Abstract
This article provides a closed form solution to the telegrapher’s equation with three space variables defined on a subset of a sphere within two radii, two azimuthal angles and one polar angle. The Dirichlet problem for general boundary conditions is solved in detail, on the basis of which Neumann and Robin conditions are easily handled. The solution to the simpler problem in cylindrical coordinates is also provided. Ways to efficiently implement the formulae are explained. Minor adjustments result in solutions to the wave equation and to the heat equation on the same domain as well, since the latter are particular cases of the more general telegrapher’s equation.
Highlights
Hyperbolic partial differential equations (PDEs) are among the three main classes of PDEs in mathematical physics
The most comprehensive compilation of known analytical formulae is the handbook by Polyanin and Nazaikinskii [4], which collects and updates all the results from mathematical physics disseminated in previous handbooks such as [5] and
The main contribution of this paper is to provide a closed form solution for a class of initial boundary value problems (IBVPs) involving the linear telegrapher’s equation with three space variables and general boundary condi
Summary
Hyperbolic partial differential equations (PDEs) are among the three main classes of PDEs in mathematical physics. The most comprehensive compilation of known analytical formulae is the handbook by Polyanin and Nazaikinskii [4], which collects and updates all the results from mathematical physics disseminated in previous handbooks such as [5] and [6] It appears that not all IBVPs associated with the three-dimensional telegrapher’s equation have had their exact solutions derived in the literature. If we denote the radius by r, the azimuthal (or longitudinal) angle by φ and the polar (or latitudinal) angle by θ , the only exact solutions to the telegrapher’s equation currently available are those that apply to the following domain [4]: Such a restriction rules out problems that require functional dependency on the boundaries of the variables φ or θ as well as on the boundary of r.
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