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Fractional-Order Derivatives and Integrals: Introductory Overview and Recent Developments
Srivastava, Hari Mohan Department of Mathematics 2020 Kyungpook mathematical journal Vol.60 No.1
The subject of fractional calculus (that is, the calculus of integrals and derivatives of any arbitrary real or complex order) has gained considerable popularity and importance during the past over four decades, due mainly to its demonstrated applications in numerous seemingly diverse and widespread fields of mathematical, physical, engineering and statistical sciences. Various operators of fractional-order derivatives as well as fractional-order integrals do indeed provide several potentially useful tools for solving differential and integral equations, and various other problems involving special functions of mathematical physics as well as their extensions and generalizations in one and more variables. The main object of this survey-cum-expository article is to present a brief elementary and introductory overview of the theory of the integral and derivative operators of fractional calculus and their applications especially in developing solutions of certain interesting families of ordinary and partial fractional "differintegral" equations. This general talk will be presented as simply as possible keeping the likelihood of non-specialist audience in mind.
A GENERALIZATION OF THE KINETIC EQUATION USING THE PRABHAKAR-TYPE OPERATORS
Dorrego, Gustavo Abel,Kumar, Dinesh The Honam Mathematical Society 2017 호남수학학술지 Vol.39 No.3
Fractional kinetic equations are investigated in order to describe the various phenomena governed by anomalous reaction in dynamical systems with chaotic motion. Many authors have provided solutions of various families of fractional kinetic equations involving special functions. Here, in this paper, we aim at presenting solutions of certain general families of fractional kinetic equations using Prabhakar-type operators. The idea of present paper is motivated by Tomovski et al. [21].
A GENERALIZATION OF THE KINETIC EQUATION USING THE PRABHAKAR-TYPE OPERATORS
( Gustavo Abel Dorrego ),( Dinesh Kumar ) 호남수학회 2017 호남수학학술지 Vol.39 No.3
Fractional kinetic equations are investigated in order to describe the various phenomena governed by anomalous reaction in dynamical systems with chaotic motion. Many authors have provided solutions of various families of fractional kinetic equations involving special functions. Here, in this paper, we aim at presenting solutions of certain general families of fractional kinetic equations using Prabhakar-type operators. The idea of present paper is motivated by Tomovski et al. [21].
ANALYTICAL AND APPROXIMATE SOLUTIONS FOR GENERALIZED FRACTIONAL QUADRATIC INTEGRAL EQUATION
Basim N. Abood,Saleh S. Redhwan,Mohammed S. Abdo 경남대학교 수학교육과 2021 Nonlinear Functional Analysis and Applications Vol.26 No.3
In this paper, we study the analyticaland approximate solutions for a fractional quadratic integral equationinvolving Katugampola fractional integral operator. The existence anduniqueness results obtained in the given arrangement are not only new butalso yield some new particular results corresponding to special values ofthe parameters $\rho $ and $\vartheta $. The main results are obtained byusing Banach fixed point theorem, Picard Method, and Adomian decompositionmethod. An illustrative example is given to justify the main results.
Rita A. Hibschweiler Korean Mathematical Society 2023 대한수학회보 Vol.60 No.4
In this paper, we study products of composition, multiplication and differentiation acting on the fractional Cauchy spaces and mapping into the Zygmund space. Characterizations are provided for boundedness and compactness of these operators.
GENERALIZED PSEUDO-DIFFERENTIAL OPERATORS INVOLVING FRACTIONAL FOURIER TRANSFORM
B. B. Waphare,P. D. Pansare 경남대학교 수학교육과 2021 Nonlinear Functional Analysis and Applications Vol.26 No.1
Generalized pseudo-differential operators (PDO) involving fractional Fourier transform associate with the symbol a(x, y) whose derivatives satisfy certain growth condition is defined. The product of two generalized pseudo-differential operators is shown to be a generalized pseudo-differential operator.
Some subclasses of multivalent functions involving a certain linear operator
M. K. Aouf,H. Silverman,H. M. Srivastava 장전수학회 2007 Advanced Studies in Contemporary Mathematics Vol.14 No.2
In this paper, we investigate the various important properties and characteristics of the subclasses Sa,c(λ;p,A,B) and Ta,c(λ;p,A,B) of p-valent functions defined by means of certain linear operators. We first establish an inclusion relation for the class Sa,c(λ;p,A,B). We then derive many results for the modied Hadamard products of functions belonging to the class Ta,c(λ;p,A,B). Finally, several applications involving an integral operator and certain fractional calculus operators are also considered.
Zuomao Yan,Xiumei Jia 제어·로봇·시스템학회 2017 International Journal of Control, Automation, and Vol.15 No.3
In this paper, we establish the existence of optimal controls of systems governed by a class of fractionalimpulsive partial neutral stochastic integro-differential systems with infinite delay in a Hilbert space. Theapproaches used are fixed point theorem, stochastic analysis theory, and approximation technique combined withproperties of the solution operator. Finally, an example is discussed to illustrate the efficiency of the results.
Deterministic analysis of distributed order systems using operational matrix
Duong, P.L.T.,Kwok, E.,Lee, M. Elsevier Science Ltd 2016 Applied mathematical modelling Vol.40 No.3
<P>Recently, distributed order systems as a generalized concept of fractional order have been a major focus in science and engineering areas, and have rapidly extended application across a wide range of disciplines. However, only a few numerical methods are available for analyzing the distributed order systems. This paper proposes a novel numerical scheme to analyze the behavior of single input single output linear systems in the time domain with a single distributed order differentiator/integrator by using operational matrix technique. The proposed method reduces different analysis problems to a system of algebraic equations by using block pulse functions, which makes it easy to handle an arbitrary input. Numerical examples were used to illustrate the accuracy and computational efficiency of the proposed method. The proposed method was found to be an efficient tool for analyzing linear distributed order systems. (C) 2015 Elsevier Inc. All rights reserved.</P>
THE (k, s)-FRACTIONAL CALCULUS OF CLASS OF A FUNCTION
Rahman, G.,Ghaffar, A.,Nisar, K.S.,Azeema, Azeema The Honam Mathematical Society 2018 호남수학학술지 Vol.40 No.1
In this present paper, we deal with the generalized (k, s)-fractional integral and differential operators recently defined by Nisar et al. and obtain some generalized (k, s)-fractional integral and differential formulas involving the class of a function as its kernels. Also, we investigate a certain number of their consequences containing the said function in their kernels.