In this paper, we develop theorems on finite and infinite summation formulas by utilizing the q (q,h) anti-difference operators, also extend these core to q(α) (q,h)α difference operators. Several integer order based operator have been published, which gave us idea derive fractional equations for construct a function known as quantum geometric alpha-quantum function, behaves class of series. We...

In this paper, we consider the second-kind Chebyshev polynomials (SKCPs) for the numerical solution of the fractional optimal control problems (FOCPs). Firstly, an introduction of the fractional calculus and properties of the shifted SKCPs are given and then operational matrix of fractional integration is introduced. Next, these properties are used together with the Legendre-Gauss quadrature fo...

Special functions play important roles in applied mathematics. It has been seen that these functions have appeared in different frameworks, such as the mathematical physics 1 , the combinatorial analysis 2 , and the statistics 3 . Indeed, the explicit relationships between special functions and generalised hypergeometric functions have been obtained and mentioned in 4, 5 . Some extension of the...

We prove a new Taylor’s theorem for generalized weighted fractional calculus with nonsingular kernels. The proof is based on the establishment of relations nth-weighted integrals and derivatives. As an application, mean value theorems operators are obtained. Direct corollaries allow one to obtain recent Caputo–Fabrizio, Atangana–Baleanu–Caputo (ABC) ABC

An extension of the general fractional calculus (GFC) an arbitrary order, proposed by Luchko, is formulated. This also based on a multi-kernel approach, in which Laplace convolutions different Sonin kernels are used. The GFC order considered for case intervals (a,b) where ??<a<b??. Examples operators orders proposed.

In the present paper certain operational formulae involving RiemannLiouville and Kober fractional q-integral operators for an analytic function are derived. The usefulness of the main results are exhibited by considering some examples which also yield q-extensions of some known results for ordinary hypergeometric functions of one and more variables.

Multiplicative calculus, also called non-Newtonian represents an alternative approach to the usual calculus of Newton (1643–1727) and Leibniz (1646–1716). This type was first introduced by Grossman Katz it provides a defined calculation, from start, for positive real numbers only. In this investigation, we propose study symmetrical fractional multiplicative inequalities Simpson type. For this, ...

Here we state the main properties of the Caputo, Riemann-Liouville and the Caputo via Riemann-Liouville fractional derivatives and give some general notes on these properties. Some properties given in some recent literatures and used to solve fractional nonlinear partial differential equations will be proved that they are incorrect by giving some counter examples.

In this paper a new function called as K-function, which is an extension of the generalization of the Mittag-Leffler function[10,11] and its generalized form introduced by Prabhakar[20], is introduced and studied by the author in terms of some special functions and derived the relations that exists between the Kfunction and the operators of Riemann-Liouville fractional integrals and derivatives.