Some Pioneers of the Application of Fractional Calculus

2013 ◽  
Author(s):  
J. A. Tenreiro Machado ◽  
Alexandra M. S. F. Galhano
Keyword(s):  
Twentieth Century ◽  
Fractional Calculus ◽  
Research And Development ◽  
Fractional Derivatives ◽  
Operational Calculus ◽  
Time Line ◽  
Seminal Work ◽  
Electrical Relaxation ◽  
Intense Research

In the last decades fractional calculus (FC) became an area of intense research and development. This paper goes back and draws the time line of FC. We recall also important pioneers that started to apply FC during the beginning of the twentieth century. First we remember Oliver Heaviside, a pioneer in several areas of mathematics, physics and engineering. His operational calculus revealed expressions involving fractional derivatives, a premonition of the progresses that emerged later in the last decades of the twentieth century. Second it is addressed the Cole-Cole electrical relaxation. This model was a seminal work in the application of FC and plays nowadays an important role in many areas of physics and engineering.

scholarly journals Operational calculus for the Riemann–Liouville fractional derivative with respect to a function and its applications

2021 ◽  
Vol 24 (2) ◽  
pp. 518-540
Author(s):  
Hafiz Muhammad Fahad ◽  
Arran Fernandez
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Fractional Derivatives ◽  
Operational Calculus ◽  
General Setting ◽  
Fractional Differentiation ◽  
Fractional Calculus Operators ◽  
Liouville Fractional Derivative ◽  
Constant Coefficients ◽  
Fractional Differential

Abstract Mikusiński’s operational calculus is a formalism for understanding integral and derivative operators and solving differential equations, which has been applied to several types of fractional-calculus operators by Y. Luchko and collaborators, such as for example [26], etc. In this paper, we consider the operators of Riemann–Liouville fractional differentiation of a function with respect to another function, and discover that the approach of Luchko can be followed, with small modifications, in this more general setting too. The Mikusiński’s operational calculus approach is used to obtain exact solutions of fractional differential equations with constant coefficients and with this type of fractional derivatives. These solutions can be expressed in terms of Mittag-Leffler type functions.

scholarly journals The Four-Parameters Wright Function of the Second kind and its Applications in FC

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 970 ◽  
Author(s):  
Yuri Luchko
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivatives ◽  
Dimensional Space ◽  
Wide Spectrum ◽  
Operational Calculus ◽  
Fractional Diffusion ◽  
Wright Function ◽  
Scale Invariant ◽  
First Case

In this survey paper, we present both some basic properties of the four-parameters Wright function and its applications in Fractional Calculus. For applications in Fractional Calculus, the four-parameters Wright function of the second kind is especially important. In the paper, three case studies illustrating a wide spectrum of its applications are presented. The first case study deals with the scale-invariant solutions to a one-dimensional time-fractional diffusion-wave equation that can be represented in terms of the Wright function of the second kind and the four-parameters Wright function of the second kind. In the second case study, we consider a subordination formula for the solutions to a multi-dimensional space-time-fractional diffusion equation with different orders of the fractional derivatives. The kernel of the subordination integral is a special case of the four-parameters Wright function of the second kind. Finally, in the third case study, we shortly present an application of an operational calculus for a composed Erdélyi-Kober fractional operator for solving some initial-value problems for the fractional differential equations with the left- and right-hand sided Erdélyi-Kober fractional derivatives. In particular, we present an example with an explicit solution in terms of the four-parameters Wright function of the second kind.

Mkhitar Djrbashian and his contribution to Fractional Calculus

2020 ◽  
Vol 23 (6) ◽  
pp. 1797-1809
Author(s):  
Sergei Rogosin ◽  
Maryna Dubatovskaya
Keyword(s):  
Cauchy Problem ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Approximation Theory ◽  
Fractional Derivatives ◽  
Complex Analysis ◽  
Integral Transforms ◽  
Modern Theory ◽  
Survey Paper ◽  
The Cauchy Problem

