Well-posedness and Ulam-Hyers stability results of solutions to pantograph fractional stochastic differential equations in the sense of conformable derivatives

Wedad Albalawi; Muhammad Imran Liaqat; Fahim Ud Din; Kottakkaran Sooppy Nisar; Abdel-Haleem Abdel-Aty; Wedad Albalawi; Muhammad Imran Liaqat; Fahim Ud Din; Kottakkaran Sooppy Nisar; Abdel-Haleem Abdel-Aty

doi:10.3934/math.2024605

AIMS Mathematics

2024, Volume 9, Issue 5: 12375-12398. doi: 10.3934/math.2024605

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Well-posedness and Ulam-Hyers stability results of solutions to pantograph fractional stochastic differential equations in the sense of conformable derivatives

1.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
2.
Abdus Salam School of Mathematical Sciences, Government College University, 68-B, New Muslim Town, Lahore 54600, Pakistan
3.
Department of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam bin Abdulaziz University, Alkharj 11942, Saudi Arabia
4.
Department of Physics, College of Sciences, University of Bisha, Bisha 61922, Saudi Arabia

Received: 25 January 2024 Revised: 11 March 2024 Accepted: 13 March 2024 Published: 29 March 2024
MSC : 34A07, 34A08, 60G22

One kind of stochastic delay differential equation in which the delay term is dependent on a proportion of the current time is the pantograph stochastic differential equation. Electric current collection, nonlinear dynamics, quantum mechanics, and electrodynamics are among the phenomena modeled using this equation. A key idea in physics and mathematics is the well-posedness of a differential equation, which guarantees that the solution to the problem exists and is a unique and meaningful solution that relies continuously on the initial condition and the value of the fractional derivative. Ulam-Hyers stability is a property of equations that states that if a function is approximately satisfying the equation, then there exists an exact solution that is close to the function. Inspired by these findings, in this research work, we established the Ulam-Hyers stability and well-posedness of solutions of pantograph fractional stochastic differential equations (PFSDEs) in the framework of conformable derivatives. In addition, we provided examples to analyze the theoretical results.
- conformable fractional derivative,
- pantograph fractional stochastic differential equations,
- well-posedness,
- Ulam-Hyers stability
Citation: Wedad Albalawi, Muhammad Imran Liaqat, Fahim Ud Din, Kottakkaran Sooppy Nisar, Abdel-Haleem Abdel-Aty. Well-posedness and Ulam-Hyers stability results of solutions to pantograph fractional stochastic differential equations in the sense of conformable derivatives[J]. AIMS Mathematics, 2024, 9(5): 12375-12398. doi: 10.3934/math.2024605

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Abstract

One kind of stochastic delay differential equation in which the delay term is dependent on a proportion of the current time is the pantograph stochastic differential equation. Electric current collection, nonlinear dynamics, quantum mechanics, and electrodynamics are among the phenomena modeled using this equation. A key idea in physics and mathematics is the well-posedness of a differential equation, which guarantees that the solution to the problem exists and is a unique and meaningful solution that relies continuously on the initial condition and the value of the fractional derivative. Ulam-Hyers stability is a property of equations that states that if a function is approximately satisfying the equation, then there exists an exact solution that is close to the function. Inspired by these findings, in this research work, we established the Ulam-Hyers stability and well-posedness of solutions of pantograph fractional stochastic differential equations (PFSDEs) in the framework of conformable derivatives. In addition, we provided examples to analyze the theoretical results.

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