volterra integral equation. In the case of boundary value problem, the
differential equations can be converted into Fredholm integral equation.

Now the Fredholm equations can be of the form

b
x
Q ( ) = f ( ) K ( , ) ( )dt ...(1)
x
x
t
Q
t
a
b
t
Q
or f ( ) = K ( , ) ( )dt ...(2)
x
t
x
a
LOVELY PROFESSIONAL UNIVERSITY 445