A high-order finite difference scheme is proposed for solving time fractional heat equations. The time fractional derivative is described in the Riemann-Liouville sense. In the proposed scheme a new second-order discretization, which is based on Crank-Nicholson method, is applied for the time fractional part and fourth-order accuracy compact approximation is applied for the second-order space d...

In this paper, we consider a time-dependent diffusion problem with two-sided Riemann-Liouville fractional derivatives. By introducing a fractional-order flux as auxiliary variable, we establish the saddle-point variational formulation, based on which we employ a locally conservative mixed finite element method to approximate the unknown function, its derivative and the fractional flux in space ...

We extend the classical treatment of the Ince equation to include the effect of a fractional derivative term of order a > 0 and amplitude c. A Fourier expansion is used to determine the eigenvalue curves að Þ in function of the parameter , the stability domains, and the periodic stable solutions of the fractional Ince equation. Two important observations are the detachment of the eigenvalue cur...

in this paper, we prove the existence and uniqueness results to the random fractional functional differential equations under assumptions more general than the lipschitz type condition. moreover, the distance between exact solution and appropriate solution, and the existence extremal solution of the problem is also considered.

and Applied Analysis 3 where n α 1, α denotes the integer part of number α, provided that the right side is pointwise defined on 0, ∞ . Definition 2.2 see 20 . The Riemann-Liouville fractional integral of order α > 0 of a function f : 0, ∞ → R is given by I 0 f t 1 Γ α ∫ t 0 t − s α−1f s ds, 2.2 provided that the right side is pointwise defined on 0, ∞ . From the definition of the Riemann-Liouv...

and Applied Analysis 3 2. Background Materials and Lemmas For the convenience of the readers, in this section, we provide definitions of RiemannLiouville fractional integral and fractional derivative and some of their basic properties which will be helpful in the forth coming investigations. Definition 2.1 see 2, 5 . For a function φ : a,∞ → R, the Riemann-Liouville fractional integral of order...

The paper is concerned with existence of mild solution of evolution equation with Hilfer fractional derivative which generalized the famous Riemann–Liouville fractional derivative. By noncompact measure method, we obtain some sufficient conditions to ensure the existence of mild solution. Our results are new and more general to known results. Nowadays, fractional calculus receives increasing at...

Anomalous dispersion is observed throughout hydrology, yielding a contaminant plume with heavy leading tails. The fractional advection dispersion equation (FADE) captures this behavior by replacing the second-order spatial derivative with a Riemann-Liouville (RL) fractional derivative. The RL fractional derivative is a nonlocal operator and models large jumps of solute particles in heterogeneou...

In this article, the modified simple equation method has been extended to celebrate the exact solutions of nonlinear partial time-space differential equations of fractional order. Firstly, the fractional complex transformation has been implemented to convert nonlinear partial fractional differential equations into nonlinear ordinary differential equations. Afterwards, modified simple equation m...