Fractional Langevin Equation Involving Two Fractional Orders: Existence and Uniqueness Revisited

Fractional Langevin Equation Involving Two Fractional Orders: Existence and Uniqueness Revisited

We consider the nonlinear fractional Langevin equation involving two fractional orders with initial conditions. Using some basic properties of Prabhakar integral operator, we find an equivalent Volterra integral equation with two parameter Mittag–Leffler function in the kernel to the mentioned equat...

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Main Authors: Hossein Fazli, HongGuang Sun, Juan J. Nieto
Format: Article
Language:English
Published: MDPI AG 2020-05-01
Series:Mathematics
Subjects:
fractional Langevin equation
Mittag–Leffler function
Prabhakar integral operator
existence
uniqueness
Online Access:https://www.mdpi.com/2227-7390/8/5/743
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spelling doaj-b471c2d2f2be47f38e7e4d239fb758f82020-11-25T02:20:13ZengMDPI AGMathematics2227-73902020-05-01874374310.3390/math8050743Fractional Langevin Equation Involving Two Fractional Orders: Existence and Uniqueness RevisitedHossein Fazli0HongGuang Sun1Juan J. Nieto2State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, College of Mechanics and Materials, Hohai University, Nanjing 210098, ChinaState Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, College of Mechanics and Materials, Hohai University, Nanjing 210098, ChinaDepartment of Statistics, Mathematical Analysis and Optimization, Institute of Mathematics, University of Santiago de Compostela, 15782 Santiago de Compostela, SpainWe consider the nonlinear fractional Langevin equation involving two fractional orders with initial conditions. Using some basic properties of Prabhakar integral operator, we find an equivalent Volterra integral equation with two parameter Mittag–Leffler function in the kernel to the mentioned equation. We used the contraction mapping theorem and Weissinger’s fixed point theorem to obtain existence and uniqueness of global solution in the spaces of Lebesgue integrable functions. The new representation formula of the general solution helps us to find the fixed point problem associated with the fractional Langevin equation which its contractivity constant is independent of the friction coefficient. Two examples are discussed to illustrate the feasibility of the main theorems.https://www.mdpi.com/2227-7390/8/5/743fractional Langevin equationMittag–Leffler functionPrabhakar integral operatorexistenceuniqueness
collection DOAJ
language English
format Article
sources DOAJ
author Hossein Fazli
HongGuang Sun
Juan J. Nieto
spellingShingle Hossein Fazli
HongGuang Sun
Juan J. Nieto
Fractional Langevin Equation Involving Two Fractional Orders: Existence and Uniqueness Revisited
Mathematics
fractional Langevin equation
Mittag–Leffler function
Prabhakar integral operator
existence
uniqueness
author_facet Hossein Fazli
HongGuang Sun
Juan J. Nieto
author_sort Hossein Fazli
title Fractional Langevin Equation Involving Two Fractional Orders: Existence and Uniqueness Revisited
title_short Fractional Langevin Equation Involving Two Fractional Orders: Existence and Uniqueness Revisited
title_full Fractional Langevin Equation Involving Two Fractional Orders: Existence and Uniqueness Revisited
title_fullStr Fractional Langevin Equation Involving Two Fractional Orders: Existence and Uniqueness Revisited
title_full_unstemmed Fractional Langevin Equation Involving Two Fractional Orders: Existence and Uniqueness Revisited
title_sort fractional langevin equation involving two fractional orders: existence and uniqueness revisited
publisher MDPI AG
series Mathematics
issn 2227-7390
publishDate 2020-05-01
description We consider the nonlinear fractional Langevin equation involving two fractional orders with initial conditions. Using some basic properties of Prabhakar integral operator, we find an equivalent Volterra integral equation with two parameter Mittag–Leffler function in the kernel to the mentioned equation. We used the contraction mapping theorem and Weissinger’s fixed point theorem to obtain existence and uniqueness of global solution in the spaces of Lebesgue integrable functions. The new representation formula of the general solution helps us to find the fixed point problem associated with the fractional Langevin equation which its contractivity constant is independent of the friction coefficient. Two examples are discussed to illustrate the feasibility of the main theorems.
topic fractional Langevin equation
Mittag–Leffler function
Prabhakar integral operator
existence
uniqueness
url https://www.mdpi.com/2227-7390/8/5/743
work_keys_str_mv AT hosseinfazli fractionallangevinequationinvolvingtwofractionalordersexistenceanduniquenessrevisited
AT hongguangsun fractionallangevinequationinvolvingtwofractionalordersexistenceanduniquenessrevisited
AT juanjnieto fractionallangevinequationinvolvingtwofractionalordersexistenceanduniquenessrevisited
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