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 ests
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 2Ï€
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