Abstract
We study a boundary value problem for fractional equations involving two fractional orders. By means of a fixed point theorem, we establish sufficient conditions for the existence and uniqueness of solutions for the fractional equations. In addition, we describe the dynamic behaviors of the fractional Langevin equation by using theG2algorithm.
Highlights
We study a boundary value problem for fractional equations involving two fractional orders
Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in various fields, such as physics, mechanics, chemical technology, population dynamics, biotechnology, and economics
As one of the important topics in the research on differential equations, the boundary value problem has attained a great deal of attention from many researchers and the references therein
Summary
Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in various fields, such as physics, mechanics, chemical technology, population dynamics, biotechnology, and economics (see, e.g., [1,2,3,4,5,6,7]). In [22], Zhong and Lin studied the existence and uniqueness of solutions in the nonlocal and multiple-point boundary value problem for fractional differential equation: cDqu (t) + f (t, u (t)) = 0, 0 < t < 1, 1 < q ≤ 2, u (0) = u0 + g (u) , m−2. Chen studied existence of solutions to nonlinear Langevin equation involving two fractional orders with boundary value conditions: cDβ ( cDα + λ) u (t) = f (t, u (t) , u (t)) ,. As far as we know, there are no papers discussing the existence and numerical simulation of solutions for fractional equations involving two fractional orders with nonlocal boundary conditions.
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