Existence, uniqueness and numerical solution of stochastic fractional differential equations with integer and non-integer orders

Seda IGRET ARAZ; Mehmet Akif CETIN; Abdon ATANGANA; Seda IGRET ARAZ; Mehmet Akif CETIN; Abdon ATANGANA

doi:10.3934/era.2024035

Electronic Research Archive

2024, Volume 32, Issue 2: 733-761. doi: 10.3934/era.2024035

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Existence, uniqueness and numerical solution of stochastic fractional differential equations with integer and non-integer orders

1.
Siirt University, Department of Mathematics Education, Siirt, Turkey
2.
Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, South Africa
3.
ALTSO Vocational School, Alanya Alaaddin Keykubat University, Antalya, Turkey
4.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan
5.
IT4Innovations, VSB–Technical University of Ostrava, Ostrava-Poruba 70800, Czech Republic

Received: 26 September 2023 Revised: 03 December 2023 Accepted: 15 December 2023 Published: 10 January 2024

The parametrized approach is extended in this study to find solutions to differential equations with fractal, fractional, fractal-fractional, and piecewise derivatives with the inclusion of a stochastic component. The existence and uniqueness of the solution to the stochastic Atangana-Baleanu fractional differential equation are established using Caratheodory's existence theorem. For the solution of differential equations using piecewise differential operators, which take into account combining deterministic and stochastic processes utilizing certain significant mathematical tools such as fractal and fractal-fractional derivatives, the applicability of the parametrized technique is being examined. We discuss the crossover behaviors of the model obtained by including these operators and we present some illustrative examples for some problems with piecewise differential operators.
- Caratheodory's conditions,
- fractal-fractional differentiation,
- piecewise calculus,
- parametrized method
Citation: Seda IGRET ARAZ, Mehmet Akif CETIN, Abdon ATANGANA. Existence, uniqueness and numerical solution of stochastic fractional differential equations with integer and non-integer orders[J]. Electronic Research Archive, 2024, 32(2): 733-761. doi: 10.3934/era.2024035

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Abstract

The parametrized approach is extended in this study to find solutions to differential equations with fractal, fractional, fractal-fractional, and piecewise derivatives with the inclusion of a stochastic component. The existence and uniqueness of the solution to the stochastic Atangana-Baleanu fractional differential equation are established using Caratheodory's existence theorem. For the solution of differential equations using piecewise differential operators, which take into account combining deterministic and stochastic processes utilizing certain significant mathematical tools such as fractal and fractal-fractional derivatives, the applicability of the parametrized technique is being examined. We discuss the crossover behaviors of the model obtained by including these operators and we present some illustrative examples for some problems with piecewise differential operators.

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