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 the notion of conjugate kernels, which will be useful to guaranty that the proposed derivative operator admits a right inverse. The proposed concept includes as special cases Riemann-Liouville fractional derivatives, Hadamard fractional derivatives, and many other fractional operators. Moreover, using our concept, new fractional operators involving certain special functions are introduced, and some of their properties are studied. Finally, an existence result for a boundary value problem involving the introduced derivative operator is proved.
The theory of fractional calculus has developed in a number of directions over the years, including: the formulation of multiple different definitions of fractional differintegration; the extension of various properties of standard calculus into the fractional scenario; the application of fractional differintegrals to assorted special functions. Recently, a new variant of fractional calculus has arisen, namely incomplete fractional calculus. In two very recent papers, incomplete versions of the Riemann-Liouville and Caputo fractional differintegrals have been formulated and applied to several …
We consider the integral and derivative operators of tempered fractional calculus, and examine their analytic properties. We discover connections with the classical Riemann-Liouville fractional calculus and demonstrate how the operators may be used to obtain special functions such as hypergeometric and Appell's functions. We also prove an analogue of Taylor's theorem and some integral inequalities to enrich the mathematical theory of tempered fractional calculus.
In this article, we illustrate the asymptotic behaviour of exponential functions of nabla fractional calculus. For this purpose, we propose a novel matrix technique to compute these functions numerically.
The operators of fractional calculus come in many different types, which can be categorised into general classes according to their nature and properties. We conduct a formal study of the class known as weighted fractional calculus and its extension to the larger class known as weighted fractional calculus with respect to functions. These classes contain tempered, Hadamard-type, and Erdélyi--Kober operators as special cases, and in general they can be related to the classical Riemann--Liouville fractional calculus via conjugation relations. Considering the corresponding modifications of the La…
This paper gives particular emphasis to the nabla tempered fractional calculus, which involves the multiplication of the rising function kernel with tempered functions, and provides a more flexible alternative with considerable promise for practical applications. Some remarkable inequalities for such nabla fractional calculus are developed and analyzed, which greatly enrich the mathematical theory of nabla tempered fractional calculus. Numerical results confirm the validity of the developed properties once again, which also reveals that the introduction of tempered function provides high value…
Many possible definitions have been proposed for fractional derivatives and integrals, starting from the classical Riemann-Liouville formula and its generalisations and modifying it by replacing the power function kernel with other kernel functions. We demonstrate, under some assumptions, how all of these modifications can be considered as special cases of a single, unifying, model of fractional calculus. We provide a fundamental connection with classical fractional calculus by writing these general fractional operators in terms of the original Riemann-Liouville fractional integral operator. W…
In this work we present a novel proof of the Saalschutz formula by using the theory of discrete fractional calculus. The proofs of some results within this theory, namely, the fractional power rule and the fractional Leibniz rule are revisited.
Many different types of fractional calculus have been proposed, which can be organised into some general classes of operators. For a unified mathematical theory, results should be proved in the most general possible setting. Two important classes of fractional-calculus operators are the fractional integrals and derivatives with respect to functions (dating back to the 1970s) and those with general analytic kernels (introduced in 2019). To cover both of these settings in a single study, we can consider fractional integrals and derivatives with analytic kernels with respect to functions, which h…
A generalization of fractional vector calculus as a self-consistent mathematical theory is proposed to take into account a general form of non-locality in kernels of fractional vector differential and integral operators. Self-consistency involves proving generalizations of all fundamental theorems of vector calculus for generalized kernels of operators. In the generalization of FVC from power-law nonlocality to the general form of nonlocality in space, we use the general fractional calculus in the Luchko approach. This paper proposed the following: (1) Self-consistent definitions of general fr…
In this paper, we study some extended hypergeometric functions from matrix point of view. We have given the integral representations of these matrix functions. Finally, we obtain some generating function relations using fractional derivative operators.
We define an analogue of the classical Mittag-Leffler function which is applied to two variables, and establish its basic properties. Using a corresponding single-variable function with fractional powers, we define an associated fractional integral operator which has many interesting properties. The motivation for these definitions is twofold: firstly their link with some fundamental fractional differential equations involving two independent fractional orders, and secondly the fact that they emerge naturally from certain applications in bioengineering.
