Analytic approximation of solutions of parabolic partial differential equations with variable coefficients

摘要

A complete family of solutions for the one-dimensional reaction-diffusionequation \[ u_{xx}(x,t)-q(x)u(x,t) = u_t(x,t) \] with a coefficient $q$depending on $x$ is constructed. The solutions represent the images of the heatpolynomials under the action of a transmutation operator. Their use allows oneto obtain an explicit solution of the noncharacteristic Cauchy problem for theconsidered equation with sufficiently regular Cauchy data as well as to solvenumerically initial boundary value problems. In the paper the Dirichletboundary conditions are considered however the proposed method can be easilyextended onto other standard boundary conditions. The proposed numerical methodis shown to reveal good accuracy.