In this paper, we introduce the nabla fractional derivative and integral on time scales in Riemann-Liouville sense. We also Gr\"unwald-Letnikov Some of basic properties theorems related to calculus are discussed.

In the present paper, the notion of generalized $(r;g,s,m,varphi)$-preinvex function is applied to establish some new generalizations of Ostrowski type integral inequalities via Caputo $k$-fractional derivatives. At the end, some applications to special means are given.

Integral inequalities have accumulated a comprehensive and prolific field of research within mathematical interpretations. In recent times, strategies fractional calculus become the subject intensive in historical contemporary generations because their applications various branches science. this paper, we concentrate on establishing Hermite–Hadamard Pachpatte-type integral with aid two differen...

Haar wavelets for the solution of fractional integral equations are applied. Fractional Vol-terra and Fredholm integral equations are considered. The proposed method also is used for analysing fractional harmonic vibrations. The efficiency of the method is demonstrated by three numerical examples. Although the conception of the fractional derivatives was introduced already in the middle of the ...

In this article, we propose a robust method to compute the output of a fractional linear system defined through a linear fractional differential equation (FDE) with timevarying coefficients, where the input can be noisy. We firstly introduce an estimator of the fractional derivative of an unknown signal, which is defined by an integral formula obtained by calculating the fractional derivative o...

We review the progress achieved in optical information processing during the last decade by applying fractional linear integral transforms. The fractional Fourier transform and its applications for phase retrieval, beam characterization, space-variant pattern recognition, adaptive filter design, encryption, watermarking, and so forth is discussed in detail. A general algorithm for the fractiona...

This paper is concerned with the discretization of the fractional-order differentiator and integrator, which is the foundation of the digital realization of fractional order controller. Firstly, the parameterized Al-Alaoui transform is presented as a general generating function with one variable parameter, which can be adjusted to obtain the commonly used generating functions (e.g. Euler operat...

In this article, we define the fractional Mellin transform by using Riemann-Liouville fractional integral operator and Caputo fractional derivative of order [Formula: see text] and study some of their properties. Further, some properties are extended to fractional way for Mellin transform.