In this paper, we first address the general fractional integrals and derivatives with the Sonine kernels that possess the integrable singularities of power function type at the point zero. Both particular cases and compositions of these operators are discussed. Then we proceed with a construction of an operational calculus of the Mikusiński type for the general fractional derivatives with the Sonine kernels. This operational calculus is applied for analytical treatment of some initial value problems for the fractional differential equations with the general fractional derivatives. The solution…
We will use analytic function theory and Fourier analysis to establish a characterization for some classical umbral calculus, which will focus on the generalization of the evaluation function. Although we cannot cover all the umbral calculus people care about, the part about Bernoulli numbers can still answer an open question about fractional sums raised by Müller and Schleicher in 2005 and synthesize some common analysis results. We will only develop the results which are sufficient to serve the purpose of this article, but at the same time, we will briefly mention some possible extensions.
In this paper, we first discuss the convolution series that are generated by the Sonine kernels from a class of functions continuous on the real positive semi-axis that have an integrable singularity of power function type at the point zero. These convolution series are closely related to the general fractional integrals and derivatives with the Sonine kernels and represent a new class of the special functions of Fractional Calculus. The Mittag-Leffler functions as solutions to the fractional differential equations with the fractional derivatives of both Riemann-Liouville and Caputo types are …
In this paper, we address the one-parameter families of the fractional integrals and derivatives defined on a finite interval. First we remind the reader of the known fact that under some reasonable conditions, there exists precisely one unique family of the fractional integrals, namely, the well-known Riemann-Liouville fractional integrals. As to the fractional derivatives, their natural definition follows from the fundamental theorem of the Fractional Calculus, i.e., they are introduced as the left-inverse operators to the Riemann-Liouville fractional integrals. Until now, three families of …
Several approaches to the formulation of a fractional theory of calculus of "variable order" have appeared in the literature over the years. Unfortunately, most of these proposals lack a rigorous mathematical framework. We consider an alternative view on the problem, originally proposed by G. Scarpi in the early seventies, based on a naive modification of the representation in the Laplace domain of standard kernels functions involved in (constant-order) fractional calculus. We frame Scarpi's ideas within recent theory of General Fractional Derivatives and Integrals, that mostly rely on the Son…
The area of fractional calculus (FC) has been fast developing and is presently being applied in all scientific fields. Therefore, it is of key relevance to assess the present state of development and to foresee, if possible, the future evolution, or, at least, the challenges identified in the scope of advanced research works. This paper gives a vision about the directions for further research as well as some open problems of FC. A number of topics in mathematics, numerical algorithms and physics are analyzed, giving a systematic perspective for future research.
This paper is concerned with analyzing a class of fractional calculus of variations problems and their associated Euler-Lagrange (fractional differential) equations. Unlike the existing fractional calculus of variations which is based on the classical notion of fractional derivatives, the fractional calculus of variations considered in this paper is based on a newly developed notion of weak fractional derivatives and their associated fractional order Sobolev spaces. Since fractional derivatives are direction-dependent, using one-sided fractional derivatives and their combinations leads to new …
In this article, we presented some properties of the Katugampola fractional integrals and derivatives. Also we studied the fractional calculus properties involving Katugampola Fractional integrals and derivatives of generalized $k-$Wright function $_{n}Φ_{m}^{k}(z).$\\[2mm]
This paper presents a self-contained new theory of weak fractional differential calculus and fractional Sobolev spaces in one-dimension. The crux of this new theory is the introduction of a weak fractional derivative notion which is a natural generalization of integer order weak derivatives; it also helps to unify multiple existing fractional derivative definitions and characterize what functions are fractionally differentiable. Various calculus rules including a fundamental theorem of calculus, product and chain rules, and integration by parts formulas are established for weak fractional deri…
Many different types of fractional calculus have been defined, which may be categorised into broad classes according to their properties and behaviours. Two types that have been much studied in the literature are the Hadamard-type fractional calculus and tempered fractional calculus. This paper establishes a connection between these two definitions, writing one in terms of the other by making use of the theory of fractional calculus with respect to functions. By extending this connection in a natural way, a generalisation is developed which unifies several existing fractional operators: Rieman…
This paper presents a self-contained new theory of weak fractional differential calculus in one-dimension. The crux of this new theory is the introduction of a weak fractional derivative notion which is a natural generalization of integer order weak derivatives; it also helps to unify multiple existing fractional derivative definitions and characterize what functions are fractionally differentiable. Various calculus rules including a fundamental theorem calculus, product and chain rules, and integration by parts formulas are established for weak fractional derivatives. Additionally, relationsh…
In this work we look at the original fractional calculus of variations problem in a somewhat different way. As a simple consequence, we show that a fractional generalization of a classical problem has a solution without any restrictions on the derivative-order $α$.
