Mathematics

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296 KiB

Open AccessArticle

A Class of Extended Mittag–Leffler Functions and Their Properties Related to Integral Transforms and Fractional Calculus

by Rakesh K. Parmar

Mathematics 2015, 3(4), 1069-1082; https://doi.org/10.3390/math3041069 - 6 Nov 2015

Cited by 21 | Viewed by 4953

Abstract

In a joint paper with Srivastava and Chopra, we introduced far-reaching generalizations of the extended Gammafunction, extended Beta function and the extended Gauss hypergeometric function. In this present paper, we extend the generalized Mittag–Leffler function by means of the extended Beta function. We [...] Read more.

In a joint paper with Srivastava and Chopra, we introduced far-reaching generalizations of the extended Gammafunction, extended Beta function and the extended Gauss hypergeometric function. In this present paper, we extend the generalized Mittag–Leffler function by means of the extended Beta function. We then systematically investigate several properties of the extended Mittag–Leffler function, including, for example, certain basic properties, Laplace transform, Mellin transform and Euler-Beta transform. Further, certain properties of the Riemann–Liouville fractional integrals and derivatives associated with the extended Mittag–Leffler function are investigated. Some interesting special cases of our main results are also pointed out. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus and Its Applications)

545 KiB

Open AccessArticle

Root Operators and “Evolution” Equations

by Giuseppe Dattoli and Amalia Torre

Mathematics 2015, 3(3), 690-726; https://doi.org/10.3390/math3030690 - 13 Aug 2015

Cited by 4 | Viewed by 4976

Abstract

Root-operator factorization à la Dirac provides an effective tool to deal with equations, which are not of evolution type, or are ruled by fractional differential operators, thus eventually yielding evolution-like equations although for a multicomponent vector. We will review the method along with [...] Read more.

Root-operator factorization à la Dirac provides an effective tool to deal with equations, which are not of evolution type, or are ruled by fractional differential operators, thus eventually yielding evolution-like equations although for a multicomponent vector. We will review the method along with its extension to root operators of degree higher than two. Also, we will show the results obtained by the Dirac-method as well as results from other methods, specifically in connection with evolution-like equations ruled by square-root operators, that we will address to as relativistic evolution equations. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus and Its Applications)

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292 KiB

Open AccessArticle

Time Automorphisms on C*-Algebras

by R. Hilfer

Mathematics 2015, 3(3), 626-643; https://doi.org/10.3390/math3030626 - 16 Jul 2015

Cited by 3 | Viewed by 3954

Abstract

Applications of fractional time derivatives in physics and engineering require the existence of nontranslational time automorphisms on the appropriate algebra of observables. The existence of time automorphisms on commutative and noncommutative C*-algebras for interacting many-body systems is investigated in this article. A mathematical [...] Read more.

Applications of fractional time derivatives in physics and engineering require the existence of nontranslational time automorphisms on the appropriate algebra of observables. The existence of time automorphisms on commutative and noncommutative C*-algebras for interacting many-body systems is investigated in this article. A mathematical framework is given to discuss local stationarity in time and the global existence of fractional and nonfractional time automorphisms. The results challenge the concept of time flow as a translation along the orbits and support a more general concept of time flow as a convolution along orbits. Implications for the distinction of reversible and irreversible dynamics are discussed. The generalized concept of time as a convolution reduces to the traditional concept of time translation in a special limit. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus and Its Applications)

302 KiB

Open AccessArticle

The Fractional Orthogonal Difference with Applications

by Enno Diekema

Mathematics 2015, 3(2), 487-509; https://doi.org/10.3390/math3020487 - 12 Jun 2015

Cited by 3 | Viewed by 4749

Abstract

This paper is a follow-up of a previous paper of the author published in Mathematics journal in 2015, which treats the so-called continuous fractional orthogonal derivative. In this paper, we treat the discrete case using the fractional orthogonal difference. The theory is illustrated [...] Read more.

