Analytic Approximation of Solutions of Parabolic Partial Differential Equations with Variable Coefficients

摘要

A complete family of solutions for the one-dimensional reaction-diffusion equation, u_(xx)(x, t) -(x)u(x, t) = u_t(x, t), with a coefficient depending on x is constructed. The solutions represent the images of the heat polynomials under the action of a transmutation operator. Their use allows one to obtain an explicit solution of the noncharacteristicCauchy problem with sufficiently regular Cauchy data as well as to solve numerically initial boundary value problems. In the paper, theDirichlet boundary conditions are considered; however, the proposed method can be easily extended onto other standard boundary conditions. The proposed numerical method is shown to reveal good accuracy.