On a Langevin equation involving Caputo fractional proportional derivatives with respect to another function

Zaid Laadjal; Fahd Jarad; Zaid Laadjal; Fahd Jarad

doi:10.3934/math.2022075

AIMS Mathematics

2022, Volume 7, Issue 1: 1273-1292. doi: 10.3934/math.2022075

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Research article

On a Langevin equation involving Caputo fractional proportional derivatives with respect to another function

Zaid Laadjal ¹,
Fahd Jarad ^{2,3
,
,}

1.
Department of Mathematics and Computer Sciences, Abbes Laghrour University, Khenchela, 40004, Algeria
2.
Department of Mathematics, Çankaya University, 06790 Ankara, Turkey
3.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan

Received: 23 August 2021 Accepted: 14 October 2021 Published: 22 October 2021
MSC : 26A33, 34A08, 58C30

In this work, we introduce and study a class of Langevin equation with nonlocal boundary conditions governed by a Caputo fractional order proportional derivatives of an unknown function with respect to another function. The qualitative results concerning the given problem are obtained with the aid of the lower regularized incomplete Gamma function and applying the standard fixed point theorems. In order to homologate the theoretical results we obtained, we present two examples.
Citation: Zaid Laadjal, Fahd Jarad. On a Langevin equation involving Caputo fractional proportional derivatives with respect to another function[J]. AIMS Mathematics, 2022, 7(1): 1273-1292. doi: 10.3934/math.2022075

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Abstract

In this work, we introduce and study a class of Langevin equation with nonlocal boundary conditions governed by a Caputo fractional order proportional derivatives of an unknown function with respect to another function. The qualitative results concerning the given problem are obtained with the aid of the lower regularized incomplete Gamma function and applying the standard fixed point theorems. In order to homologate the theoretical results we obtained, we present two examples.

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