  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.
