Application of the Laplace Method to the Construction of Solutions of the Helmholtz Equation

Abstract

A technique closely related to the theory of integral transforms is used in many problems of mathematical physics. The method of integral transforms makes it possible to successfully study processes of wave propagation and diffraction [1–9]; it is widely used in solving problems of the theory of heat conduction [10–14] and also in a number of other cases [15–17], However, together with the well-known advantages of this method, there are also shortcomings which appear in the solution of boundary value problems of mathematical physics with complicated conditions of the boundary. This latter fact is the reason that integral transforms have been used mainly in problems with time variables, while problems with spatial variables have been little considered by this method. Because of this, the necessity has long been felt of working out methods of integral transforms which take into account not only the particular features of the processes described but also the shape of the region in which these processes take place. Such methods have been proposed in [18–20], It must be noted that to obtain new transforms, as a rule, is not an elementary problem. Moreover, in choosing a transform for some group of problems there is always a certain amount of freedom; this can either restrict the possibilities for applying the transform chosen or, on the other hand, extend them.