Galerkin Finite Element Approximation of a Stochastic Semilinear Fractional Wave Equation Driven by Fractionally Integrated Additive Noise
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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/M...
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MDPI AG
2023-05-01
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| Series: | Foundations |
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| Online Access: | https://www.mdpi.com/2673-9321/3/2/23 |
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| author | Bernard A. Egwu Yubin Yan |
| author_facet | Bernard A. Egwu Yubin Yan |
| author_sort | Bernard A. Egwu |
| collection | DOAJ |
| description | 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>. 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. |
| first_indexed | 2024-03-11T02:28:14Z |
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| last_indexed | 2024-03-11T02:28:14Z |
| publishDate | 2023-05-01 |
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| spelling | doaj.art-ff0026a5d67541a5a176864c60737c4c2023-11-18T10:28:57ZengMDPI AGFoundations2673-93212023-05-013229032210.3390/foundations3020023Galerkin Finite Element Approximation of a Stochastic Semilinear Fractional Wave Equation Driven by Fractionally Integrated Additive NoiseBernard A. Egwu0Yubin Yan1Department of Physical, Mathematical and Engineering Sciences, University of Chester, Chester CH1 4BJ, UKDepartment of Physical, Mathematical and Engineering Sciences, University of Chester, Chester CH1 4BJ, UKWe 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>. 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.https://www.mdpi.com/2673-9321/3/2/23stochastic fractional wave equationintegrated additive noiseCaputo derivativefinite element methodoptimal error estimates |
| spellingShingle | Bernard A. Egwu Yubin Yan Galerkin Finite Element Approximation of a Stochastic Semilinear Fractional Wave Equation Driven by Fractionally Integrated Additive Noise Foundations stochastic fractional wave equation integrated additive noise Caputo derivative finite element method optimal error estimates |
| title | Galerkin Finite Element Approximation of a Stochastic Semilinear Fractional Wave Equation Driven by Fractionally Integrated Additive Noise |
| title_full | Galerkin Finite Element Approximation of a Stochastic Semilinear Fractional Wave Equation Driven by Fractionally Integrated Additive Noise |
| title_fullStr | Galerkin Finite Element Approximation of a Stochastic Semilinear Fractional Wave Equation Driven by Fractionally Integrated Additive Noise |
| title_full_unstemmed | Galerkin Finite Element Approximation of a Stochastic Semilinear Fractional Wave Equation Driven by Fractionally Integrated Additive Noise |
| title_short | Galerkin Finite Element Approximation of a Stochastic Semilinear Fractional Wave Equation Driven by Fractionally Integrated Additive Noise |
| title_sort | galerkin finite element approximation of a stochastic semilinear fractional wave equation driven by fractionally integrated additive noise |
| topic | stochastic fractional wave equation integrated additive noise Caputo derivative finite element method optimal error estimates |
| url | https://www.mdpi.com/2673-9321/3/2/23 |
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