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Sachin Kaushal, Lovely Professional University Unit 25: Volterra Equations and L Kernels and Functions
2

Unit 25: Volterra Equations and L 2 Notes
Kernels and Functions

CONTENTS
Objectives

Introduction
25.1 Classification of Integral Equations

25.2 Volterra Integral Equations
25.3 L Kernels and Functions
2
25.4 Solution of Volterra Integral Equation of Second Kind

25.5 Summary
25.6 Keywords

25.7 Review Questions
25.8 Further Readings

Objectives

After studying this unit, you should be able to:

 Know that integral equations can be of Volterra type equations of first or second kind or
they can be Fredholm type of first or second kind.
 See that in the case of Volterra integral equations the upper limit depends upon the
independent variable while in the case of Fredholm integral equations the limits are
fixed.

 Understand that there are certain conditions on the Kernels as well on the functions for the
existence of the solution. Here it is seen that the Kernels as well as the functions are L class
2
and so the solution does exist.
Introduction

L class Kernels as well as functions are square integrable. So if the iteration procedure is applied
2
one can see that product of two L class Kernels is also L -class.
2 2
This method enables us to find the resolvent Kernels by L -class method and the solution of the
2
integral equation is obtainable.
25.1 Classification of Integral Equations

In the last unit we studied the integral equations by converting a differential equation with
boundary conditions or initial conditions. We see that the boundary conditions lead us to
integral equations of the type

b
( , ) ( ) u du
y ( ) x f ( ) x K x u y ...(1)
a

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