This Special Issue of the scientific journal Axioms, entitled âRecent Advances in Fractional Calculusâ, is dedicated to one of the most dynamic areas of mathematical sciences today [...]
This Special Issue of the scientific journal Axioms, entitled âRecent Advances in Fractional Calculusâ, is dedicated to one of the most dynamic areas of mathematical sciences today [...]
With the rapid development of information technology,fractional calculus as an important branch of mathematics in signal analysis and processing and other fields has been widely studied and applied.In the course âŠ
With the rapid development of information technology,fractional calculus as an important branch of mathematics in signal analysis and processing and other fields has been widely studied and applied.In the course of handling the many problems,fractional calculus have advantages are gradually revealed.This paper expounds the fractional calculus in time domain and frequency domain definition,introduces its application in engineering.
Fractional Calculus (FC) was a bright idea of Gottfried Leibniz originating in the end of the seventeenth century. The topic was developed mainly in a mathematical framework, but during the âŠ
Fractional Calculus (FC) was a bright idea of Gottfried Leibniz originating in the end of the seventeenth century. The topic was developed mainly in a mathematical framework, but during the last decades FC was recognized to represent an useful tool for understanding and modeling many natural and artificial phenomena. Scientific areas including not only mathematics and physics, but also engineering, biology, finance, economy, chemistry and human sciences successfully applied FC concepts. The huge progress can be measured by the increasing number of papers, books, and conferences. This chapter presents a brief historical sketch and some bibliographic metrics of the evolution that occurred during the previous five decades.
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 âŠ
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.
The article gives the historical background and a brief introduction to fractional calculus. An overview of fractional calculus was also given as well as its potential applications. Examples of complex âŠ
The article gives the historical background and a brief introduction to fractional calculus. An overview of fractional calculus was also given as well as its potential applications. Examples of complex system modeled by means of fractional calculus was given.
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In this paper, we study some types of fractional differential equations which can be transformed into separable variables, regarding the Jumarie type of modified Riemann-Liouville fractional derivatives. We use a âŠ
In this paper, we study some types of fractional differential equations which can be transformed into separable variables, regarding the Jumarie type of modified Riemann-Liouville fractional derivatives. We use a new multiplication of fractional functions and product rule for fractional derivatives to obtain the solutions of these fractional differential equations. Furthermore, some examples are given to demonstrate our results.
The paper presents analysis of the second order band-pass and notch filter with a dynamic damping factor ÎČd of fractional order. Factor ÎČd is given in the form of fractional âŠ
The paper presents analysis of the second order band-pass and notch filter with a dynamic damping factor ÎČd of fractional order. Factor ÎČd is given in the form of fractional differentiator of order a, i.e. ÎČd=ÎČ/sa , where ÎČ and a are adjustable parameters. The aim of the paper is to exploit an extra degree of freedom of presented filters to achieve the desired filter specifications and obtain a desired response in the frequency and time domain. Shaping of the frequency response enables achieving a better phase response compared to the integer-order counterparts which is of great concern in many applications. For the implementation purpose, the paper presents a comparison of four discretization techniques: the Osutaloup's Recursive Algorithm (ORA+Tustin), Continued Fractional Expansion (CFE+Tustin), Interpolation of Frequency Characteristic (IFC+Tustin) and recently proposed AutoRegressive with eXogenous input (ARX)-based direct discretization method.
Abstract In this paper, without requiring the complete continuity of integral operators and the existence of upperâlower solutions, by means of the sum-type mixed monotone operator fixed point theorem based âŠ
Abstract In this paper, without requiring the complete continuity of integral operators and the existence of upperâlower solutions, by means of the sum-type mixed monotone operator fixed point theorem based on the cone $P_{h}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>P</mml:mi><mml:mi>h</mml:mi></mml:msub></mml:math> , we investigate a kind of p -Laplacian differential equation RiemannâStieltjes integral boundary value problem involving a tempered fractional derivative. Not only the existence and uniqueness of positive solutions are obtained, but also we can construct successively sequences for approximating the unique positive solution. As an application of our fundamental aims, we offer a realistic example to illustrate the effectiveness and practicability of the main results.
