The residual variable method applied to the diffusion equation in cylindrical coordinates
We consider a partial differential equation in cylindrical coordinates which describes a dynamic process going on in an infinite medium with an inner cylindrical boundary. If an analytic solution to the problem is not possible to obtain then one would have to resort to numerical techniques. It becomes necessary to discretize the infinite domain even if the solution is required on the inner cylindrical surface only. The numerical difficulties associated with the discretization of an infinite medium are well known. The residual variable method (RVM) is a rigorous method which essentially reduces the number of the spatial dimensions of the problem by one. The governing differential equation is integrated once in the radial direction and, thus, it becomes possible to determine the solution on the inner boundary without having to deal with the infinite domain. In this work, the classical cylindrical diffusion equation is expressed in terms of a residual concentration applying the RVM. Thus, it became possible to eliminate the second derivative of the concentration with respect to the radial coordinate from the diffusion equation. The diffusion on the surface of the cylindrical cavity was studied by considering the cavity surface only. The analytical results obtained from the application of the RVM and the numerical results obtained from the application of the finite difference techniques are compared for the flux at the surface of the region bounded internally by a circular cylinder with zero initial conditions and surface concentration in the form of the Heaviside step function.
scientific article; zbMATH DE number 4159229