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Article

Galerkin Finite Element Approximation of a Stochastic Semilinear Fractional Wave Equation Driven by Fractionally Integrated Additive Noise

by
Bernard A. Egwu
† and
Yubin Yan
*,†
Department of Physical, Mathematical and Engineering Sciences, University of Chester, Chester CH1 4BJ, UK
*
Author to whom correspondence should be addressed.
†
These authors contributed equally to this work.
Foundations 2023, 3(2), 290-322; https://doi.org/10.3390/foundations3020023
Submission received: 1 May 2023 / Revised: 21 May 2023 / Accepted: 25 May 2023 / Published: 29 May 2023
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions II)
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Abstract

We investigate the application of the Galerkin finite element method to approximate a stochastic semilinear space–time fractional wave equation. The equation is driven by integrated additive noise, and the time fractional order α∈(1,2). The existence of a unique solution of the problem is proved by using the Banach fixed point theorem, and the spatial and temporal regularities of the solution are established. The noise is approximated with the piecewise constant function in time in order to obtain a stochastic regularized semilinear space–time wave equation which is then approximated using the Galerkin finite element method. The optimal error estimates are proved based on the various smoothing properties of the Mittag–Leffler functions. Numerical examples are provided to demonstrate the consistency between the theoretical findings and the obtained numerical results.
Keywords: stochastic fractional wave equation; integrated additive noise; Caputo derivative; finite element method; optimal error estimates stochastic fractional wave equation; integrated additive noise; Caputo derivative; finite element method; optimal error estimates

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MDPI and ACS Style

Egwu, B.A.; Yan, Y. Galerkin Finite Element Approximation of a Stochastic Semilinear Fractional Wave Equation Driven by Fractionally Integrated Additive Noise. Foundations 2023, 3, 290-322. https://doi.org/10.3390/foundations3020023

AMA Style

Egwu BA, Yan Y. Galerkin Finite Element Approximation of a Stochastic Semilinear Fractional Wave Equation Driven by Fractionally Integrated Additive Noise. Foundations. 2023; 3(2):290-322. https://doi.org/10.3390/foundations3020023

Chicago/Turabian Style

Egwu, Bernard A., and Yubin Yan. 2023. "Galerkin Finite Element Approximation of a Stochastic Semilinear Fractional Wave Equation Driven by Fractionally Integrated Additive Noise" Foundations 3, no. 2: 290-322. https://doi.org/10.3390/foundations3020023

APA Style

Egwu, B. A., & Yan, Y. (2023). Galerkin Finite Element Approximation of a Stochastic Semilinear Fractional Wave Equation Driven by Fractionally Integrated Additive Noise. Foundations, 3(2), 290-322. https://doi.org/10.3390/foundations3020023

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