Systems of fractional Langevin equations of Riemann-Liouville and Hadamard types

Advances in Difference Equations

Jessada Tariboon¹  Weerawat Sudsutad³  Sotiris K Ntouyas⁴ 

[1] Centre of Excellence in Mathematics, CHE, Bangkok, Thailand;Department of Mathematics, University of Ioannina, Ioannina, Greece;Nonlinear Dynamic Analysis Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok, Thailand

关键词: Riemann-Liouville fractional derivative;    Hadamard fractional derivative;    fractional integral boundary conditions;    Langevin equation;    coupled system;    existence;    uniqueness;    fixed point theorems;    34A08;    34A12;    34B15;   

DOI  :  10.1186/s13662-015-0566-8

学科分类：数学（综合）

来源: SpringerOpen

【 摘 要 】

Fractional differential equations have been shown to be very useful in the study of models of many phenomena in various fields of science and engineering, such as physics, chemistry, biology, signal and image processing, biophysics, blood flow phenomena, control theory, economics, aerodynamics, and fitting of experimental data. Much of the work on the topic deals with the governing equations involving Riemann-Liouville- and Caputo-type fractional derivatives. Another kind of fractional derivative is the Hadamard type, which was introduced in 1892. This derivative differs from the aforementioned derivatives in the sense that the kernel of the integral in the definition of the Hadamard derivative contains a logarithmic function of arbitrary exponent. In the present paper we introduce a new class of boundary value problems for Langevin fractional differential systems. The Langevin equation is widely used to describe the evolution of physical phenomena in fluctuating environments. We combine Riemann-Liouville- and Hadamard-type Langevin fractional differential equations subject to Hadamard and Riemann-Liouville fractional integral boundary conditions, respectively. Some new existence and uniqueness results for coupled and uncoupled systems are obtained by using fixed point theorems. The existence and uniqueness of solutions is established by Banach’s contraction mapping principle, while the existence of solutions is derived by using the Leray-Schauder’s alternative. The obtained results are well illustrated with the aid of examples.

【 授权许可】

CC BY