A novel spectral framework for stochastic differential equations: leveraging shifted Vieta-Fibonacci polynomials

Ahmed G. Khattab; D.A. Hammad; Mourad S. Semary; Emad A. Mohamed; Iram Malik; Aisha F. Fareed; Ahmed G. Khattab; D.A. Hammad; Mourad S. Semary; Emad A. Mohamed; Iram Malik; Aisha F. Fareed

doi:10.3934/math.20251324

AIMS Mathematics

2025, Volume 10, Issue 12: 30134-30161. doi: 10.3934/math.20251324

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Research article

A novel spectral framework for stochastic differential equations: leveraging shifted Vieta-Fibonacci polynomials

1.
Basic Engineering Sciences Department, Benha Faculty of Engineering, Benha University, Benha 13512, Egypt
2.
Basic Sciences Department, Faculty of Engineering, Badr University in Cairo BUC, Cairo 11829, Egypt
3.
Department of Electrical Engineering, College of Engineering, Prince Sattam bin Abdulaziz University, Al Kharj 11942, Saudi Arabia

Received: 28 September 2025 Revised: 08 December 2025 Accepted: 19 December 2025 Published: 24 December 2025
MSC : 65C30, 65L12

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.
- Vieta-Fibonacci polynomials,
- orthogonal polynomials,
- heat equation,
- Poisson equation,
- stochastic differential equations,
- white noise
Citation: Ahmed G. Khattab, D.A. Hammad, Mourad S. Semary, Emad A. Mohamed, Iram Malik, Aisha F. Fareed. A novel spectral framework for stochastic differential equations: leveraging shifted Vieta-Fibonacci polynomials[J]. AIMS Mathematics, 2025, 10(12): 30134-30161. doi: 10.3934/math.20251324

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Abstract

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.

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