Sparse series solutions of random boundary and initial value problems

The study investigates the utilization of sparse representations through dictionary elements to address the classical Dirichlet and Cauchy types of stochastic boundary value problems (BVPs) and initial value problems (IVPs). A novel approach is introduced based on the recently developed stochastic pre-orthogonal adaptive Fourier decomposition (SPOAFD) technique. By employing SPOAFD, both analytic and numerical solutions for the stochastic BVPs and IVPs are formulated. Furthermore, the scope of the study is extended to include BVPs and IVPs associated with a specific class of fractional heat equations and fractional Poisson equations. In addition to establishing the theoretical framework, important computational aspects are thoroughly discussed to enable the implementation of practical algorithms. The proposed methodology is validated through numerical examples, demonstrating its effectiveness and computational efficiency.