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Extension of the residual variable method to propagation problems and its application to the diffusion equation in spherical coordinates

Tokdemir, Turgut
We consider a partial differential equation in spherical (cylindrical) coordinates describing a dynamic process in an infinite medium with an inner spherical (cylindrical) boundary. If an analytical solution is not possible to obtain, then one resorts to numerical techniques. In this case it becomes necessary to discretize the infinite domain even if the solution is required on the inner spherical (cylindrical) surface or at a limited number of points in the domain only. The Residual Variable Method (RVM) circumvents the difficulty of discretizing the infinite domain. The governing equation is integrated once in radial direction. The number of the spatial dimensions is, thus, reduced by one. It is now possible to determine the solution on the inner boundary without having to deal with the infinite domain. The RVM is amenable to ''marching'' solutions in a finite-difference implementation and it is suitable for the analysis of propagation into the infinite medium from the inner surface. There is no need to discretize the infinite domain in its entirety at all. The propagation analysis can be terminated at any point in the radial direction without having to consider the rest.