This paper is a follow-up of a previous paper of the author published in Mathematics journal in 2015, which treats the so-called continuous fractional orthogonal derivative. In this paper, we treat the discrete case using the fractional orthogonal difference. The theory is illustrated with an application of a fractional differentiating filter. In particular, graphs are presented of the absolutel value of the modulus of the frequency response. These make clear that for a good insight into the behavior of a fractional differentiating filter, one has to look for the modulus of its frequency response in a log-log plot, rather than for plots in the time domain. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus and Its Applications)

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705 KiB

Open AccessArticle

Sinc-Approximations of Fractional Operators: A Computing Approach

by Gerd Baumann and Frank Stenger

Mathematics 2015, 3(2), 444-480; https://doi.org/10.3390/math3020444 - 5 Jun 2015

Cited by 12 | Viewed by 5950

Abstract

We discuss a new approach to represent fractional operators by Sinc approximation using convolution integrals. A spin off of the convolution representation is an effective inverse Laplace transform. Several examples demonstrate the application of the method to different practical problems. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus and Its Applications)

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216 KiB

Open AccessArticle

Subordination Principle for a Class of Fractional Order Differential Equations

by Emilia Bazhlekova

Mathematics 2015, 3(2), 412-427; https://doi.org/10.3390/math3020412 - 26 May 2015

Cited by 25 | Viewed by 5747

Abstract

The fractional order differential equation \(u'(t)=Au(t)+\gamma D_t^{\alpha} Au(t)+f(t), \ t>0\), \(u(0)=a\in X\) is studied, where \(A\) is an operator generating a strongly continuous one-parameter semigroup on a Banach space \(X\), \(D_t^{\alpha}\) is the Riemann–Liouville fractional derivative of order \(\alpha \in (0,1)\), \(\gamma>0\) and [...] Read more.

The fractional order differential equation \(u'(t)=Au(t)+\gamma D_t^{\alpha} Au(t)+f(t), \ t>0\), \(u(0)=a\in X\) is studied, where \(A\) is an operator generating a strongly continuous one-parameter semigroup on a Banach space \(X\), \(D_t^{\alpha}\) is the Riemann–Liouville fractional derivative of order \(\alpha \in (0,1)\), \(\gamma>0\) and \(f\) is an \(X\)-valued function. Equations of this type appear in the modeling of unidirectional viscoelastic flows. Well-posedness is proven, and a subordination identity is obtained relating the solution operator of the considered problem and the \(C_{0}\)-semigroup, generated by the operator \(A\). As an example, the Rayleigh–Stokes problem for a generalized second-grade fluid is considered. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus and Its Applications)

227 KiB

Open AccessArticle

Implicit Fractional Differential Equations via the Liouville–Caputo Derivative

by Juan J. Nieto, Abelghani Ouahab and Venktesh Venktesh

Mathematics 2015, 3(2), 398-411; https://doi.org/10.3390/math3020398 - 25 May 2015

Cited by 48 | Viewed by 6397

Abstract

We study an initial value problem for an implicit fractional differential equation with the Liouville–Caputo fractional derivative. By using fixed point theory and an approximation method, we obtain some existence and uniqueness results. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus and Its Applications)

265 KiB

Open AccessArticle

The Role of the Mittag-Leffler Function in Fractional Modeling

by Sergei Rogosin

Mathematics 2015, 3(2), 368-381; https://doi.org/10.3390/math3020368 - 13 May 2015

Cited by 58 | Viewed by 6562

Abstract

This is a survey paper illuminating the distinguished role of the Mittag-Leffler function and its generalizations in fractional analysis and fractional modeling. The content of the paper is connected to the recently published monograph by Rudolf Gorenflo, Anatoly Kilbas, Francesco Mainardi and Sergei [...] Read more.

This is a survey paper illuminating the distinguished role of the Mittag-Leffler function and its generalizations in fractional analysis and fractional modeling. The content of the paper is connected to the recently published monograph by Rudolf Gorenflo, Anatoly Kilbas, Francesco Mainardi and Sergei Rogosin. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus and Its Applications)

314 KiB

Open AccessArticle

The Fractional Orthogonal Derivative

by Enno Diekema

Mathematics 2015, 3(2), 273-298; https://doi.org/10.3390/math3020273 - 22 Apr 2015

Cited by 5 | Viewed by 4310

Abstract

This paper builds on the notion of the so-called orthogonal derivative, where an n-th order derivative is approximated by an integral involving an orthogonal polynomial of degree n. This notion was reviewed in great detail in a paper by the author and Koornwinder [...] Read more.

