Solvability of Nonlinear Langevin Equation Involving Two Fractional Orders with Dirichlet Boundary Conditions

Bashir Ahmad,Juan J Nieto

doi:10.1155/2010/649486

Abstract

We study a Dirichlet boundary value problem for Langevin equation involving two fractional orders. Langevin equation has been widely used to describe the evolution of physical phenomena in fluctuating environments. However, ordinary Langevin equation does not provide the correct description of the dynamics for systems in complex media. In order to overcome this problem and describe dynamical processes in a fractal medium, numerous generalizations of Langevin equation have been proposed. One such generalization replaces the ordinary derivative by a fractional derivative in the Langevin equation. This gives rise to the fractional Langevin equation with a single index. Recently, a new type of Langevin equation with two different fractional orders has been introduced which provides a more flexible model for fractal processes as compared with the usual one characterized by a single index. The contraction mapping principle and Krasnoselskii′s fixed point theorem are applied to prove the existence of solutions of the problem in a Banach space.

Highlights

Fractional differential equations have recently gained much importance and attention
We study a Dirichlet boundary value problem for Langevin equation involving two fractional orders
The study of fractional differential equations ranges from the theoretical aspects of existence and uniqueness of solutions to the analytic and numerical methods for finding solutions

Summary

Introduction

Fractional differential equations have recently gained much importance and attention. The study of fractional differential equations ranges from the theoretical aspects of existence and uniqueness of solutions to the analytic and numerical methods for finding solutions. One of the possible generalizations of Langevin equation is to replace the ordinary derivative by a fractional derivative in it. In 18 , the authors studied a new type of Langevin equation with two different fractional orders. We study a Dirichlet boundary value problem of Langevin equation with two different fractional orders. We first use the contraction mapping principle to prove the existence and uniqueness of the solution of problem 1.1 in a Banach space. A function x ∈ C with its Caputo derivative of fractional order existing on 0, 1 is a solution of 1.1 if it satisfies 1.1. There exists z ∈ M such that z Az Bz

Existence of Solutions

Lr M ΓαΓβ 1

Conclusions