The Experts below are selected from a list of 14847 Experts worldwide ranked by ideXlab platform
Delfim F M Torres - One of the best experts on this subject based on the ideXlab platform.
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A Stochastic Fractional Calculus with Applications to Variational Principles
Fractal and Fractional, 2020Co-Authors: Houssine Zine, Delfim F M TorresAbstract:We introduce a stochastic Fractional Calculus. As an application, we present a stochastic Fractional Calculus of variations, which generalizes the Fractional Calculus of variations to stochastic processes. A stochastic Fractional Euler-Lagrange equation is obtained, extending those available in the literature for the classical, Fractional, and stochastic Calculus of variations. To illustrate our main theoretical result, we discuss two examples: one derived from quantum mechanics, the second validated by an adequate numerical simulation.
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SpringerBriefs in Applied Sciences and Technology - The Variable-Order Fractional Calculus of Variations
SpringerBriefs in Applied Sciences and Technology, 2019Co-Authors: Ricardo Almeida, Dina Tavares, Delfim F M TorresAbstract:This book intends to deepen the study of the Fractional Calculus, giving special emphasis to variable-order operators. It is organized in two parts, as follows. In the first part, we review the basic concepts of Fractional Calculus (Chapter 1) and of the Fractional Calculus of variations (Chapter 2). In Chapter 1, we start with a brief overview about Fractional Calculus and an introduction to the theory of some special functions in Fractional Calculus. Then, we recall several Fractional operators (integrals and derivatives) definitions and some properties of the considered Fractional derivatives and integrals are introduced. In the end of this chapter, we review integration by parts formulas for different operators. Chapter 2 presents a short introduction to the classical Calculus of variations and review different variational problems, like the isoperimetric problems or problems with variable endpoints. In the end of this chapter, we introduce the theory of the Fractional Calculus of variations and some Fractional variational problems with variable-order. In the second part, we systematize some new recent results on variable-order Fractional Calculus of (Tavares, Almeida and Torres, 2015, 2016, 2017, 2018). In Chapter 3, considering three types of Fractional Caputo derivatives of variable-order, we present new approximation formulas for those Fractional derivatives and prove upper bound formulas for the errors. In Chapter 4, we introduce the combined Caputo Fractional derivative of variable-order and corresponding higher-order operators. Some properties are also given. Then, we prove Fractional Euler-Lagrange equations for several types of Fractional problems of the Calculus of variations, with or without constraints.
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computational methods in the Fractional Calculus of variations
2015Co-Authors: Ricardo Almeida, Shakoor Pooseh, Delfim F M TorresAbstract:This book fills a gap in the literature by introducing numerical techniques to solve problems of Fractional Calculus of variations (FCV). In most cases, finding the analytic solution to such problems is extremely difficult or even impossible, and numerical methods need to be used.The authors are well-known researchers in the area of FCV and the book contains some of their recent results, serving as a companion volume to Introduction to the Fractional Calculus of Variations by A B Malinowska and D F M Torres, where analytical methods are presented to solve FCV problems. After some preliminaries on the subject, different techniques are presented in detail with numerous examples to help the reader to better understand the methods. The techniques presented may be used not only to deal with FCV problems but also in other contexts of Fractional Calculus, such as Fractional differential equations and Fractional optimal control. It is suitable as an advanced book for graduate students in mathematics, physics and engineering, as well as for researchers interested in Fractional Calculus.
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A Fractional Calculus on arbitrary time scales
Signal Processing, 2015Co-Authors: Nadia Benkhettou, Artur M. C. Brito Da Cruz, Delfim F M TorresAbstract:We introduce a general notion of Fractional (noninteger) derivative for functions defined on arbitrary time scales. The basic tools for the time-scale Fractional Calculus (Fractional differentiation and Fractional integration) are then developed. As particular cases, one obtains the usual time-scale Hilger derivative when the order of differentiation is one, and a local approach to Fractional Calculus when the time scale is chosen to be the set of real numbers. HighlightsWe introduce a general Fractional Calculus on an arbitrary time scale.The basic tools for the time-scale Fractional Calculus are rigorously developed.The time-scale Hilger derivative is obtained when the order of differentiation is one.Kolwankar-Gangal approach is obtained when the time scale is the set of real numbers.
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introduction to the Fractional Calculus of variations
2012Co-Authors: Agnieszka B Malinowska, Delfim F M TorresAbstract:The Classical Calculus of Variations Fractional Calculus of Variations via Riemann - Liouville Operators Fractional Calculus of Variations via Caputo Operators Other Approaches to the Fractional Calculus of Variations Towards a Combined Fractional Mechanics and Quantization.
Paul Anthony Williams - One of the best experts on this subject based on the ideXlab platform.
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Fractional Calculus on time scales with Taylor’s theorem
Fractional Calculus and Applied Analysis, 2012Co-Authors: Paul Anthony WilliamsAbstract:We present a definition of the Riemann-Liouville Fractional Calculus for arbitrary time scales through the use of time scales power functions, unifying a number of theories including continuum, discrete and Fractional Calculus. Basic properties of the theory are introduced including integrability conditions and index laws. Special emphasis is given to extending Taylor’s theorem to incorporate our theory.
Arran Fernandez - One of the best experts on this subject based on the ideXlab platform.
