Source solutions for the nonlinear diffusion-convection equation

In the paper King [8], a new class of source solutions was derived for the nonlinear diffusion equation for diffusivities of the form D(c) = D0cm /(1 -νc)m+2. Here we extend this method for the nonlinear diffusion and convection equation ∂c/∂t = ∂/∂z [D(c)∂c/∂z - K(c)], to obtain mass-conserving source solutions for a nonlinear conductivity function K(c) = K0cm+2/(1 - νc)m+1. In particular we consider the cases m = -1, 0, and 1, where fully analytical solutions are available. Furthermore we provide source solutions for the exponential forms of the diffusivity and conductivity as given by D(c) = D0c-2e-n/c and K(c) = K0ce-n/c. © Australian Mathematical Society, 1997

Finite Difference/Spectral Approximations for the Fractional Cable Equation

In this paper, we consider the numerical solution of the fractional Cable equation, which is a generalization of the classical Cable equation by taking into account the anomalous diffusion in the movement of the ions in neuronal system. A schema combining a finite difference approach in the time direction and a spectral method in the space direction is proposed and analyzed. The main contribution of this work is three-fold:1) We construct a finite difference/Legendre spectral schema for discretization of the fractional Cable equation. 2) We give a detailed analysis of the proposed schema by providing some stability and error estimates. Based on this analysis, the convergence of the method is rigourously established. We prove that the overall schema is unconditionally stable, and the numerical solution converges to the exact one with order O(Delta t(2-max{alpha,beta})) , where Delta 4t is the time step size, alpha and beta are two different exponents between 0 and 1 involved in the fractional derivatives. 3) Finally, some numerical experiments are carried out to support the theoretical claims