Related papers: Computing Hadamard type operators of variable frac…

In this paper, we introduce a new method for calculating fractional integrals and differentials. The method involves an equation that we have obtained from infinite applied integration by parts. The equation works for special class of…

We introduce three types of partial fractional operators of variable order. An integration by parts formula for partial fractional integrals of variable order and an extension of Green's theorem are proved. These results allow us to obtain…

In this paper, we propose a new concept of derivative with respect to an arbitrary kernel-function. Several properties related to this new operator, like inversion rules, integration by parts, etc. are studied. In particular, we introduce…

In the present article, a new method for the evaluation of fractional derivatives of arbitrary real order is proposed. Numerous but inequivalent formulations have been given in the past. Some of them exhibit unsatisfactory properties such…

We introduce the linear operators of fractional integration and fractional differentiation in the framework of the Riemann-Liouville fractional calculus. Particular attention is devoted to the technique of Laplace transforms for treating…

In this paper, we propose a numerical method of computing a Hadamard finite-part integral with a non-integral power singularity at an endpoint, that is, a finite part of a divergent integral as a limiting procedure. In the proposed method,…

Diffusive representations of fractional differential and integral operators can provide a convenient means to construct efficient numerical algorithms for their approximate evaluation. In the current literature, many different variants of…

This paper is in concern with Cauchy problems involving the fractional derivatives with respect to another function. Results of existence, uniqueness, and Taylor series among others are established in appropriate functional spaces. We prove…

We introduce new fractional operators of variable order on isolated time scales with Mittag-Leffler kernels. This allows a general formulation of a class of fractional variational problems involving variable-order difference operators. Main…

Through duality it is possible to transform left fractional operators into right fractional operators and vice versa. In contrast to existing literature, we establish integration by parts formulas that exclusively involve either left or…

Some extremalities for quadrature operators are proved for convex functions of higher order. Such results are known in the numerical analysis, however they are often proved under suitable differentiability assumptions. In our considerations…

In this paper, we propose a numerical method for computing Hadamard finite-part integrals with an integral-power singularity at an endpoint, the part of the divergent integral which is finite as a limiting procedure. In the proposed method,…

In this paper we provide a definition of fractional gradient operators, related to directional derivatives. We develop a fractional vector calculus, providing a probabilistic interpretation and mathematical tools to treat multidimensional…

We present an extension of a previously developed method employing the formalism of the fractional derivatives to solve new classes of integral equations. This method uses different forms of integral operators that generalizes the…

In this paper, we introduce the new construction of fractional derivatives and integrals with respect to a function, based on a matrix approach. We believe that this is a powerful tool in both analytical and numerical calculations. We begin…

In this article we study the basic theoretical properties of Mellin-type fractional integrals, known as generalizations of the Hadamard-type fractional integrals. We give a new approach and version, specifying their semigroup property,…

We study a numerical approximation for a nonlinear variable-order fractional differential equation via an integral equation method. Due to the lack of the monotonicity of the discretization coefficients of the variable-order fractional…

Recently, fractional differential equations have been investigated via the famous variational iteration method. However, all the previous works avoid the term of fractional derivative and handle them as a restricted variation. In order to…

Considering the large number of fractional operators that exist, and since it does not seem that their number will stop increasing soon at the time of writing this paper, it is presented for the first time, as far as the authors know, a…

Some mathematical models of applied problems lead to the need of solving boundary value problems with a fractional power of an elliptic operator. In a number of works, approximations of such a nonlocal operator are constructed on the basis…