scholarly journals Boundary value problem of a nonlinear Langevin equation with two different fractional orders and impulses

2012 ◽  
Vol 2012 (1) ◽  
pp. 200 ◽  
Author(s):  
Guotao Wang ◽  
Lihong Zhang ◽  
Guangxing Song
Keyword(s):  
Boundary Value Problem ◽  
Langevin Equation ◽  
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.

scholarly journals On implicit impulsive Langevin equation involving mixed order derivatives

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Akbar Zada ◽  
Rizwan Rizwan ◽  
Jiafa Xu ◽  
Zhengqing Fu
Keyword(s):  
Boundary Value Problem ◽  
Langevin Equation ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Sufficient Conditions ◽  
Nonlocal Boundary Value Problem ◽  
Qualitative Properties ◽  
Mixed Order

AbstractIn this paper, we consider a nonlocal boundary value problem of nonlinear implicit impulsive Langevin equation involving mixed order derivatives. Sufficient conditions are constructed to discuss the qualitative properties like existence and Ulam’s stability of the proposed problem. The main result is verified by an example.

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
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