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On Fractional Calculus with analytic kernels with respect to functions
arXiv: Classical Analysis and ODEs, 2020Co-Authors: Christian Maxime Steve Oumarou, Hafiz Muhammad Fahad, Jean-daniel Djida, Arran FernandezAbstract: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 have never been studied in detail before. Here we establish the basic properties of these general operators, including series formulae, composition relations, function spaces, and Laplace transforms. The tools of convergent series, from Fractional Calculus with analytic kernels, and of operational Calculus, from Fractional Calculus with respect to functions, are essential ingredients in the analysis of the general class that covers both.
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On some analytic properties of tempered Fractional Calculus
Journal of Computational and Applied Mathematics, 2020Co-Authors: Arran Fernandez, Ceren UstaoğluAbstract:Abstract 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.
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Analytical Development of Incomplete Riemann-Liouville Fractional Calculus
arXiv: Classical Analysis and ODEs, 2019Co-Authors: Arran Fernandez, Ceren Ustaoğlu, Mehmet Ali ÖzarslanAbstract: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 important special functions. In the current work, we develop the theory of incomplete Fractional Calculus in more depth, investigating further properties of the incomplete Riemann-Liouville Fractional differintegrals and answering some fundamental questions about these operators. By considering appropriate function spaces, we formulate rigorously the definitions of incomplete Riemann-Liouville Fractional integration, and justify how this model may be used to analyse a wider class of functions than classical Fractional Calculus. By using analytic continuation, we formulate definitions for incomplete Riemann-Liouville Fractional differentiation, hence extending the incomplete integrals to a fully-fledged model of Fractional Calculus. We also investigate and analyse these operators further, in order to prove new properties. These include a Leibniz rule for incomplete differintegrals of products, and composition properties of incomplete differintegrals with classical Calculus operations. These are natural and expected issues to investigate in any new model of Fractional Calculus, and in the incomplete Riemann-Liouville model the results emerge naturally from the definition previously proposed.
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On Fractional Calculus with general analytic kernels
Applied Mathematics and Computation, 2019Co-Authors: Arran Fernandez, Mehmet Ali Özarslan, Dumitru BaleanuAbstract:Abstract 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. We also consider inversion properties of the new operators, prove analogues of the Leibniz and chain rules in this model of Fractional Calculus, and solve some Fractional differential equations using the new operators.
R. N. Ingle - One of the best experts on this subject based on the ideXlab platform.
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Note on Integral Transform of Fractional Calculus
International Journal of Research, 2019Co-Authors: S. V. Nakade, R. N. IngleAbstract:In recent years Fractional Calculus is highly growing field in research because of its wide applicability and interdisciplinary approach. In this article we study various integral transform particularly Laplace Transform, Mellin Transform, of Fractional Calculus i.e. Fractional derivative and Fractional Integral particularly of Riemann-Liouville Fractional derivative, Riemann-Liouville Fractional integral, Caputo’s Fractional derivative and their properties. AMS subject classification 2000: 26A33, 44A10
Francesco Mainardi - One of the best experts on this subject based on the ideXlab platform.
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Fractional Calculus: Theory and Applications
Mathematics, 2018Co-Authors: Francesco MainardiAbstract:Fractional Calculus is allowing integrals and derivatives of any positive order (the term Fractional is kept only for historical reasons).[...]
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An historical perspective on Fractional Calculus in linear viscoelasticity
Fractional Calculus and Applied Analysis, 2012Co-Authors: Francesco MainardiAbstract:The article provides an historical survey of the early contributions on the applications of Fractional Calculus in linear viscoelasticty. The period under examination covers four decades, since 1930’s up to 1970’s, and authors are from both Western and Eastern countries. References to more recent contributions may be found in the bibliography of the author’s book. This paper reproduces, with Publisher’s permission, Section 3.5 of the book: F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity , Imperial College Press-London and World Scienific-Singapore, 2010.
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Fractional Calculus in Wave Propagation Problems
arXiv: Mathematical Physics, 2012Co-Authors: Francesco MainardiAbstract:Fractional Calculus, in allowing integrals and derivatives of any positive order (the term "Fractional" kept only for historical reasons), can be considered a branch of mathematical physics which mainly deals with integro-differential equations, where integrals are of convolution form with weakly singular kernels of power law type. In recent decades Fractional Calculus has won more and more interest in applications in several fields of applied sciences. In this lecture we devote our attention to wave propagation problems in linear viscoelastic media. Our purpose is to outline the role of Fractional Calculus in providing simplest evolution processes which are intermediate between diffusion and wave propagation. The present treatment mainly reflects the research activity and style of the author in the related scientific areas during the last decades.
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recent history of Fractional Calculus
Communications in Nonlinear Science and Numerical Simulation, 2011Co-Authors: J Tenreiro A Machado, Virginia Kiryakova, Francesco MainardiAbstract:Abstract This survey intends to report some of the major documents and events in the area of Fractional Calculus that took place since 1974 up to the present date.
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Professor Rudolf Gorenflo and his contribution to Fractional Calculus
Fractional Calculus and Applied Analysis, 2011Co-Authors: Yury Luchko, Francesco Mainardi, Sergei RogosinAbstract:This paper presents a brief overview of the life story and professional career of Prof. R. Gorenflo — a well-known mathematician, an expert in the field of Differential and Integral Equations, Numerical Mathematics, Fractional Calculus and Applied Analysis, an interesting conversational partner, an experienced colleague, and a real friend. Especially his role in the modern Fractional Calculus and its applications is highlighted.
