This paper investigates a new approach to studying several fractional derivatives in the distributional sense based on the products of distributions and the delta sequence with compact support. Furthermore, we consider an asymptotic expression to the fractional derivative of the delta function and show that it is the first-order approximation in the Schwartz space. At the end of paper, we provide several asymptotic formulas to more complicated fractional derivatives of distributions.

The main purpose of this paper is to study the existence of solutions for the nonlinear fractional partial integrodifferential equations with Dirichlet boundary condition. Under suitable assumption the results are established by using the Leray-Schauder fixed point theorem and Arzela-Ascoli theorem. An example is provided to illustrate the main result.

Fractional calculus and the fractional differential equations have appeared in many physical and engineering processes. Therefore, an efficient and suitable method to solve them is very important. In this paper, novel numerical methods are introduced based on the fractional order of the Chebyshev orthogonal functions (FCF) with Tau and collocation methods to solve differential equations of the arbitrary (integer or fractional) order. The FCFs are obtained from the classical Chebyshev polynomials of the first kind. Also, the operational matrices of the fractional derivative and the product for the FCFs have been constructed. To show the efficiency and capability of these methods we have solved some well-known problems: the momentum, the Bagley-Torvik, and the Lane-Emden differential equations, then have compared our results with the famous methods in other papers.

Projection equations arise in several optimization problems and possess significant applications in many areas of science and engineering. In this paper, we propose a fractional-order projection neural network to solve quadratic programming problems. We study stability and synchronization for a class of delayed projection neural networks of mixed type via impulsive control. Using concepts of fractional calculus, we investigate the existence of solution and study its global asymptotic stability. Moreover, we propose an effective impulsive control scheme to achieve synchronization for the system. We demonstrate the validity and transient behaviour of the proposed neural network with the help of suitable examples.

The aim of this research paper is to establish the Hardy-type inequalities for Hilfer fractional derivative and generalized fractional integral involving Mittag-Leffer function in its kernel using convex and increasing functions.

In this paper, we introduce $\alpha-\psi$ type contractive mapping in rectangular metric space satisfying certain admissibility conditions and prove a fixed point result for such mapping in complete and Hausdorff rectangular metric space. Some examples are given to justify our result. Also we have shown that the existence of solution of a nonlinear fractional differential equation can be guaranteed, as an application of our result.

In this paper, the one-dimensional Laplace transform method is evolved to solve linear one-dimensional fractional Volterra integro-differential equations with initial conditions and extend this study by taking multi-dimensional Laplace transform method of the linear fractional multidimensional Volterra integro-differential equations to find solution of the initial and boundary value problems. It is noticed that the suggested methods is suitable to find solution for problems. The results of the noticed methods are supporter, easily and active.

Modeling the motion and propagation characteristics of waves have importance in coastal, ocean and maritime engineering. Especially, waves are the major source of environmental actions on beaches or on man-made fixed or floating structures in most geographical areas. So Maccari system has great application in mentioned areas. The modified KdV is ion acoustic perturbations evolution model in a plasma with two negative ion components which have different temperatures. As for the KdV equation, the modified ZK (mZK) equation arises naturally as weakly two-dimensional variations of the mKdV equation. In this paper authors used functional variable method(FVM) for the first time to obtain exact travelling wave and soliton solutions of conformable fractional modified KdV-Zakharov-Kuznetsov(mKdv-ZK) equation and Maccari system. As a consequence, new solutions are obtained and it is seen that FVM is an valuable and efficient tool for solving nonlinear equations and systems where the derivatives defined by means of conformable fractional derivative.

In this paper, the simulation of the time-fractional-forced-damped-wave equation (the diffusion advection forced wave) is given for different parameters. The common finite difference rules beside the backward Grünwald–Letnikov scheme are used to find the approximation solution of this model. The paper discusses also the effects of the memory, the internal force (resistance) and the external force on the travelling wave. We follow the waves till they reach their stationary waves. The Von-Neumann stability condition is also considered and discussed. Besides the simulation of the time evolution of the approximation solution of the classical and time-fractional model, the stationary solutions are also simulated. All the numerical results are compared for different values of time.

