scholarly journals Fractional integrals and derivatives: mapping properties

2016 ◽  
Vol 19 (3) ◽  
Author(s):  
Humberto Rafeiro ◽  
Stefan Samko
Keyword(s):  
Large Scale ◽  
Fractional Derivatives ◽  
Fractional Integration ◽  
Function Spaces ◽  
Fractional Integrals ◽  
Multidimensional Case ◽  
Simple Fact ◽  
Hypersingular Integrals ◽  
Fractional Integrals And Derivatives ◽  
The One

AbstractThis survey is aimed at the audience of readers interested in the information on mapping properties of various forms of fractional integration operators, including multidimensional ones, in a large scale of various known function spaces.As is well known, the fractional integrals defined in this or other forms improve in some sense the properties of the functions, at least locally, while fractional derivatives to the contrary worsen them. With the development of functional analysis this simple fact led to a number of important results on the mapping properties of fractional integrals in various function spaces.In the one-dimensional case we consider both Riemann-Liouville and Liouville forms of fractional integrals and derivatives. In the multidimensional case we consider in particular mixed Liouville fractional integrals, Riesz fractional integrals of elliptic and hyperbolic type and hypersingular integrals. Among the function spaces considered in this survey, the reader can find Hölder spaces, Lebesgue spaces, Morrey spaces, Grand spaces and also weighted and/or variable exponent versions.

scholarly journals Fractional derivatives and the fundamental theorem of fractional calculus

2020 ◽  
Vol 23 (4) ◽  
pp. 939-966 ◽  
Author(s):  
Yuri Luchko
Keyword(s):  
Fractional Calculus ◽  
Fundamental Theorem ◽  
Fractional Derivatives ◽  
Finite Interval ◽  
Fractional Integrals ◽  
Important Class ◽  
Left Inverse ◽  
Caputo Fractional Derivatives ◽  
Fractional Integrals And Derivatives ◽  
The One

AbstractIn 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 such derivatives were suggested in the literature: the Riemann-Liouville fractional derivatives, the Caputo fractional derivatives, and the Hilfer fractional derivatives. We clarify the interconnections between these derivatives on different spaces of functions and provide some of their properties including the formulas for their projectors and the Laplace transforms. However, it turns out that there exist infinitely many other families of the fractional derivatives that are the left-inverse operators to the Riemann-Liouville fractional integrals. In this paper, we focus on an important class of these fractional derivatives and discuss some of their properties.

scholarly journals Forward: There Are Still Plenty of Opportunities On the Boundaries

2007 ◽  
Vol 1 (2) ◽  
Author(s):  
Yijun Liu
Keyword(s):  
Large Scale ◽  
Anisotropic Elasticity ◽  
Boundary Integral ◽  
Stress Concentrations ◽  
Modeling Tools ◽  
Hypersingular Integrals ◽  
Solid Foundation ◽  
The 1960S ◽  
The One ◽  
Local Integral Equations

<table width="530" border="0" cellspacing="0" cellpadding="0"> <tr> <td align="left" valign="top"> <a name="abstract"></a> <span class="subtitle" style="font-weight:bold">Abstract</span><br /> <p><img src="http://ejbe.libraries.rutgers.edu/files/rizzo.gif" align=left HSPACE=20>This is the second of the two special issues of Electronic Journal of Boundary Elements in honor of Professor Frank Rizzo. There are thirteen technical papers in this issue, contributed by Professor Rizzo’s colleagues, friends and former students. These papers cover a broad range of topics in the boundary integral equation and boundary element method (BIE/BEM), including Galerkin BEM for anisotropic elasticity (Gray, Griffith, et al.), evaluations of hypersingular integrals in Galerkin BEM (Bonnet and Guiggiani), Green’s function BEM for bimaterials (Denda), new 3-D Green’s functions for piezoelectric bimaterials (Pan), new formulations using local integral equations (Sladek and Sladek), BEM in sensitivity analysis with stress concentrations (Burczynski and Habarta), fracture of thermopiezoelectric materials (Qin), BEM for 3-D gradient elastodynamics (Polyzos, Tsepoura and Beskos), time-domain large-scale elastodynamic analysis (Yoshikawa and Nishimura), acoustic BEM for analyzing mufflers and silencers (Wu and Cheng), analysis of solids with randomly distributed inclusions (Yao, Kong and Zheng), thermal and stress analyses of thermal barrier coatings (Lu and Dong), and finally, modeling of carbon nanotube-based composites (Liu and Chen). These authors are gratefully acknowledged for their excellent contributions, and for their patience and cooperation in the process of preparing this special issue. It is interesting to note that the wide applications of the elasticity BIE/BEM in engineering all started with a simple idea. That is, boundary-value problems can be solved by boundary-only methods. The first result in this direction is also amazingly concise. During a recent trip to Urbana, Illinois, I checked out Professor Rizzo’s Ph.D. dissertation from the UIUC library. The thirty-page dissertation is without doubt a masterpiece that many current and future Ph.D. candidates may like to follow, for its originality and succinct writing. The dissertation laid a solid foundation for what is now called the BEM for elasticity and many other problems, and eventually led to the seminal paper of 1967. Behind this masterpiece are Professor Rizzo’s affection and conviction in the BIE/BEM and his willingness to explore a different route in research. This spirit of exploration and his serious attitude in research have inspired and influenced many of his former students and colleagues in the last forty years. Researchers in the pursuit of boundary-only methods can be described as explorers in a Flatland (see Professor Rizzo’s article in Issue No. 1). They can have different perspectives, but can also discover treasures that others could not uncover. As younger researchers come into this playground, new breakthroughs, just like the one made by Professor Rizzo in the 1960s, may not be far away. There will certainly be more innovative boundary-only methods emerging in the near future. More special numerical tools will be developed and more emerging problems will be solved by these new modeling tools. The fields of computational mechanics will be further diversified and thus prosperous. There are still plenty of opportunities on the boundaries! <br /><br /><br /> </td> </tr> </table>

