This manuscript mainly focused on the nonlocal controllability of Hilfer fractional stochastic differential equations via almost sectorial operators. The key ideas study are illustrated by using from calculus, fixed point technique, and measures noncompactness. Then, authors establish new criteria for mild existence solutions derive fundamental characteristics a system. In addition, researchers...

Abstract. We present relatively simple and direct proofs of the integral representations established recently in [7]. An algorithm is then furnished and applied to obtain new classes of integral formulas for the multivariable hypergeometric functions, thereby, providing generalizations to the results of [7]. Also, an operational formula involving fractional calculus operators for an analytic fu...

By making use of the familiar Salagean derivatives, a systematic investigation of a certain subclass of uniformly convex and starlike functions with negative coefficients is presented. In addition to finding various coefficient bounds and a number of characterization and distortion theorems, a number of other potentially interesting properties of this class of functions are investigated. Finall...

Abstract. By means of the extended beta function B (κl) b,d , we introduce new extensions of the generalized hypergeometric functions and present some new integral and series representations (including the one obtained by adopting the well-known Ramanujan’s Master Theorem). Further generalizations are also considered. We point out the usefulness of some of the the results by showing their conne...

Stochastic diierential equations in R n with random coeecients are considered where one continuous driving process admits a generalized quadratic variation process. The latter and the other driving processes are assumed to possess sample paths in the fractional Sobolev space W 2 for some > 1=2. The stochastic integrals are determined as anticipating forward integrals. A pathwise solution proced...

Let A be a sectorial operator on a non-atomic Lp-space, 1 ≤ p < ∞, whose resolvent consists of integral operators, or more generally, has a diffuse representation. Then the fractional domain spaces D(Aα) for α ∈ (0, 1) do not coincide with the real interpolation spaces of (Lq , D(A)). As a consequence, we obtain that no such operator A has a bounded H∞-calculus if p = 1.

Fractional calculus and fractional differential equations are popular in describing anomalous diffusion, ground water flow and transport, and the price fluctuation in finance, etc. Some numerical methods are developed to solve the fractional ordinary differential equations. However, for most of these methods it seems that we always have to make a trade-off between efficiency and accuracy becaus...

Applying a bi-local mean field approximation to the fractional quantum Hall state of ν = 1/3, we obtain charged and neutral vortex mean field solutions numerically. We calculate the mean field energy and the fluctuation corrections. The charged vortex has a fractional charge and a fractional angular momentum. The neutral vortex is a bound state of two charged vortices and has a zero charge and ...

Fractional calculus, which has almost the same history as classic calculus, did not attract enough attention for a long time. However, in recent decades, fractional calculus and fractional differential equations become more and more popular because of its powerful potential applications. A large number of new differential equations (models) that involve fractional calculus are developed. These ...