In this paper, we introduced a novel numerical approach for solving stochastic heat
equations and multi-dimensional stochastic Poisson equations using shifted Vieta-Fibonacci
polynomials (SVFPs), marking their first application in stochastic differential equations. The proposed
method leveraged the orthogonality and recurrence properties of SVFPs to approximate solutions with
high precision. By normalizing the polynomial basis and their derivatives, the technique ensured
numerical stability and convergence, addressing challenges encountered in earlier implementations.
The method was rigorously validated through comparisons with the fast discrete Fourier transform
approach, other methods in the literature, and, where applicable, exact solutions, demonstrating
superior accuracy. Five illustrative problems were analyzed, with results showcasing significantly
reduced variance and absolute errors, particularly for higher-order approximations. The numerical
simulations, executed using Mathematica 12, highlighted the robustness of the SVFPs-based algorithm
in handling stochastic variability. This work not only extended the applicability of SVFPs to stochastic
domains but also provided a reliable framework for future research on fractional and nonlinear
stochastic systems. |