Operational method for solving fractional differential equations with the left-and right-hand sided Erdélyi-Kober fractional derivatives

2020 ◽  
Vol 23 (1) ◽  
pp. 103-125 ◽  
Author(s):  
Latif A-M. Hanna ◽  
Maryam Al-Kandari ◽  
Yuri Luchko
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Fractional Derivatives ◽  
Fractional Integrals ◽  
Operational Method ◽  
Initial Value ◽  
Right Hand ◽  
Fractional Differential ◽  
Left And Right ◽  
Fractional Integrals And Derivatives

AbstractIn this paper, we first provide a survey of some basic properties of the left-and right-hand sided Erdélyi-Kober fractional integrals and derivatives and introduce their compositions in form of the composed Erdélyi-Kober operators. Then we derive a convolutional representation for the composed Erdélyi-Kober fractional integral in terms of its convolution in the Dimovski sense. For this convolution, we also determine the divisors of zero. These both results are then used for construction of an operational method for solving an initial value problem for a fractional differential equation with the left-and right-hand sided Erdélyi-Kober fractional derivatives defined on the positive semi-axis. Its solution is obtained in terms of the four-parameters Wright function of the second kind. The same operational method can be employed for other fractional differential equation with the left-and right-hand sided Erdélyi-Kober fractional derivatives.

scholarly journals Two index laws for fractional integrals and derivatives

1972 ◽  
Vol 14 (4) ◽  
pp. 385-410 ◽  
Author(s):  
E. R. Love
Keyword(s):  
Integral Equations ◽  
Fractional Derivatives ◽  
Addition Theorem ◽  
Positive Real Part ◽  
Generalized Functions ◽  
Fractional Integrals ◽  
Positive Real ◽  
Integrable Functions ◽  
Hypergeometric Integral ◽  
Fractional Integrals And Derivatives

SummaryThe first index law, or addition theorem, is well known. The second is much less well known; but both have been found to be of importance in recent studies of hypergeometric integral equations. The first law has usually been considered only in the simple case of orders of integration which have positive real part, or in the context of generalized functions. Arising out of the need to manipulate expressions involving several fractional integrals and derivatives, our aim here is to establish both laws for all combinations of complex orders of integration and differentiation, and for nearly all functions for which the fractional derivatives involved exist as locally integrable functions.

A distributional theory of fractional calculus

1985 ◽  
Vol 99 (3-4) ◽  
pp. 347-357 ◽  
Author(s):  
W. Lamb
Keyword(s):  
Fractional Calculus ◽  
Spectral Properties ◽  
Fractional Integration ◽  
Integral Operators ◽  
Fractional Integrals ◽  
Fractional Powers ◽  
Fractional Powers Of Operators ◽  
Fractional Integrals And Derivatives

SynopsisIn this paper, a theory of fractional calculus is developed for certain spacesD′p,μof generalised functions. The theory is based on the construction of fractionalpowers of certain simple differential and integral operators. With the parameter μ suitably restricted, these fractional powers are shown to coincide with the Riemann-Liouville and Weyl operators of fractional integration and differentiation. Standard properties associated with fractional integrals and derivatives follow immediately from results obtained previously by the author on fractional powers of operators; see [6], [7]. Some spectral properties are also obtained.

