Introduction

Integral representations play a central role in various fields of pure and applied mathematics, theoretical physics, and engineering. Many boundary value problems and initial boundary value problems can be solved using integral kernels. In such a case, we usually have an explicit formula for the solution as a (finite) sum of integrals involving integral kernels and the associated side conditions (which are the given data). The integral kernel depends on the differential operator, the type of given boundary condition, and the domain. It should be computed just once for any given type of problem. Hence, given an integral representation for a differential problem, we can find the solution for a specific choice of associated conditions by carrying out just a small number of integrations.

A typical example of an integral representation is the Poisson formula (7.81) which is an explicit integral representation for solutions of the Dirichlet problem for the Laplace equation in a disk. Note that the d'Alembert formula (4.17) for solving the (one-dimensional) Cauchy problem for the nonhomogeneous wave equation with zero initial conditions is also an integral representation.

In this chapter, we present some further examples of integral representations for the Laplace operator and for the heat equation. The integral kernel for the Laplace operator is called Green's function in honor of the great English mathematician George Green (1793–1841).