440 12 Integral Transform Methods with Applications
particularly near rapidly-varying features of a function, where the conver-
gence of an eigenfunction expansion is expected to b e slow. Third, in view
of the Gibbs phenomenon discussed in Chapter 6, the integral representa-
tion seems to be less stringent requirements on the functions that describ e
the initial conditions or the values of a solution are required to assume on
a given boundary than expansions based on eigenfunctions.
Integral transform methods are found to b e very useful for finding so-
lutions of initial and/or boundary-value problems governed by partial dif-
ferential equations for the fol lowing reason. The differential equations can
readily be replaced by algebraic equations that are inverted by the inverse
transform so that the solution of the differential equations can then be
obtained in terms of the original variables. The aim of this chapter is to
provide an introduction to the use of integral transform methods for stu-
dents of applied mathematics, physics, and engineering. Since our major
interest is the application of integral transforms, no attempt will be made
to discuss the basic results and theorems relating to transforms in their
general forms. The present treatment is restricted to classes of functions
which usually occur in physical and engineering applications.
12.2 Fourier Transforms
We first give a formal definition of the Fourier transform by using the
complex Fourier integral formula (6.13.10).
Definition 12.2.1. If u (x, t) is a continuous, piecewise smooth, and abso-
lutely integrable function, then the Fourier transform of u (x, t) with respect
to x ∈ R is denoted by U (k, t) and is defined by
F{u (x, t)} = U (k, t)=
1
√
2π
∞
−∞
e
−ikx
u (x , t) dx, (12.2.1)
where k is called the Fourier transform variable and exp (−ikx) is called
the kernel of the transform.
Then, for all x ∈ R, the inverse Fourier transform of U (k, t) is defined
by
F
−1
{U (k, t)} = u (x, t)=
1
√
2π
∞
−∞
e
ikx
U (k, t) dk. (12.2.2)
We may note that the factor (1/2π) in the Fourier integral formula
(6.13.9) has been split and placed in front of the integrals (12.2.1) and
(12.2.2). Often the factor (1/2π) can be placed in only one of the relations
(12.2.1) and (12.2.2). It is not uncommon to adopt the kernel exp (ikx) in
(12.2.1) instead of exp (−ikx), and as a consequence, exp (−ikx) would be
replaced by exp (ikx) in (12.2.2).