Equivalence of Initialized Fractional Integrals and the Diffusive Model

2018 ◽  
Vol 13 (3) ◽  
Author(s):  
Jian Yuan ◽  
Youan Zhang ◽  
Jingmao Liu ◽  
Bao Shi
Keyword(s):  
Differential Equations ◽  
Fractional Derivatives ◽  
Fractional Integration ◽  
Input Function ◽  
Fractional Integrals ◽  
Initial Condition ◽  
Engineering Structures ◽  
Viscoelastic Dampers ◽  
Fractional Derivatives And Integrals ◽  
Long Time

Fractional calculus is viewed as a novel and powerful tool to describe the stress and strain relations in viscoelastic materials. Consequently, the motions of engineering structures incorporated with viscoelastic dampers can be described by fractional-order differential equations. To deal with the fractional differential equations, initialization for fractional derivatives and integrals is considered to be a fundamental and unavoidable problem. However, this issue has been an open problem for a long time and controversy persists. The initialization function approach and the infinite state approach are two effective ways in initialization for fractional derivatives and integrals. By comparing the above two methods, this technical brief presents equivalence and unification of the Riemann–Liouville fractional integrals and the diffusive representation. First, the equivalence is proved in zero initialization case where both of the initialization function and the distributed initial condition are zero. Then, by means of initialized fractional integration, equivalence and unification in the case of arbitrary initialization are addressed. Connections between the initialization function and the distributed initial condition are derived. Besides, the infinite dimensional distributed initial condition is determined by means of input function during historic period.

scholarly journals Fractional Sums and Differences with Binomial Coefficients

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Thabet Abdeljawad ◽  
Dumitru Baleanu ◽  
Fahd Jarad ◽  
Ravi P. Agarwal
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Fractional Integrals ◽  
Binomial Coefficients ◽  
Discrete Version ◽  
Binomial Theorem ◽  
Left And Right ◽  
Fractional Integrals And Derivatives ◽  
Dual Identities

In fractional calculus, there are two approaches to obtain fractional derivatives. The first approach is by iterating the integral and then defining a fractional order by using Cauchy formula to obtain Riemann fractional integrals and derivatives. The second approach is by iterating the derivative and then defining a fractional order by making use of the binomial theorem to obtain Grünwald-Letnikov fractional derivatives. In this paper we formulate the delta and nabla discrete versions for left and right fractional integrals and derivatives representing the second approach. Then, we use the discrete version of the Q-operator and some discrete fractional dual identities to prove that the presented fractional differences and sums coincide with the discrete Riemann ones describing the first approach.

scholarly journals Finding the fractional value of 2pi

2020 ◽  
Author(s):  
Anurag Vaidya
Keyword(s):  
Fractional Derivatives ◽  
Fractional Integrals ◽  
Explorative Study ◽  
Detection Technology ◽  
Fractional Integrals And Derivatives ◽  
Definition Of ◽  
Rational Order ◽  
Derivatives Of ◽  
Sine And Cosine Functions ◽  
Cosine Functions

Rational order integral and derivative of a myriad of functions—ln(x); e^(ax) and t^(ax)—are known. Nevertheless, the investigation focuses on rational order integrals and derivatives of sine and cosine as these functions follow a cyclic order and determiningwhether properties of sine and cosine extend to their fractional integrals and derivativescould expand current applications of sine and cosine, which are extensively in engineeringand economics. For example, Mehdi and Majid outline in, "Applications of FractionalCalculus" how fractional integrals and derivatives come handy while modeling ultrasonicwave propagation in human cancellous bones and bettering edge detection technology. Thus, this explorative study investigates the properties of fractional derivatives and fractional integrals of standard sine and cosine functions. THe study also looks at how to generalize the definition of the 2pi using fractional calculus.

scholarly journals An Expansion Formula with Higher-Order Derivatives for Fractional Operators of Variable Order

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Ricardo Almeida ◽  
Delfim F. M. Torres
Keyword(s):  
Differential Equations ◽  
Approximation Method ◽  
Fractional Derivatives ◽  
Fractional Integrals ◽  
Variable Order ◽  
Fractional Operators ◽  
Expansion Formula ◽  
Approximation Formulas ◽  
Fractional Integrals And Derivatives ◽  
Derivatives Of

We obtain approximation formulas for fractional integrals and derivatives of Riemann-Liouville and Marchaud types with a variable fractional order. The approximations involve integer-order derivatives only. An estimation for the error is given. The efficiency of the approximation method is illustrated with examples. As applications, we show how the obtained results are useful to solve differential equations, and problems of the calculus of variations that depend on fractional derivatives of Marchaud type.

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