Solution of the fractional Langevin equation and the Mittag–Leffler functions

2009 ◽  
Vol 50 (6) ◽  
pp. 063507 ◽  
Author(s):  
R. Figueiredo Camargo ◽  
Ary O. Chiacchio ◽  
R. Charnet ◽  
E. Capelas de Oliveira
Keyword(s):  
Langevin Equation ◽  
Fractional Langevin Equation

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. 

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