In this paper, we study the existence and nonexistence of positive solutions for the nonlinear differential equation of fractional order 0 ( ) ( ) ( ( )) 0, 0 1, 2 3, D u t a t f u t t α λ α + + = ≤ ≤ < ≤ where 0 D α + is the standard Riemann-Liouville fractional derivative, subject to the boundary conditions ( ) ( ) (0) (0) 0, (1) . u u u u u β η γ ξ ′ ′ ′ ′ = = = + Our analysis relies on Kras...

In this article, we survey the asymptotic stability analysis of fractional differential systems with the Prabhakar fractional derivatives. We present the stability regions for these types of fractional differential systems. A brief comparison with the stability aspects of fractional differential systems in the sense of Riemann-Liouville fractional derivatives is also given. 

A universal approach by Laplace transform to the variational iteration method for fractional derivatives with the nonsingular kernel is presented; in particular, the Caputo-Fabrizio fractional derivative and the Atangana-Baleanu fractional derivative with the non-singular kernel is considered. The analysis elaborated for both non-singular kernel derivatives is shown the necessity of considering...

A system of nonlinear fractional differential equations with the Riemann–Liouville derivative is considered. Lipschitz stability in time for studied defined and studied. This connected singularity at initial point. Two types derivatives Lyapunov functions among are applied to obtain sufficient conditions property. Some examples illustrate results.

where D−y is the Liouville right-sided fractional derivative of order a Î (0,1) of y and h >0 is a quotient of odd positive integers. We establish some oscillation criteria for the equation by using a generalized Riccati transformation technique and an inequality. Examples are shown to illustrate our main results. To the best of author’s knowledge, nothing is known regarding the oscillatory beh...

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 modied 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 p...

This survey paper is concerned with some of the most recent results on Lyapunov-type inequalities for fractional boundary value problems involving a variety derivative operators and conditions. Our work deals Caputo, Riemann-Liouville, ?-Caputo, ?-Hilfer, hybrid, Caputo-Fabrizio, Hadamard, Katugampola, Hilfer-Katugampola, p-Laplacian, proportional operators.

This article is in continuation of the authors research attempts to derive computational solutions of an unified reaction-diffusion equation of distributed order associated with Caputo derivatives as the time-derivative and Riesz-Feller derivative as space derivative. This article presents computational solutions of distributed order fractional reaction-diffusion equations associated with Riema...

The fractional diffusion equation that is constructed replacing the time derivative with a fractional derivative, (0)D(alpha)(t) f = C(alpha) theta(2) f/theta x(2), where (0)D(alpha)(t) is the Riemann-Liouville derivative operator, is characterized by a probability density that decays with time as t(alpha -1) (alpha < 1) and an initial condition that diverges as t -->0 [R. Hilfer, J. Phys. Chem...

In this paper fractional generalization of Liouville equation is considered. We derive fractional analog of normalization condition for distribution function. Fractional generalization of the Liouville equation for dissipative and Hamiltonian systems was derived from the fractional normalization condition. This condition is considered as a normalization condition for systems in fractional phase...