Analyzing the continuity of the mild solution in finite element analysis of semilinear stochastic subdiffusion problems

Fang Cheng; Ye Hu; Mati ur Rahman; Fang Cheng; Ye Hu; Mati ur Rahman

doi:10.3934/math.2024456

AIMS Mathematics

2024, Volume 9, Issue 4: 9364-9379. doi: 10.3934/math.2024456

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Research article

Analyzing the continuity of the mild solution in finite element analysis of semilinear stochastic subdiffusion problems

1.
School of Statistics and Applied Mathematics, Anhui University of Finance & Economics, Bengbu 233030, China
2.
Department of Mathematics and Artificial Intelligence, Lyuliang University, Lishi 033000, China
3.
School of Mathematical Sciences, Jiangsu University, Zhenjiang 212013, China
4.
Department of Computer Science and Mathematics, Lebanese American University, Beirut Lebanon

Received: 02 January 2024 Revised: 04 February 2024 Accepted: 18 February 2024 Published: 07 March 2024
MSC : 65C60, 65J15, 65M70, 65N35

This paper aimed to further introduce the finite element analysis of non-smooth data for semilinear stochastic subdiffusion problems driven by fractionally integrated additive noise. The mild solution of this stochastic model consisted of three different Mittag-Leffler functions. We analyzed the smoothness of the solution and utilized complex integration to approximate the error of the solution operator under non-smooth data. Consequently, optimal convergence estimates were obtained, and we also obtained the continuity conditions of the mild solution. Finally, the influence of the fractional parameters $ \alpha $ and $ \gamma $ on the convergence rates were accurately demonstrated through numerical examples.
- stochastic time-fractional equation,
- nonsmooth data analysis,
- continuity,
- error estimate
Citation: Fang Cheng, Ye Hu, Mati ur Rahman. Analyzing the continuity of the mild solution in finite element analysis of semilinear stochastic subdiffusion problems[J]. AIMS Mathematics, 2024, 9(4): 9364-9379. doi: 10.3934/math.2024456

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Abstract

This paper aimed to further introduce the finite element analysis of non-smooth data for semilinear stochastic subdiffusion problems driven by fractionally integrated additive noise. The mild solution of this stochastic model consisted of three different Mittag-Leffler functions. We analyzed the smoothness of the solution and utilized complex integration to approximate the error of the solution operator under non-smooth data. Consequently, optimal convergence estimates were obtained, and we also obtained the continuity conditions of the mild solution. Finally, the influence of the fractional parameters $ \alpha $ and $ \gamma $ on the convergence rates were accurately demonstrated through numerical examples.

References

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