A fourier method for initial value problems with mixed boundary conditions

Abstract

Works concerning solutions of initial problems with mixed boundary conditions are very rare in the literature and essentially depend on the geometry of some individual problems. K.B. Ranger, for example, considered the cooling of a finite needle (and strip) with prescribed cooling temperature in a medium bounded by two parallel planes perpendicular to the axis and maintained at zero temperature [1]. Because of the axial symmetries of these problems, Ranger could employ integral operators which map 2-D harmonics into the solution of the reduced wave equation and into that of the heat equation thereby. In this work, a systematic method for solving initial plane problems with mixed boundary conditions is proposed. It can be considered as a natural generalization of the Fourier method that deals with the uniform initial problems on the one hand and of that started by Ju. I. Čersiǐ [2] dealing with steady mixed boundary problems on the other. By gradual modifications, the latter method has become of wide application to solutions of problems in several branches of mathematical physics. This has recently been expounded [3]. We shall start with presenting the method by means of a concrete example that, geometrically as well as physically, is rather simplified: the heat conductivity in the unit circle with Dirichlet-Neumann boundary conditions. The method is applicable to problems with different configurations. Further, it can be extended to deal with more complicated physical situations of mixed character, such as dynamical problems of the theory of elasticity with the typical stress-strain boundary conditions.