Analytic approximation of solutions of parabolic partial differential equations with variable coefficients
This work provides a new analytic and numerical approach for solving parabolic PDEs with variable coefficients, which is relevant for applied mathematics and physics.
The paper constructs a complete family of solutions for the one-dimensional reaction-diffusion equation with variable coefficients, enabling explicit solutions for the Cauchy problem and accurate numerical solutions for initial-boundary value problems.
A complete family of solutions for the one-dimensional reaction-diffusion equation \[ u_{xx}(x,t)-q(x)u(x,t) = u_t(x,t) \] with a coefficient $q$ depending on $x$ is constructed. The solutions represent the images of the heat polynomials under the action of a transmutation operator. Their use allows one to obtain an explicit solution of the noncharacteristic Cauchy problem for the considered equation with sufficiently regular Cauchy data as well as to solve numerically initial boundary value problems. In the paper the Dirichlet 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.