In this article we examine the densities of a product and a ratio of two real positive scalar random variables $x_1$ and $x_2$, which are statistically independently distributed, and we consider the density of the product $u_1=x_1x_2$ as well as the density of the ratio $u_2=\frac{x_2}{x_1}$ and show that Kober operator of the second kind is available as the density of $u_1$ and Kober operator of the first kind is available as the density of $u_2$ when $x_1$ has a type-1 beta density and $x_2$ has an arbitrary density. We also give interpretations of Kober operators of the second and first kind as Mellin convolution for a product and ratio respectively. Then we look at various types of generalizations of the idea thereby obtaining a large collection of operators which can all be called generalized Kober operators. One of the generalizations considered is the pathway idea where one can move from one family of operators to another family and yet another family and eventually end up with an exponential form. Common generalizations in terms of a Gauss' hypergeometric series is also given a statistical interpretation and put on a more general structure so that the standard generalizations given by various authors, including Saigo operators, are given statistical interpretations and are derivable as special cases of the general structure considered in this article.

A new multiple Čebyšev type functional is introduced by making use of the generalized fractional integral operator with hypergeometric kernel and the notion of permanent of matrix analysis. Inequalities for this functional are established for synchronous functions.

In this paper, we study convergence characteristics of a class of continuous time fractional-order cellular neural network containing delay. Using the method of Lyapunov and Mittag-Leffer functions, we derive sufficient condition for global Mittag-Leffer stability, which further implies global asymptotic stability of the system equilibrium. Based on the theory of fractional calculus and the generalized Gronwall inequality, we approximate the solution of the corresponding neural network model using discretization method by piecewise constant argument and obtain some easily verifiable conditions, which ensures that the discrete-time analogues preserve the convergence dynamics of the continuous-time networks. In the end, we give appropriate examples to validate the proposed results, illustrating advantages of the discrete-time analogue over continuous-time neural network for numerical simulation.

In this paper, we introduce and define a new metric on the space of fuzzy continuous functions in the fractional calculus. Regarding this metric and using the well-known Banach fixed point theorem, we provide some conditions that guarantee the existence and uniqueness of solution to a nonlinear fuzzy fractional differential equation in the proposed metric. Finally, two examples are given to illustrate the results.

A survey for studies on boundary value problems of higher order ordinary differential equations is given firstly. Secondly a simple review for studies on solvability of boundary value problems for impulsive fractional differential equations is presented. Thirdly by using a general method for converting an impulsive fractional differential equation with the Riemann-Liouville fractional derivatives to an equivalent integral equation and employing fixed point theorems in Banach space, we establish existence results of solutions for three classes of boundary value problems ($(n,n-k)$ type BVPs) of impulsive higher order fractional differential equations. Some examples are presented to illustrate the efficiency of the results obtained and some mistakes are also corrected at the end of the paper finally. A conclusion section is given at the end of the paper.

The study of fuzzy fractional variational problems in terms of a fractional Liouville–Caputo derivative is introduced. Necessary optimality conditions for problems of the fuzzy fractional calculus of variations with free end–points are proved, as well as transversality conditions.

In this paper, a fractional model for population growth of species within a closed system is considered. A numerical method which is based on the implementation of the fractional Legendre functions with a pseudospectral approach is applied. The aim is to show the effectiveness of fractional Legendre functions for the numerical simulation of fractional models.

In this work, we formulate a numerical scheme for the solution of the Bagley-Torvik equation using an integral representation in complex plane. The resultant integral is approximated to high order accuracy using quadrature. The accuracy of numerical algorithm depends on the selection of optimal contour of integration. Several contour have been developed in the literature for solving FDEs. In the present work, we will investigate the applicability of optimal contours for solving Bagley-Torvik equation. We compared our results with other methods available in the literature to validate the efficiency and accuracy of the method for various optimal contour of integrations.

In this paper, by using the variational principle of Ekeland, we prove the existence of at least one solution to the fractional advection dispersion equation in $\mathbb{R}^N$.

This is an expository article that describes, in brief, one of the preeminent branch of applicable mathematics, roots of which lie in the nucleus of pure mathematics that ruled the research since past six decades. In writing this article though several important research papers were excised yet attempt is made to retain the beauty of fractional calculus. This article, accommodates Stieltjes transform and fractional integral operator on spaces of generalized functions, distributional Laplace-Hankel transform by fractional integral operators, and wavelet transform of fractional integrals for the integral Boehmians.

In this work, we introduce fractional Mellin transform of order $ \alpha, \ \ 0 \lt \alpha \leq1$ on a function which belongs to the Lizorkin space. Further, some properties and applications of fractional Mellin transform are given.