This paper builds on the notion of the so-called orthogonal derivative, where an n-th order derivative is approximated by an integral involving an orthogonal polynomial of degree n. This notion was reviewed in great detail in a paper by the author and Koornwinder in 2012. Here, an approximation of the Weyl or Riemann–Liouville fractional derivative is considered by replacing the n-th derivative by its approximation in the formula for the fractional derivative. In the case of, for instance, Jacobi polynomials, an explicit formula for the kernel of this approximate fractional derivative can be given. Next, we consider the fractional derivative as a filter and compute the frequency response in the continuous case for the Jacobi polynomials and in the discrete case for the Hahn polynomials. The frequency response in this case is a confluent hypergeometric function. A different approach is discussed, which starts with this explicit frequency response and then obtains the approximate fractional derivative by taking the inverse Fourier transform. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus and Its Applications)

1856 KiB

Open AccessArticle

Fractional Euler-Lagrange Equations Applied to Oscillatory Systems

by Sergio Adriani David and Carlos Alberto Valentim, Jr.

Mathematics 2015, 3(2), 258-272; https://doi.org/10.3390/math3020258 - 20 Apr 2015

Cited by 17 | Viewed by 7297

Abstract

In this paper, we applied the Riemann-Liouville approach and the fractional Euler-Lagrange equations in order to obtain the fractional nonlinear dynamic equations involving two classical physical applications: “Simple Pendulum” and the “Spring-Mass-Damper System” to both integer order calculus (IOC) and fractional order calculus [...] Read more.

In this paper, we applied the Riemann-Liouville approach and the fractional Euler-Lagrange equations in order to obtain the fractional nonlinear dynamic equations involving two classical physical applications: “Simple Pendulum” and the “Spring-Mass-Damper System” to both integer order calculus (IOC) and fractional order calculus (FOC) approaches. The numerical simulations were conducted and the time histories and pseudo-phase portraits presented. Both systems, the one that already had a damping behavior (Spring-Mass-Damper) and the system that did not present any sort of damping behavior (Simple Pendulum), showed signs indicating a possible better capacity of attenuation of their respective oscillation amplitudes. This implication could mean that if the selection of the order of the derivative is conveniently made, systems that need greater intensities of damping or vibrating absorbers may benefit from using fractional order in dynamics and possibly in control of the aforementioned systems. Thereafter, we believe that the results described in this paper may offer greater insights into the complex behavior of these systems, and thus instigate more research efforts in this direction. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus and Its Applications)

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336 KiB

Open AccessArticle

Asymptotic Expansions of Fractional Derivatives andTheir Applications

by Tohru Morita and Ken-ichi Sato

Mathematics 2015, 3(2), 171-189; https://doi.org/10.3390/math3020171 - 15 Apr 2015

Cited by 5 | Viewed by 5093

Abstract

We compare the Riemann–Liouville fractional integral (fI) of a function f(z)with the Liouville fI of the same function and show that there are cases in which theasymptotic expansion of the former is obtained from those of the latter and the differenceof the two [...] Read more.

We compare the Riemann–Liouville fractional integral (fI) of a function f(z)with the Liouville fI of the same function and show that there are cases in which theasymptotic expansion of the former is obtained from those of the latter and the differenceof the two fIs. When this happens, this fact occurs also for the fractional derivative (fD).This method is applied to the derivation of the asymptotic expansion of the confluenthypergeometric function, which is a solution of Kummer’s differential equation. In thepresent paper, the solutions of the equation in the forms of the Riemann–Liouville fI orfD and the Liouville fI or fD are obtained by using the method, which Nishimoto used insolving the hypergeometric differential equation in terms of the Liouville fD. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus and Its Applications)

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305 KiB

Open AccessArticle

Analytical Solution of Generalized Space-Time Fractional Cable Equation

by Ram K. Saxena, Zivorad Tomovski and Trifce Sandev

Mathematics 2015, 3(2), 153-170; https://doi.org/10.3390/math3020153 - 9 Apr 2015

Cited by 6 | Viewed by 5313

Abstract

In this paper, we consider generalized space-time fractional cable equation in presence of external source. By using the Fourier-Laplace transform we obtain the Green function in terms of infinite series in H-functions. The fractional moments of the fundamental solution are derived and their [...] Read more.

