and Dn denotes the derivative operator ∂/∂x1, . . . ,∂xn. The operators in (1.1) provide multidimensional generalizations to the well-known one-dimensional Riemann-Liouville andWeyl fractional integral operators defined in [5] (see also [1]). The paper [7] considers several formulas and interesting properties of (1.1). By invoking the Gauss hypergeometric function 2F1(α,β;γ;x), the following ge...

where 1 < α < 2, 0 < βi < 1, i = 1, 2, . . . ,m – 2, 0 < η1 < η2 < · · · < ηm–2 < 1, ∑m–2 i=1 βiη α–1 i < 1, D α 0+ is the standard Riemann–Liouville derivative. Here our nonlinearity f may be singular at u = 0. As an application of Green’s function, we give some multiple positive solutions for singular positone and semipositone boundary value problems by means of the Leray–Schauder nonlinear a...

This paper aims to obtain the approximate solution of time-fractional advection-dispersion equation (FADE) involving Jumarie's modification of Riemann-Liouville derivative by the fractional variational iteration method (FVIM). FVIM provides an analytical approximate solution in the form of a convergent series. Some examples are given and the results indicate that the FVIM is of high accuracy, m...

The paper is devoted to studying the solutions of boundary value problem for nonlinear fractional differential equation with Riemann-Liouville type history-state-based variable-order derivative. Using Schauder fixed point theorem and Banach thoerem, we prove existence uniqueness in Hölder space. Lastly, two examples are given show applicability theorems.

This research determines an unknown source term in the fractional diffusion equation with Riemann–Liouville derivative. problem is ill-posed. Conditional stability for inverse can be given. Further, a Tikhonov regularization method was applied to regularize problem. In theoretical results, we propose priori and posteriori parameter choice rules obtain convergence estimates.

We establish some properties of the bilateral Riemann–Liouville fractional derivative Ds. set notation, and study associated Sobolev spaces order s, denoted by Ws,1(a,b), bounded variation BVs(a,b). Examples, embeddings compactness related to these are addressed, aiming a functional framework suitable for variational models image analysis.

In this paper, based on certain variable transformation, we apply the known (G’/G) method to seek exact solutions for three fractional partial differential equations: the space fractional (2+1)-dimensional breaking soliton equations, the space-time fractional Fokas equation, and the spacetime fractional Kaup-Kupershmidt equation. The fractional derivative is defined in the sense of modified Rie...

In this paper, we deal with a class of nonlinear fractional nonautonomous evolution equations with delay by using Hilfer fractional derivative, which generalizes the famous Riemann-Liouville fractional derivative. The definition of mild solutions for the studied problem was given based on an operator family generated by the operator pair [Formula: see text] and probability density function. Com...

This paper explicitly computes the transition densities of a spectrally negative stable process with index greater than one, reflected at its infimum. First we derive the forward equation using the theory of sun-dual semigroups. The resulting forward equation is a boundary value problem on the positive half-line that involves a negative Riemann-Liouville fractional derivative in space, and a fr...