On the Porous Medium Equation with Modified Fourier's Law: Symmetries and Integrability

Abstract

The paper considers the general form of the porous Medium Equation f(x)u_t=(g(x)D(u)(u_x)^m)_x+H(u)u_x, where the probability density and flux obey the modified Fourier's law and the drift terms are not omitted. The conditions for invariance are applied to determine the functional forms of the arbitrary functions f, g and H which admit the symmetry reduction transformations. We have studied models depending on the functional form of D(u): 1-D(u)=uⁿ and 2-D(u)=e^nu. A catalogue of symmetry reduction, to ordinary differential equation ODE, is given to each system including its integrability.