Many different types of fractional calculus have been defined, which may be categorised into broad classes according to their properties and behaviours. Two that much studied in the literature are Hadamard-type tempered calculus. This paper establishes a connection between these two definitions, writing one terms other by making use theory with respect functions. By extending this natural way, ...

The application of q-calculus in operator theory is an active area of research in the last two decades. Several new q-operators were introduced and their approximation behavior was discussed. In the present paper, we study the Stancu type generalization of Beta-Szasz type operators for their q-analogues. We obtain moments and convergence results in terms of higher order modulus of continuity. A...

The applications of q calculus in operator theory is an active area of research in the last two decades. Several new q -operators were introduced and their approximation behavior were discussed. This paper is extension of the paper [South. Asian Bul. Math. (36) (2012), 343–352]. We propose here the Stancu variant of q -Beta-Szász type operators. We estimate the moments and also establish direct...

The aim of this paper is to deduce a discrete version of the fractional Laplacian in matrix form defined on the 1D periodic (cyclically closed) linear chain of finite length. We obtain explicit expressions for this fractional Laplacian matrix and deduce also its periodic continuum limit kernel. The continuum limit kernel gives an exact expression for the fractional Laplacian (Riesz fractional d...

This paper is devoted to the study of discrete fractional calculus; the particular goal is to define and solve well-defined discrete fractional difference equations. For this purpose we first carefully develop the commutativity properties of the fractional sum and the fractional difference operators. Then a ν-th (0 < ν ≤ 1) order fractional difference equation is defined. A nonlinear problem wi...

In this paper, we focus on the existence of Hilfer fractional stochastic differential systems via almost sectorial operators. The main results are obtained by using concepts and ideas from calculus, multivalued maps, semigroup theory, operators, fixed-point technique. We start confirming mild solution Dhage’s theorem. Finally, an example is provided to demonstrate considered Hilferr theory.

In this paper‎, ‎a modern method is presented to solve a class of fractional optimal control problems (FOCPs) indirectly‎. ‎First‎, ‎the necessary optimality conditions for the FOCP are obtained in the form of two fractional differential equations (FDEs)‎. ‎Then‎, ‎the unknown functions are approximated by the hybrid functions‎, ‎including Bernoulli polynomials and Block-pulse functions based o...