Abstract This survey paper is devoted to the description of the results by M.M. Djrbashian related to the modern theory of Fractional Calculus. M.M. Djrbashian (1918-1994) is a well-known expert in complex analysis, harmonic analysis and approximation theory. Anyway, his contributions to fractional calculus, to boundary value problems for fractional order operators, to the investigation of properties of the Queen function of Fractional Calculus (the Mittag-Leffler function), to integral transforms’ theory has to be understood on a better level. Unfortunately, most of his works are not enough popular as in that time were published in Russian. The aim of this survey is to fill in the gap in the clear recognition of M.M. Djrbashian’s results in these areas. For same purpose, we decided also to translate in English one of his basic papers [21] of 1968 (joint with A.B. Nersesian, “Fractional derivatives and the Cauchy problem for differential equations of fractional order”), and were invited by the “FCAA” editors to publish its re-edited version in this same issue of the journal.

scholarly journals Moving out of the Attic: Susan Glaspell and the American She-Tragedy

2018 ◽  
Vol 8 (4) ◽  
pp. 1
Author(s):  
Zaid Ibrahim Ismael ◽  
Jinan Waheed Jassim
Keyword(s):  
Twentieth Century ◽  
Feminist Theory ◽  
Gender Issues ◽  
Vicious Circle ◽  
American Theater ◽  
Seminal Work ◽  
Male Dominated ◽  
Susan Glaspell

Midwestern American dramatist Susan Glaspell (1876-1948) was one of the early voices in the American theater who explored gender issues and woman’s rights at the beginning of the twentieth century. She portrays women in distress, trying to find an outlet from the vicious circle of loneliness and abuse in which they live. Despite the fact that her characters resist oppression and degradation and try to defy the patriarchal authority that restrains them, they are often overwhelmed by this powerful male-dominated system. As a result, they grow defensive and resort to violence and murder in order to avenge themselves from the society that dehumanizes them. They ultimately fall prey to their tragic fate, not necessarily death, but psychological disintegration and incarceration. Glaspell’s tragic heroines are outsiders, living in a world that thwarts their dreams to have a free life beyond the prescribed roles and social demands of house management, domesticity, and social propriety. This study applies the feminist theory of Sandra M. Gilbert and Susan Gubar, introduced in their seminal work The Madwoman in the Attic, to two of Glaspell’s major plays, namely Trifles and The Verge. It also aims at tracing the elements of the Restoration ‘she-tragedy’ in these plays to prove to what extent Glaspell is a master of this form of writing.

On Fractional Hamilton Formulation Within Caputo Derivatives

2007 ◽  
Author(s):  
Dumitru Baleanu ◽  
Sami I. Muslih ◽  
Eqab M. Rabei
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivatives ◽  
Variational Principles ◽  
Hamiltonian Dynamics ◽  
Fractional Integration ◽  
Fractional Dynamics ◽  
Classical Dynamics ◽  
Lagrange Equations ◽  
Caputo Fractional Derivatives ◽  
Non Locality

The fractional Lagrangian and Hamiltonian dynamics is an important issue in fractional calculus area. The classical dynamics can be reformulated in terms of fractional derivatives. The fractional variational principles produce fractional Euler-Lagrange equations and fractional Hamiltonian equations. The fractional dynamics strongly depends of the fractional integration by parts as well as the non-locality of the fractional derivatives. In this paper we present the fractional Hamilton formulation based on Caputo fractional derivatives. One example is treated in details to show the characteristics of the fractional dynamics.

Properties of Spatial-Rotation Transformations of Fractional Derivatives

2020 ◽  
pp. 198-223
Author(s):  
Ehab Malkawi
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivatives ◽  
Dimensional Space ◽  
Scalar Fields ◽  
Two Dimensional ◽  
Spatial Rotation ◽  
Essential Step ◽  
Derivatives Of

The transformation properties of the fractional derivatives under spatial rotation in two-dimensional space and for both the Riemann-Liouville and Caputo definitions are investigated and derived in their general form. In particular, the transformation properties of the fractional derivatives acting on scalar fields are studied and discussed. The study of the transformation properties of fractional derivatives is an essential step for the formulation of fractional calculus in multi-dimensional space. The inclusion of fractional calculus in the Lagrangian and Hamiltonian dynamical formulation relies on such transformation. Specific examples on the transformation of the fractional derivatives of scalar fields are discussed.

