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Boundary-value problems associated with either ordinary or partial differential equations arise most frequently in mathematics, mathematical physics and engineering science. The linear superposition principle is one of the most elegant and effective methods to represent solutions of boundary-value problems in terms of an auxiliary function known as Green’s function . Such a function was first introduced by George Green as early as 1828. Subsequently, the method of Green’s functions became a very useful analytical method in mathematics and in many of the applied sciences.

The linear superposition principle is one of the most effective and elegant methods to represent solutions of partial differential equations in terms of eigenfunctions or Green’s functions. More precisely, the eigenfunction expansion method expresses the solution as an infinite series, whereas the integral solution can be obtained by integral superposition or by using Green’s functions with initial and boundary conditions. The latter offers several advantages over eigenfunction expansion. First, an integral representation provides a direct way of describing the general analytical structure of a solution that may be obscured by an infinite series representation. Second, from a practical point of view, the evaluation of a solution from an integral representation may prove simpler than finding the sum of an infinite series, particularly near rapidly-varying features of a function, where the convergence of an eigenfunction expansion is expected to be slow. Third, in view of the Gibbs phenomenon discussed in Chapter 6, the integral representation seems to be less stringent requirements on the functions that describe the initial conditions or the values of a solution are required to assume on a given boundary than expansions based on eigenfunctions.

The three-dimensional linear wave equation (13.1.1) $$ u_{tt} = s^2 \nabla ^2 u, $$ arises in the areas of elasticity, fluid dynamics, acoustics, magnetohydrodynamics, and electromagnetism.

The preceding chapters have been devoted to the analytical treatment of linear and nonlinear partial differential equations. Several analytical methods to find the exact analytical solution of these equations within simple domains have been discussed. The boundary and initial conditions in these problems were also relatively simple, and were expressible in simple mathematical form. In dealing with many equations arising from the modelling of physical problems, the determination of such exact solutions in a simple domain is a formidable task even when the boundary and/or initial data are simple. It is then necessary to resort to numerical or approximation methods in order to deal with the problems that cannot be solved analytically. In view of the widespread accessibility of today’s high speed electronic computers, numerical and approximation methods are becoming increasingly important and useful in applications.

In this chapter we provide a set of short tables of integral transforms of the functions that are either cited in the text or are in most common use in mathematical, physical, and engineering applications. For exhaustive lists of integral transforms, the reader is referred to Erdélyi et al. (1954), Campbell and Foster (1948), Ditkin and Prudnikov (1965), Doetsch (1970), Marichev (1983), Debnath (1995), and Oberhettinger (1972).