Langevin Equations with Generalized Proportional Hadamard-Caputo Fractional Derivative

M A Barakat; Ahmed H Soliman; Abd-Allah Hyder

doi:10.1155/2021/6316477

Langevin Equations with Generalized Proportional Hadamard-Caputo Fractional Derivative

Comput Intell Neurosci. 2021 Dec 8:2021:6316477. doi: 10.1155/2021/6316477. eCollection 2021.

Authors

M A Barakat^{1

2}, Ahmed H Soliman², Abd-Allah Hyder^{3

4}

Affiliations

¹ Department of Computer Science, College of Al Wajh, University of Tabuk, Tabuk, Saudi Arabia.
² Department of Mathematics, Faculty of Sciences, Al-Azhar University, Assiut 71524, Egypt.
³ King Khalid University, College of Science, Department of Mathematics, P.O. Box 9004, Abha 61413, Saudi Arabia.
⁴ Department of Engineering Mathematics and Physics, Faculty of Engineering, Al-Azhar University, Cairo, Egypt.

Abstract

We look at fractional Langevin equations (FLEs) with generalized proportional Hadamard-Caputo derivative of different orders. Moreover, nonlocal integrals and nonperiodic boundary conditions are considered in this paper. For the proposed equations, the Hyres-Ulam (HU) stability, existence, and uniqueness (EU) of the solution are defined and investigated. In implementing our results, we rely on two important theories that are Krasnoselskii fixed point theorem and Banach contraction principle. Also, an application example is given to bolster the accuracy of the acquired results.