The failure of certain fractional calculus operators in two physical models

2019 ◽  
Vol 22 (2) ◽  
pp. 255-270 ◽  
Author(s):  
Manuel D. Ortigueira ◽  
Valeriy Martynyuk ◽  
Mykola Fedula ◽  
J. Tenreiro Machado
Keyword(s):  
Fractional Calculus ◽  
Human Body ◽  
Real World ◽  
Fractional Derivatives ◽  
Electrical Impedance ◽  
Real Data ◽  
Physical Models ◽  
Data Sets ◽  
Real World Data ◽  
Fractional Calculus Operators

Abstract The ability of the so-called Caputo-Fabrizio (CF) and Atangana-Baleanu (AB) operators to create suitable models for real data is tested with real world data. Two alternative models based on the CF and AB operators are assessed and compared with known models for data sets obtained from electrochemical capacitors and the human body electrical impedance. The results show that the CF and AB descriptions perform poorly when compared with the classical fractional derivatives.

A Fractional Calculus Perspective in Electromagnetics

2005 ◽  
Author(s):  
J. A. Tenreiro Machado ◽  
Isabel S. Jesus ◽  
Alexandra Galhano
Keyword(s):  
Twentieth Century ◽  
Fractional Calculus ◽  
Approximation Method ◽  
Electric Potential ◽  
Constant Velocity ◽  
Order Approximation ◽  
Electrical Potential ◽  
Late 19Th Century ◽  
Electric And Magnetic Fields ◽  
Electricity And Magnetism

Some experimentation with magnets was beginning in the late 19th century. By then reliable batteries had been developed and the electric current was recognized as a stream of charge particles. Maxwell developed a set of equations expressing the basic laws of electricity and magnetism, and demonstrated that these two phenomena are complementary aspects of electromagnetism. He showed that electric and magnetic fields travel through space, in the form of waves, at a constant velocity. Maxwell is generally regarded as the nineteenth century scientist who had the greatest influence on twentieth century physics, making contributions to the fundamental models of nature. Bearing these ideas in mind, in this study we apply the concept of fractional calculus and some aspects of electromagnetism, to the static electric potential, and we develop a new fractional order approximation method to the electrical potential.

scholarly journals Fractional Sobolev’s Spaces on Time Scales via Conformable Fractional Calculus and Their Application to a Fractional Differential Equation on Time Scales

2016 ◽  
Vol 2016 ◽  
pp. 1-21 ◽  
Author(s):  
Yanning Wang ◽  
Jianwen Zhou ◽  
Yongkun Li
Keyword(s):  
Differential Equation ◽  
Fractional Calculus ◽  
Fractional Differential Equation ◽  
Time Scales ◽  
Fractional Derivatives ◽  
Point Theory ◽  
Existence Results ◽  
Recent Approach ◽  
Equation Boundary ◽  
Fractional Differential

Using conformable fractional calculus on time scales, we first introduce fractional Sobolev spaces on time scales, characterize them, and define weak conformable fractional derivatives. Second, we prove the equivalence of some norms in the introduced spaces and derive their completeness, reflexivity, uniform convexity, and compactness of some imbeddings, which can be regarded as a novelty item. Then, as an application, we present a recent approach via variational methods and critical point theory to obtain the existence of solutions for ap-Laplacian conformable fractional differential equation boundary value problem on time scaleT:  Tα(Tαup-2Tα(u))(t)=∇F(σ(t),u(σ(t))),Δ-a.e.  t∈a,bTκ2,u(a)-u(b)=0,Tα(u)(a)-Tα(u)(b)=0,whereTα(u)(t)denotes the conformable fractional derivative ofuof orderαatt,σis the forward jump operator,a,b∈T,  0<a<b,  p>1, andF:[0,T]T×RN→R. By establishing a proper variational setting, we obtain three existence results. Finally, we present two examples to illustrate the feasibility and effectiveness of the existence results.

scholarly journals Recent applications of fractional calculus to science and engineering

2003 ◽  
Vol 2003 (54) ◽  
pp. 3413-3442 ◽  
Author(s):  
Lokenath Debnath
Keyword(s):  
Dynamical Systems ◽  
Fractional Calculus ◽  
Control Theory ◽  
Numerical Computation ◽  
Fractional Derivatives ◽  
Fluid Flows ◽  
Voltage Divider ◽  
Science And Engineering ◽  
Electrical Circuits ◽  
Fractional Derivatives And Integrals

This paper deals with recent applications of fractional calculus to dynamical systems in control theory, electrical circuits with fractance, generalized voltage divider, viscoelasticity, fractional-order multipoles in electromagnetism, electrochemistry, tracer in fluid flows, and model of neurons in biology. Special attention is given to numerical computation of fractional derivatives and integrals.

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