A robust dual spectral-spline framework for stochastic fractional equations perturbed by white noise

Aisha F. Fareed; Mourad S. Semary; Dokhyl M. Alqahtani; Emad A. Mohamed; Ahmed G. Khattab; Aisha F. Fareed; Mourad S. Semary; Dokhyl M. Alqahtani; Emad A. Mohamed; Ahmed G. Khattab

doi:10.3934/math.2026370

AIMS Mathematics

2026, Volume 11, Issue 4: 8966-8987. doi: 10.3934/math.2026370

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Research article

A robust dual spectral-spline framework for stochastic fractional equations perturbed by white noise

1.
Department of Electrical Engineering, College of Engineering, Prince Sattam bin Abdulaziz University, Al Kharj 16278, Saudi Arabia
2.
Basic Engineering Sciences Department, Benha Faculty of Engineering, Benha University, Benha 13512, Egypt
3.
Basic Sciences Department, Faculty of Engineering, Badr University in Cairo BUC, Cairo 11829, Egypt

Received: 31 January 2026 Revised: 09 March 2026 Accepted: 18 March 2026 Published: 02 April 2026
MSC : 60H15, 65C30, 65N35

In this research, we introduced dual numerical frameworks for stochastic time-fractional partial differential equations (STFPDEs) driven by additive white noise. The sifted Vieta–Fibonacci polynomials (SVFPs) and the ten non-polynomial cubic spline (TNPCS) approach are the foundation for our spectral-spline frameworks. The suggested methods were used to solve two STFPDEs: The nonlinear STF Burger's equation and the linear STF heat equation, and they approximated stochastic trajectories with high precision. This analysis showed a substantial decrease in variance, and the mean profile remained constant despite stochastic effects. The key contribution of this paper is the introduction of two separate spectral-spline numerical approaches derived from the SVFPs and the TNPCS formulations for handling stochastic fractional models. By applying each approach independently and comparing their results side by side, the study provides clearer insight into their numerical performance, stability, and practical usefulness in simulating stochastic fractional dynamics. All computations were implemented using Mathematica 12 and Jupyter Notebook, which ensured the validity, dependability, and reliability of our algorithms against noise propagation.
- white noise,
- Burger's equation,
- heat equation,
- orthogonal polynomials,
- Vieta–Fibonacci polynomials,
- stochastic differential equations
Citation: Aisha F. Fareed, Mourad S. Semary, Dokhyl M. Alqahtani, Emad A. Mohamed, Ahmed G. Khattab. A robust dual spectral-spline framework for stochastic fractional equations perturbed by white noise[J]. AIMS Mathematics, 2026, 11(4): 8966-8987. doi: 10.3934/math.2026370

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Abstract

In this research, we introduced dual numerical frameworks for stochastic time-fractional partial differential equations (STFPDEs) driven by additive white noise. The sifted Vieta–Fibonacci polynomials (SVFPs) and the ten non-polynomial cubic spline (TNPCS) approach are the foundation for our spectral-spline frameworks. The suggested methods were used to solve two STFPDEs: The nonlinear STF Burger's equation and the linear STF heat equation, and they approximated stochastic trajectories with high precision. This analysis showed a substantial decrease in variance, and the mean profile remained constant despite stochastic effects. The key contribution of this paper is the introduction of two separate spectral-spline numerical approaches derived from the SVFPs and the TNPCS formulations for handling stochastic fractional models. By applying each approach independently and comparing their results side by side, the study provides clearer insight into their numerical performance, stability, and practical usefulness in simulating stochastic fractional dynamics. All computations were implemented using Mathematica 12 and Jupyter Notebook, which ensured the validity, dependability, and reliability of our algorithms against noise propagation.

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