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Sachin Kaushal, Lovely Professional University Unit 26: Volterra Integral Equation of the First Kind

Unit 26: Volterra Integral Equation of the First Kind Notes

CONTENTS
Objectives
Introduction

26.1 Volterra Equations of First Kind, function and Kernel Classes
26.2 Reduction of Volterra Equations of the First Kind to Volterra Equations of the
Second Kind

26.3 Summary
26.4 Keyword
26.5 Review Question
26.6 Further Readings

Objectives

After studying this unit, you should be able to:
 Know various types of Kernels and how they help in solving the integral equations.
 Understand that it is difficult to solve Volterra integral equation of the first kind. It can
first be converted to Volterra integral equation of the second kind and the methods discussed
earlier in units can be employed to solve it.

Introduction

Volterra integral equations of the first kind is by suitable method converted into Volterra
integral equations to solve it by suitable method.
The resolvent Kernel can be found easily in the case of Volterra integral equation of the second
kind.

26.1 Volterra Equations of First Kind, function and Kernel Classes

In this unit we present methods for solving Volterra linear equations of the first kind which
have the form
x
y
u
K ( , ) ( )du = f(x) ...(1)
u
x
0
Here y(x) is unknown function on the interval a x b K(x, u) is the Kernel of the equation and
f(x) is a given known function. The functions y(x), f(x) are usually assumed to be continuous or
square integrable on (a, b). The Kernel K(x, u) is assumed to either continuous or the square
a x b, a u b or it satisfies the condition
b b
2
u
x
K 2 ( , )dxdu = N <
a a
i.e. K(x, u) is of class L . Also K(x, u) = 0 for u > x.
2

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