Perturbed Lie Symmetry and Systems of Non-Linear Diffusion Equations

Abstract

The method of one parameter, point symmetric, approximate Lie group invariants is applied to the problem of determining solutions of systems of pure one-dimensional, diffusion equations. The equations are taken to be non-linear in the dependent variables but otherwise homogeneous. Moreover, the matrix of diffusion coefficients is taken to differ from a constant matrix by a linear perturbation with respect to an infinitesimal parameter. The conditions for approximate Lie invariance are developed and are applied to the coupled system. The corresponding prolongation operator is derived and it is shown that this places a power law and logarithmic constraints on the nature of the perturbed diffusion matrix. The method is used to derive an approximate solution of the perturbed diffusion equation corresponding to impulsive boundary conditions.