(37b) Semi-Analytical Solution of Nonlinear Diffusion Equations

The nonlinear diffusion equations that describe many practically-important phenomena offer very limited scope for the development of analytical solutions. While application of the well known Kirchhoff integral transformation allows the corresponding steady state concentration profiles to be expressed in terms of harmonic functions, nonlinearities remain in the transient diffusion equation. The pseudotime integral transformation discovered by Brebbia et al. (1987) reduces the nonlinear equation for the Kirchhoff transformation variableinto a linear diffusion equation. The purpose of this paper is to show how the Kirchhoff and pseudotime transformations can be used to relate the desired nonlinear solution to the solution of the corresponding linear equation, for which many analytical results are available in domains of simple geometry. The concentration-dependence of the diffusivity gives rise to a local distortion of the time scale, which can be used to establish a nonlinear mapping between the solutions of the nonlinear equation and the corresponding linear equation. If an analytical solution is available for the linear equation, the solution of the nonlinear equation thus reduces to a one-dimensional quadrature and solution of a single nonlinear algebraic equation, thereby providing a highly efficient and accurate computational method that does not rely on any finite-difference approximations.