In this paper, we consider generalized space-time fractional cable equation in presence of external source. By using the Fourier-Laplace transform we obtain the Green function in terms of infinite series in H-functions. The fractional moments of the fundamental solution are derived and their asymptotic behavior in the short and long time limit is analyzed. Some previously obtained results are compared with those presented in this paper. By using the Bernstein characterization theorem we find the conditions under which the even moments are non-negative. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus and Its Applications)

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339 KiB

Open AccessArticle

Fractional Diffusion in Gaussian Noisy Environment

by Guannan Hu and Yaozhong Hu

Mathematics 2015, 3(2), 131-152; https://doi.org/10.3390/math3020131 - 31 Mar 2015

Cited by 8 | Viewed by 4912

Abstract

We study the fractional diffusion in a Gaussian noisy environment as described by the fractional order stochastic heat equations of the following form: \(D_t^{(\alpha)} u(t, x)=\textit{B}u+u\cdot \dot W^H\), where \(D_t^{(\alpha)}\) is the Caputo fractional derivative of order \(\alpha\in (0,1)\) with respect to the [...] Read more.

We study the fractional diffusion in a Gaussian noisy environment as described by the fractional order stochastic heat equations of the following form: \(D_t^{(\alpha)} u(t, x)=\textit{B}u+u\cdot \dot W^H\), where \(D_t^{(\alpha)}\) is the Caputo fractional derivative of order \(\alpha\in (0,1)\) with respect to the time variable \(t\), \(\textit{B}\) is a second order elliptic operator with respect to the space variable \(x\in\mathbb{R}^d\) and \(\dot W^H\) a time homogeneous fractional Gaussian noise of Hurst parameter \(H=(H_1, \cdots, H_d)\). We obtain conditions satisfied by \(\alpha\) and \(H\), so that the square integrable solution \(u\) exists uniquely. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus and Its Applications)

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1133 KiB

Open AccessArticle

Basic Results for Sequential Caputo Fractional Differential Equations

by Bhuvaneswari Sambandham and Aghalaya S. Vatsala

Mathematics 2015, 3(1), 76-91; https://doi.org/10.3390/math3010076 - 19 Mar 2015

Cited by 26 | Viewed by 5716

Abstract

We have developed a representation form for the linear fractional differential equation of order q when 0 < q < 1, with variable coefficients. We have also obtained a closed form of the solution for sequential Caputo fractional differential equation of order [...] Read more.

We have developed a representation form for the linear fractional differential equation of order q when 0 < q < 1, with variable coefficients. We have also obtained a closed form of the solution for sequential Caputo fractional differential equation of order 2q, with initial and boundary conditions, for 0 < 2q < 1. The solutions are in terms of Mittag–Leffler functions of order q only. Our results yield the known results of integer order when q = 1. We have also presented some numerical results to bring the salient features of sequential fractional differential equations. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus and Its Applications)

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222 KiB

Open AccessCommunication

Existence Results for Fractional Neutral Functional Differential Equations with Random Impulses

by Annamalai Anguraj, Mullarithodi C. Ranjini, Margarita Rivero and Juan J. Trujillo

Mathematics 2015, 3(1), 16-28; https://doi.org/10.3390/math3010016 - 21 Jan 2015

Cited by 10 | Viewed by 5375

Abstract

In this paper, we investigate the existence of solutions for the fractional neutral differential equations with random impulses. The results are obtained by using Krasnoselskii’s fixed point theorem. Examples are added to show applications of the main results. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus and Its Applications)

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