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Studies of convolution play an important role in Geometric Function Theory (GFT). Such studies attracted a large number of researchers in recent years. By making use of the Hadamard product (or convolution), several new and interesting subclasses of analytic and univalent functions have been introduced and investigated in the direction of well-known concepts such as the subordination and superordination inequalities, integral mean and partial sums, and so on. In this article, we apply the Hadamard product (or convolution) by utilizing some special functions. Our contribution in this paper includes defining a new linear operator in the form of the generalized Mittag-Leffler function in terms of the extensively-investigated Fox-Wright pΨq-function in the right-half of the open unit disk where where ℜ(z) > 0. We then show that the new linear convolution operator is bounded in some spaces. In particular, several boundedness properties of this linear convolution operator under mappings from a weighted Bloch space into a weighted-log Bloch space are also investigated. For uniformity and convenience, the Fox-Wright pΨq-notation is used in our results. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of solutions for a boundary value problem involving both left Riemann-Liouville and right Caputo-type fractional derivatives. For this, we convert the posed problem to a sum of two integral operators, then we apply Krasnoselskii's fixed point theorem to conclude the existence of nontrivial solutions. [ABSTRACT FROM AUTHOR]

In this article, we study nonlinear quadratic iterative integral equations and establish sufficient conditions for the existence of Volterra solutions for fractional iterative integral equations and solvency in Banach space and Clα. In the present work we use the principle of contraction, Schaefer's fixed point theorem and the non-expansive operator method as essential tools. In this study we consider Riemann-Liouville differential operator and prove some related theorems, further provide an example as an application. [ABSTRACT FROM AUTHOR]

In this work, we study and extend the one-dimensional fractional derivative to the multidimensional space-time fractional derivative, determine the exact solution via the Laplace transform, and develop mathematical foundations of the respective operators. The multidimensional space-time fractional derivative is developed to augment the equations. The results show that this method is effective, convenient, and easily executable. The main advantage of the proposed approach is that it is an accurate analytical method (the Laplace transform method) that can be implemented for both space and time discretizations of the fractional derivatives and allows us to present new solutions to problems by certain applications for solving space-time fractional derivatives. [ABSTRACT FROM AUTHOR]

Recently, the study of the fractional formal (operators, polynomials and classes of special functions) has been increased. This study not only in mathematics but extended to another topics. In this effort, we investigate a generalized integro-differential operator $$\mathfrak {J}_{m}(z)$$ defined by a fractional formal (fractional differential operator) and study some its geometric properties by employing it in new subclasses of analytic univalent functions. [ABSTRACT FROM AUTHOR]

In the present paper, we study the integro-differential equations which are combination of differential and Fredholm–Volterra equations that have the fractional order with constant coefficients by the homotopy perturbation and the variational iteration. The fractional derivatives are described in Caputo sense. Some illustrative examples are presented. [ABSTRACT FROM AUTHOR]

The purpose of this article is to present some new inequalities for products of generalized convex and generalized s-convex functions on fractal sets. Furthermore, some applications are given. [ABSTRACT FROM AUTHOR]

This research aims to investigate the strategy of fair insurance premium actuarial approach for pricing currency option, when the value of foreign currency option follows the mixed fractional Brownian motion with jumps and the European call and put currency option are presented. It has certain reference significance to avoiding foreign exchange risk. [ABSTRACT FROM AUTHOR]

We apply the homotopy perturbation method to obtain the solution of partial differential equations of fractional order. This method is powerful tool to find exact and approximate solution of many linear and nonlinear partial differential equations of fractional order. Convergence of the method is proved and the convergence analysis is reliable enough to estimate the maximum absolute truncated error of the series solution. The fractional derivatives are described in the Caputo sense. Some examples are presented to verify convergence hypothesis and simplicity of the method. [ABSTRACT FROM AUTHOR]

We are concerned here with singular partial differential equations of fractional order (FSPDEs). The variational iteration method (VIM) is applied to obtain approximate solutions of this type of equations. Convergence analysis of the VIM is discussed. This analysis is used to estimate the maximum absolute truncated error of the series solution. A comparison between the results of VIM solutions and exact solution is given. The fractional derivatives are described in Caputo sense. [ABSTRACT FROM AUTHOR]

We make use of the properties of the Sumudu transform to solve nonlinear fractional partial differential equations describing heat-like equation with variable coefficients. The method, namely, homotopy perturbation Sumudu transform method, is the combination of the Sumudu transform and the HPM using He's polynomials. This method is very powerful, and professional techniques for solving different kinds of linear and nonlinear fractional differential equations arising in different fields of science and engineering. [ABSTRACT FROM AUTHOR]

The fractional complex transformation is used to transform nonlinear partial differential equations to nonlinear ordinary differential equations. The improved (G'/G)-expansion method is suggested to solve the space and time fractional foam drainage andKdVequations. The obtained results showthat the presentedmethod is effective and appropriate for solving nonlinear fractional differential equations. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the Chebyshev polynomials to solve analytically the fractional neutron transport equation in one-dimensional plane geometry are used. The procedure is based on the expansion of the angular flux in terms of the Chebyshev polynomials. The obtained system of fractional linear differential equation is solved analytically by using fractional Sumudu transform. [Copyright &y& Elsevier]

In this paper, we establish sixteen interesting generalized fractional integral and derivative formulas including their composition formulas by using certain integral transforms involving generalized (p , q) -Mathieu-type series. [ABSTRACT FROM AUTHOR]