The Variable-Order Fractional Calculus of Variations
这本书打算深化分数微积分的研究，特别强调变阶算子。报告分为两部分，内容如下。在第一部分中，我们回顾了分数阶微积分（第1章）和分数阶变分微积分（第2章）的基本概念。在第一章中，我们首先对分数阶微积分做了一个简单的概述，并介绍了分数阶微积分中的一些特殊函数的理论。然后，我们回顾了几个分数阶算子（积分和导数）的定义和所考虑的分数阶导数和积分的一些性质。在本章的最后，我们回顾了不同算子的分部积分法公式。第二章简要介绍了经典的变分法，并回顾了不同的变分问题，如等周问题或端点可变的问题。在本章的最后，我们介绍了分数阶变分法的理论和几个变阶分数阶变分问题。在第二部分中，我们系统化了（Tavares，Almeida and Torres，2015，2016，2017，2018）中关于变阶分数阶微积分的一些最新结果。在第三章中，我们考虑了三种不同阶数的分数阶Caputo导数，给出了这三种分数阶Caputo导数的逼近公式，并证明了误差的上界公式。第四章介绍了变阶算子的组合Caputo分数阶导数和相应的高阶算子。并给出了一些性质。然后，我们证明了几种类型的分数阶变分问题的分数阶欧拉-拉格朗日方程，有或无约束。
The prime aim of the present paper is to continue developing the theory of tempered fractional integrals and derivatives of a function with respect to another function. This theory combines the tempered fractional calculus with the $Ψ$-fractional calculus, both of which have found applications in topics including continuous time random walks. After studying the basic theory of the $Ψ$-tempered operators, we prove mean value theorems and Taylor's theorems for both Riemann--Liouville type and Caputo type cases of these operators. Furthermore, we study some nonlinear fractional differential equat…
We focus on eventually non-linear abstract Cauchy problems with a generalized fractional derivative in time. First we prove a local existence and uniqueness result, then we focus on a generalized Grönwall inequality. Before addressing the inequality, we study some properties of eigenvalues and eigenfunctions of the generalized fractional derivatives. Finally, we prove some consequences of the generalized Grönwall inequality.
An important class of fractional differential and integral operators is given by the theory of fractional calculus with respect to functions, sometimes called $Ψ$-fractional calculus. The operational calculus approach has proved useful for understanding and extending this topic of study. Motivated by fractional differential equations, we present an operational calculus approach for Laplace transforms with respect to functions and their relationship with fractional operators with respect to functions. This approach makes the generalised Laplace transforms much easier to analyse and to apply in …
The standard definition for the Atangana-Baleanu fractional derivative involves an integral transform with a Mittag-Leffler function in the kernel. We show that this integral can be rewritten as a complex contour integral which can be used to provide an analytic continuation of the definition to complex orders of differentiation. We discuss the implications and consequences of this extension, including a more natural formula for the Atangana-Baleanu fractional integral and for iterated Atangana-Baleanu fractional differintegrals.
We derive Itô-type change of variable formulas for smooth functionals of irregular paths with non-zero $p-$th variation along a sequence of partitions where $p \geq 1$ is arbitrary, in terms of fractional derivative operators, extending the results of the Föllmer-Ito calculus to the general case of paths with 'fractional' regularity. In the case where $p$ is not an integer, we show that the change of variable formula may sometimes contain a non-zero a 'fractional' Itô remainder term and provide a representation for this remainder term. These results are then extended to paths with non-zero $ϕ-…
Fractional $h$-difference equations arising from the calculus of variations
本文介绍了[N. R. O.巴塞河A. C. Ferreira，D. F. M. Torres：离散时间分数变分问题，信号处理。91（2011），no. 3，513- 524]，其丰富了用于涉及左和右分数差分算子的离散方程的显式求解的有用工具。证明了右分数$h$和的新结果。算例表明了所得结果在求解分数阶离散Euler-Lagrange方程中的有效性。
Inequalities play an important role in pure and applied mathematics. In particular, Opial inequality plays a main role in the study of the existence and uniqueness of initial and boundary value problems for differential equations. It has several interesting generalizations. In this work we prove some new Opial-type inequalities, and we apply them to generalized Riemann-Liouville-type integral operators.
