and Applied Analysis 3 Definition 2.2. The Riemann-Liouville fractional derivative of order α > 0 of a function f : 0,∞ → R is given by D 0 f t 1 Γ n − α ( d dt )n ∫ t 0 f s t − s α−n 1 ds, 2.2 where n α 1 and α denotes the integer part of α. The following two lemmas can be found in 17, 22 . Lemma 2.3. Let α > 0 and u ∈ C 0, 1 ∩ L1 0, 1 . Then fractional differential equation D 0 u t 0 2.3

Several oscillation criteria are established for nonlinear fractional differential equations of the form{ a(t) [( r(t)g ( D−x(t) ))′]η}′ −F(t,∫ ∞ t (v− t)−αx(v)dv ) = 0, where D−x(t) is the Liouville right-side fractional derivative of order α ∈ (0,1) of x(t),η = 2n+1 2m+1 , and n,m∈N . F(t,G)∈C([t0,∞)×R;R) , and there exists function q(t)∈ C([t0,∞);R+) such that F(t,G) Gη q(t) for G = 0 and x ...

in this paper, the homotopy perturbation method (hpm) is applied to obtain an approximate solution of the fractional bratu-type equations. the convergence of the method is also studied. the fractional derivatives are described in the modied riemann-liouville sense. the results show that the proposed method is very ecient and convenient and can readily be applied to a large class of fractional...

Abstract We study distributed optimal control problems, governed by space-time fractional parabolic equations (STFPEs) involving time-fractional Caputo derivatives and spatial of Sturm-Liouville type. first prove existence uniqueness solutions STFPEs on an open bounded interval their regularity. Then we show to a quadratic problem. derive adjoint problem using the right-Caputo derivative in tim...

A fractional wave equation with a Riemann–Liouville derivative is considered. An arbitrary self-adjoint operator discrete spectrum was taken as the elliptic part. We studied inverse problem of determining order time derivative. By setting value projection solution onto first eigenfunction at fixed point in an additional condition, uniquely restored. The abstract allows us to include many models...

in this article we implement an operational matrix of fractional integration for legendre polynomials. we proposed an algorithm to obtain an approximation solution for fractional differential equations, described in riemann-liouville sense, based on shifted legendre polynomials. this method was applied to solve linear multi-order fractional differential equation with initial conditions, and the...

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...