Separation of variables for the 1-dimensional nonlinear diffusion equation

Research Report MRR97-011

Separation of variables for the 1-dimensional nonlinear diffusion equation

Philip W. Doyle and Peter J. Vassiliou

Abstract: The class of separable solutions of a 1-dimensional sourceless diffusion equation is stabilized by the action of the generic symmetry group. It includes all solutions invariant under a subgroup of the generic group. An equation which admits separation of variables in some field coordinate has separable solutions which are not invariant under any subgroup, as in the linear case. The class of separable equations significantly extends the class of equations having nongeneric symmetry, that is, those with exponential and power law diffusivities, for which separation of variables is a trivial process. We derive a complete list of canonical forms for diffusion equations which admit separation of variables in some field coordinate, and we describe the separation process for these equations. It involves the integration of a fixed third order ordinary differential equation, generally nonlinear, and the subsequent integration of a first order ordinary differential equation which depends on the particular solution of the third order equation. This procedure yields a 3-parameter family of separable solutions of the given diffusion equation. Several nonsymmetric examples are analysed in detail, leading to explicit noninvariant solutions.

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