Propagation of boundary of inhomogeneous heat conduction equation

Abstract

We consider the problem of determining analytically the exact solutions of the heat conduction equation in an inhomogeneous medium, described by the diffusion equation ∂ t T( x, t)= r 1− s ∂ r ( k( r) r s−1 ∂ r T( r, t)) with a position-dependent thermal diffusivity K( r). The unsteady one-dimensional heat conduction equation is transformed into an ordinary differential equation called Kummer’s equation unifiedly in the linear, cylindrical and spherical coordinate systems. Kummer’s equation is solved in terms of the confluent hypergeometric functions. These solutions exist on the conditions that boundaries move with their positions proportional to some functions of time. Progress has been made in this direction by introducing similarity variables and transformations.