scholarly journals Existence of Solutions to Nonlinear Langevin Equation Involving Two Fractional Orders with Boundary Value Conditions

2011 ◽  
Vol 2011 ◽  
pp. 1-17 ◽  
Author(s):  
Anping Chen ◽  
Yi Chen
Keyword(s):  
Langevin Equation ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Nonlinear Langevin Equation ◽  
Fractional Orders

scholarly journals On fractional Langevin equation involving two fractional orders in different intervals

2019 ◽  
Vol 24 (6) ◽  
Author(s):  
Hamid Baghani ◽  
J. Nieto
Keyword(s):  
Langevin Equation ◽  
Existence And Uniqueness ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Uniqueness Of Solutions ◽  
Numerical Examples ◽  
Fractional Equations ◽  
Fractional Langevin Equation ◽  
Nonlinear Langevin Equation ◽  
Fractional Orders

In this paper, we study a nonlinear Langevin equation involving two fractional orders  α ∈ (0; 1] and β ∈ (1; 2] with initial conditions. By means of an interesting fixed point theorem, we establish sufficient conditions for the existence and uniqueness of solutions for the fractional equations. Some illustrative numerical examples are also discussed. 

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

2010 ◽  
Vol 2010 ◽  
pp. 1-10 ◽  
Author(s):  
Bashir Ahmad ◽  
Juan J. Nieto
Keyword(s):  
Langevin Equation ◽  
Dirichlet Boundary ◽  
Contraction Mapping Principle ◽  
Dirichlet Boundary Conditions ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Single Index ◽  
New Type ◽  
Nonlinear Langevin Equation ◽  
Physical Phenomena ◽  
Fractional Orders

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.

A study of nonlinear Langevin equation involving two fractional orders in different intervals

2012 ◽  
Vol 13 (2) ◽  
pp. 599-606 ◽  
Author(s):  
Bashir Ahmad ◽  
Juan J. Nieto ◽  
Ahmed Alsaedi ◽  
Moustafa El-Shahed
Keyword(s):  
Langevin Equation ◽  
Nonlinear Langevin Equation ◽  
Fractional Orders

scholarly journals Langevin equation in terms of conformable differential operators

2020 ◽  
Vol 28 (2) ◽  
pp. 5-14
Author(s):  
Bashir Ahmad ◽  
Ravi P. Agarwal ◽  
Madeaha Alghanmi ◽  
Ahmed Alsaedi
Keyword(s):  
Functional Analysis ◽  
Boundary Conditions ◽  
Langevin Equation ◽  
Differential Operators ◽  
Existence Of Solutions ◽  
Integral Boundary Conditions ◽  
Integral Boundary ◽  
Nonlinear Langevin Equation

AbstractIn this paper, we establish sufficient criteria for the existence of solutions for a new kind of nonlinear Langevin equation involving conformable differential operators of different orders and equipped with integral boundary conditions. We apply the modern tools of functional analysis to derive the desired results for the problem at hand. Examples are constructed for the illustration of the obtained results.

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