In this article we present the notions of adjoint differential expressions for fractional-order differential expressions, adjoint boundary conditions for fractional differential equations, and adjoint fractional-order operators. These notions are based on new formulas obtained for various types of fractional derivatives. The introduced notions can be used in many fields of modelling and control...

We consider the existence of at least one positive solution of the problem –D0+u(t) = f (t,u(t)), 0 < t < 1, under the circumstances that u(0) = 0, u(1) = H1(φ(u)) + ∫ E H2(s,u(s))ds, where 1 < α < 2, D α 0+ is the Riemann-Liouville fractional derivative, and u(1) = H1(φ(u)) + ∫ E H2(s,u(s))ds represents a nonlinear nonlocal boundary condition. By imposing some relatively mild structural condit...

At the end of the 19th century Liouville and Riemann introduced the notion of a fractional-order derivative, and in the latter half of the 20th century the concept of the so-called Grünewald-Letnikov fractional-order discrete difference has been put forward. In the paper a predictive controller for MIMO fractional-order discrete-time systems is proposed, and then the concept is extended to nonl...

In this paper, by using fixed-point methods, we study the existence and uniqueness of a solution for the nonlinear fractional differential equation boundary-value problem D(α)u(t)=f(t,u(t)) with a Riemann-Liouville fractional derivative via the different boundary-value problems u(0)=u(T), and the three-point boundary condition u(0)=β(1)u(η) and u(T)=β(2)u(η), where T>0, t∈I=[0,T], 0<α<1, 0<η<T,...

In this paper, we consider a type of fractional diffusion equation (FDE) with variable coefficient on a finite domain. Firstly, we utilize a second-order scheme to approximate the Riemann-Liouville fractional derivative and present the finite difference scheme. Specifically, we discuss the Crank-Nicolson scheme and solve it in matrix form. Secondly, we prove the stability and convergence of the...

In this article, we verify the existence and uniqueness of a positive and nondecreasing solution for nonlinear boundary value problem of fractional differential equation in the form Dα 0+ x(t) + f(t, x(t)) = 0, 0 < t < 1, 2 < α ≤ 3, x(0) = x′(0) = 0, x′(1) = βx(ξ), where Dα 0+ denotes the standard Riemann-Liouville fractional derivative, 0 < ξ < 1 and 0 < β ξ < α− 1. Our analysis relies on the ...

In this paper, we devote to investigation of the existence of positive solutions for the boundary value problem of nonlinear fractional differential equations { D0+u(t)+ f (t,u(t)) = 0, 0 < t < 1, u(0) = u′(0) = · · ·u(n−2)(0) = D0+u(1), where D0+ , D β 0+ are the standard Riemann-Liouville fractional derivative, n− 1 < α n , n−2 β n−1 , n 3 . By means of constructing an exact cone of the Banac...