APPLICATIONS OF INTEGRAL TRANSFORMS | MARTINS LIBRARY

Abstract In this seminar work we study the properties of integral transforms, and their application to the solution of differential equations. In particular we consider Laplace and Fourier Transform.

Table of contents

Title page

Certification

Approval

Dedication

Acknowledgement

Abstract

Table of contents

Chapter one

Introduction

Chapter two

Literature review

Chapter three

Research methodology 

1.0 Introduction

Integral transforms are fundamental to the solution of differential (ordinary and partial) as well as integral equations. They sometimes provide a means of obtaining closed form solutions to some nonlinear equations. In this seminar work we examine the properties of Laplace and Fourier transforms and their applications to the solution of differential equations.

1.1 Definition (Integral Transform)

Given a known function K(a,x) of two variables a and x such that the integral

(1)

is convergent, then the integral (1) is termed as the integral transform of the function

/(x) denoted /(:e)or T{f(x)}, i.e.

f(x) = T{f(x)}=[K(a,x)f(x)dx                            (2)

The function K(a,x) is known as the Kernel of the transformation and a is a parameter (real or complex) independent of x.

1.2 Example: Suppose we take the Kernel,

K(a,x) = K(s,t) = 0   for  t<0

= e^-st for t>0 Then the transform

is known as the Laplace Transform.

1.3 Definition (Laplace Transform)

Let /(Obe a function defined for all positive values oft, then the Laplace transform

of f(t) denoted by

Where s is a parameter.

1.4  Remark: if  the integral e^sts f(f)dt converges for some value of s, then the Laplace transform is said to exist, otherwise it does not exist, which gives inverting the transform

g(a)= Ae

-iwo 3/3

(A = arbitrary constant)

Thus

                    dw        w³

Y(x) =   A     2π       exp I  wx- 3

References

[1] John B. Conway. A Course in Functional Analysis. Springer-Verlag, New York,

second edition, 1990. [2] V. Hutson and J.S. Pym, Applications of Functional Analysis and Operator

Theory, Academic Press, London, 1980.

[3] R. Kress, Linear Integral Equations, Springer-Verlag, Berlin, 1989. [4] S. Mikhlin, Mathematical Physics: An Advanced Course, North-Holland

Pub., Amsterdam, 1970