Non-stationary statistical solutions of a class of random diffusion equations: Analytical and numerical considerations

Non-stationary, behaviour of statistical moments up to the second order, of solutions of linear one-dimensional diffusion equations having random initial conditions and random external excitations the latter of which is represented by non-stationary Gaussian white noises, is considered. Three approaches are presented: the first is concerned with the analytical solution procedure based on separation of variables together with the superposition principle; the second deals with a semi-analytical approach by the use of finite element and finite difference approximations in space; and the third is related to numerical analysis using the simplest explicit and implicit finite differences. Comparison is made for the results obtained by these three solution procedures. Convergence behaviour of the analytical solutions is investigated, and the consideration of stability of the finite difference solutions is also given.