Page 467 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation Richa Nandra, Lovely Professional University

Notes Unit 31: Fredholm Equations with
Poincere Goursat Kernels

CONTENTS

Objectives
Introduction

31.1 The Poincere Goursat Kernels
31.2 Resolvent Kernel H(x, u, )
31.3 Eigenvalues and Eigenfunctions

31.4 Summary
31.5 Keywords

31.6 Review Question
31.7 Further Readings

Objectives

After studying this unit, you should be able to:

 Know that Fredholm equations may have varieties of Kernels. Among them the Poincere-
Goursat Kernel also plays an important part.
 Observe that in this type of Fredholm equation the resolvent Kernel is a quotient of two
polynomials of the nth degree in and the denominator is independent of the variables of
the Kernel.

 Understand the nature of singular points of resolvent Kernel in terms of zeros of the
denominator polynomial D( ).

Introduction

In this unit we saw that resolvent Kernel has a structure that helps in understanding the nature
of the solution of non-homogeneous as well as homogeneous equations.
Fredholm integral equation as well as its conjugate equation can be studied together to understand
the structure of the solutions.

31.1 The Poincere Goursat Kernels

In the unit we consider again the Fredholm integral equation of the second kind i.e.
1
y
u
x
x
x
y ( ) K ( , ) ( )du f ( ) ...(1)
u
0
Here we take the structure of the Kernel to be of the form
n
( ) ( ) u
K x g x h i ...(2)
( , ) u
i
i i
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