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

Ahmed S. Hendy; Mahmoud A. Zaky; Durvudkhan Suragan

doi:10.1016/j.matcom.2021.10.013

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

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

Research output: Contribution to journal › Article › peer-review

11 Citations (Scopus)

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.

Original language	English
Pages (from-to)	269-279
Number of pages	11
Journal	Mathematics and Computers in Simulation
Volume	193
DOIs	https://doi.org/10.1016/j.matcom.2021.10.013
Publication status	Published - Mar 2022

Funding

The second and the third authors were supported by the Nazarbayev University Program 091019CRP2120 . The second and the third authors wish also to acknowledge the partial support of the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant “Dynamical Analysis and Synchronization of Complex Neural Networks with Its Applications”). The second and the third authors were supported by the Nazarbayev University Program091019CRP2120. The second and the third authors wish also to acknowledge the partial support of the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant “Dynamical Analysis and Synchronization of Complex Neural Networks with Its Applications”).

Keywords

A priori estimate
Discrete stochastic fractional Grönwall inequalities
Interpolation schemes
Martingale
Multi-term time-fractional derivatives
Time delay

ASJC Scopus subject areas

Theoretical Computer Science
General Computer Science
Numerical Analysis
Modelling and Simulation
Applied Mathematics

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10.1016/j.matcom.2021.10.013

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title = "Discrete fractional stochastic Gr{\"o}nwall inequalities arising in the numerical analysis of multi-term fractional order stochastic differential equations",

abstract = "This paper is devoted to the rigorous derivation of some discrete versions of stochastic Gr{\"o}nwall inequalities involving a martingale, which are commonly used in the numerical analysis of multi-term stochastic time-fractional diffusion equations. A Gr{\"o}nwall 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 Gr{\"o}nwall 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.",

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author = "Hendy, \{Ahmed S.\} and Zaky, \{Mahmoud A.\} and Durvudkhan Suragan",

note = "Funding Information: The second and the third authors were supported by the Nazarbayev University Program091019CRP2120. The second and the third authors wish also to acknowledge the partial support of the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant ?Dynamical Analysis and Synchronization of Complex Neural Networks with Its Applications?). Funding Information: The second and the third authors were supported by the Nazarbayev University Program 091019CRP2120 . The second and the third authors wish also to acknowledge the partial support of the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant “Dynamical Analysis and Synchronization of Complex Neural Networks with Its Applications”). Publisher Copyright: {\textcopyright} 2021 International Association for Mathematics and Computers in Simulation (IMACS)",

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doi = "10.1016/j.matcom.2021.10.013",

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AU - Zaky, Mahmoud A.

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N1 - Funding Information: The second and the third authors were supported by the Nazarbayev University Program091019CRP2120. The second and the third authors wish also to acknowledge the partial support of the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant ?Dynamical Analysis and Synchronization of Complex Neural Networks with Its Applications?). Funding Information: The second and the third authors were supported by the Nazarbayev University Program 091019CRP2120 . The second and the third authors wish also to acknowledge the partial support of the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant “Dynamical Analysis and Synchronization of Complex Neural Networks with Its Applications”). Publisher Copyright: © 2021 International Association for Mathematics and Computers in Simulation (IMACS)

PY - 2022/3

Y1 - 2022/3

N2 - 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.

AB - 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.

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KW - Discrete stochastic fractional Grönwall inequalities

KW - Interpolation schemes

KW - Martingale

KW - Multi-term time-fractional derivatives

KW - Time delay

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JO - Mathematics and Computers in Simulation

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

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