The analysis of fractional neutral stochastic differential equations in <inline-formula id="math-09-07-845-M1"><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" href="math-09-07-845-M1.jpg"/></inline-formula> space

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.2024845

AIMS Mathematics

2024, Volume 9, Issue 7: 17386-17413. doi: 10.3934/math.2024845

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The analysis of fractional neutral stochastic differential equations in space

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 MuslimTown, Lahore 54600, Pakistan
3.
Department of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam Bin Abdulaziz University, Alkharj 11942, Saudi Arabia
4.
Saveetha School of Engineering, SIMATS, Chennai, India
5.
Department of Physics, College of Sciences, University of Bisha, Bisha 61922, Saudi Arabia

Received: 02 February 2024 Revised: 11 April 2024 Accepted: 08 May 2024 Published: 20 May 2024
MSC : 34A07, 34A08, 60G22

After extensive examination, scholars have determined that many dynamic systems exhibit intricate connections not only with their current and past states but also with the delay function itself. As a result, their focus shifts towards fractional neutral stochastic differential equations, which find applications in diverse fields such as biology, physics, signal processing, economics, and others. The fundamental principles of existence and uniqueness of solutions to differential equations, which guarantee the presence of a solution and its uniqueness for a specified equation, are pivotal in both the mathematical and physical realms. A crucial approach for analyzing complex systems of differential equations is the utilization of the averaging principle, which simplifies problems by approximating existing ones. Applying contraction mapping principles, we present results concerning the concepts of existence and uniqueness for the solutions of fractional neutral stochastic differential equations. Additionally, we present Ulam-type stability and the averaging principle results within the framework of space. This exploration involved the utilization of Jensen's, Gröenwall-Bellman's, Hölder's, Burkholder-Davis-Gundy's inequalities, and the interval translation technique. Our findings are established within the context of the conformable fractional derivative, and we provide several examples to aid in comprehending the theoretical outcomes.
- fractional neutral stochastic differential equations,
- conformable fractional derivative,
- averaging principle,
- existence and uniqueness
Citation: Wedad Albalawi, Muhammad Imran Liaqat, Fahim Ud Din, Kottakkaran Sooppy Nisar, Abdel-Haleem Abdel-Aty. The analysis of fractional neutral stochastic differential equations in space[J]. AIMS Mathematics, 2024, 9(7): 17386-17413. doi: 10.3934/math.2024845

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Abstract

After extensive examination, scholars have determined that many dynamic systems exhibit intricate connections not only with their current and past states but also with the delay function itself. As a result, their focus shifts towards fractional neutral stochastic differential equations, which find applications in diverse fields such as biology, physics, signal processing, economics, and others. The fundamental principles of existence and uniqueness of solutions to differential equations, which guarantee the presence of a solution and its uniqueness for a specified equation, are pivotal in both the mathematical and physical realms. A crucial approach for analyzing complex systems of differential equations is the utilization of the averaging principle, which simplifies problems by approximating existing ones. Applying contraction mapping principles, we present results concerning the concepts of existence and uniqueness for the solutions of fractional neutral stochastic differential equations. Additionally, we present Ulam-type stability and the averaging principle results within the framework of space. This exploration involved the utilization of Jensen's, Gröenwall-Bellman's, Hölder's, Burkholder-Davis-Gundy's inequalities, and the interval translation technique. Our findings are established within the context of the conformable fractional derivative, and we provide several examples to aid in comprehending the theoretical outcomes.

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