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Discrete fractional stochastic Grönwall inequalities arising in the numerical analysis of multi-term fractional order stochastic differential equations

Author

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  • Hendy, Ahmed S.
  • Zaky, Mahmoud A.
  • Suragan, Durvudkhan

Abstract

This paper is devoted to the rigorous derivation of some discrete versions of stochastic Grönwall inequalities involving a martingale, which are commonly used in the numerical analysis of multi-term stochastic time-fractional diffusion equations. A Grönwall lemma is also established to deal with the numerical analysis of multi-term stochastic fractional diffusion equations with delay. The proofs of the established inequalities are based on a corresponding deterministic version of the discrete fractional Grönwall lemma in case of smooth solutions and an inequality bounding the supremum in terms of the infimum for discrete time martingales. A numerical application is introduced finally in which the constructed inequalities are handled to derive a priori estimates for a discrete fractional stochastic model.

Suggested Citation

  • Hendy, Ahmed S. & Zaky, Mahmoud A. & Suragan, Durvudkhan, 2022. "Discrete fractional stochastic Grönwall inequalities arising in the numerical analysis of multi-term fractional order stochastic differential equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 193(C), pages 269-279.
    • Handle: RePEc:eee:matcom:v:193:y:2022:i:c:p:269-279
    DOI: 10.1016/j.matcom.2021.10.013
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    References listed on IDEAS

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    1. Kruse, Raphael & Scheutzow, Michael, 2018. "A discrete stochastic Gronwall lemma," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 143(C), pages 149-157.
    2. Yang, Zhiwei & Zheng, Xiangcheng & Zhang, Zhongqiang & Wang, Hong, 2021. "Strong convergence of a Euler-Maruyama scheme to a variable-order fractional stochastic differential equation driven by a multiplicative white noise," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).
    3. Pedjeu, Jean-C. & Ladde, Gangaram S., 2012. "Stochastic fractional differential equations: Modeling, method and analysis," Chaos, Solitons & Fractals, Elsevier, vol. 45(3), pages 279-293.
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    Cited by:

    1. Sarita Nandal & Mahmoud A. Zaky & Rob H. De Staelen & Ahmed S. Hendy, 2021. "Numerical Simulation for a Multidimensional Fourth-Order Nonlinear Fractional Subdiffusion Model with Time Delay," Mathematics, MDPI, vol. 9(23), pages 1-15, November.

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