In this article, we study a model problem for the advectionâreactionâdiffusion equation involving a new nonsingular time-fractional derivative with Rabotnov fractional-exponential (RFE) kernel. In order to solve this model numerically, âŠ
In this article, we study a model problem for the advectionâreactionâdiffusion equation involving a new nonsingular time-fractional derivative with Rabotnov fractional-exponential (RFE) kernel. In order to solve this model numerically, we first obtain the numerical approximation of RFE fractional derivative for a simple polynomial function, which gives rise to an operational matrix of fractional differentiation. We illustrate the accuracy and validity of this operational matrix with the aid of an example. We use Legendre collocation technique together with the newly developed operational matrix to find the numerical solution of the given model. The numerical results depict the feasibility and efficacy of our method. The error estimates show that our method is valid with great accuracy and is applicable to a fractional ODE system and an integral equation with RFE kernel fractional derivative.
The Cattaneo-Vernotte equation is a generalization of the heat and particle diffusion equations; this mathematical model combines waves and diffusion with a finite velocity of propagation. In disordered systems the âŠ
The Cattaneo-Vernotte equation is a generalization of the heat and particle diffusion equations; this mathematical model combines waves and diffusion with a finite velocity of propagation. In disordered systems the diffusion can be anomalous. In these kinds of systems, the mean-square displacement is proportional to a fractional power of time not equal to one. The anomalous diffusion concept is naturally obtained from diffusion equations using the fractional calculus approach. In this paper we present an alternative representation of the Cattaneo-Vernotte equation using the fractional calculus approach; the spatial-time derivatives of fractional order are approximated using the Caputo-type derivative in the range<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mo stretchy="false">(</mml:mo><mml:mn mathvariant="normal">0,2</mml:mn><mml:mo stretchy="false">]</mml:mo></mml:math>. In this alternative representation we introduce the appropriate fractional dimensional parameters which characterize consistently the existence of the fractional space-time derivatives into the fractional Cattaneo-Vernotte equation. Finally, consider the Dirichlet conditions, the Fourier method was used to find the full solution of the fractional Cattaneo-Vernotte equation in analytic way, and Caputo and Riesz fractional derivatives are considered. The advantage of our representation appears according to the comparison between our model and models presented in the literature, which are not acceptable physically due to the dimensional incompatibility of the solutions. The classical cases are recovered when the fractional derivative exponents are equal to<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:math>.
Abstract Fractional derivatives have non-local character, although they are not mathematical derivatives, according to differential topology. New fractional derivatives satisfying the requirements of differential topology are proposed, that have non-local âŠ
Abstract Fractional derivatives have non-local character, although they are not mathematical derivatives, according to differential topology. New fractional derivatives satisfying the requirements of differential topology are proposed, that have non-local character. A new space, the Î -space corresponding to the initial space is proposed, where the derivatives are local. Transferring the results to the initial space through Riemann-Liouville fractional derivatives, the non-local character of the analysis is shown up. Since fractional derivatives have been established, having the mathematical properties of the derivatives, the linearly elastic fractional deformation of an elastic bar is presented. The fractional axial stress along the distributed body force is discussed. Fractional analysis with horizon is also introduced and the deformation of an elastic bar is also presented.
Fractional calculus tools have been exploited to effectively model variety of engineering, physics and applied sciences problems. The concept of fractional derivative has been incorporated in the optimization process of âŠ
Fractional calculus tools have been exploited to effectively model variety of engineering, physics and applied sciences problems. The concept of fractional derivative has been incorporated in the optimization process of least mean square (LMS) iterative adaptive method. This study exploits the recently introduced enhanced fractional derivative based LMS (EFDLMS) for parameter estimation of power signal formed by the combination of different sinusoids. The EFDLMS addresses the issue of fractional extreme points and provides faster convergence speed. The performance of EFDLMS is evaluated in detail by taking different levels of noise in the composite sinusoidal signal as well as considering various fractional orders in the EFDLMS. Simulation results reveal that the EDFLMS is faster in convergence speed than the conventional LMS (i.e., EFDLMS for unity fractional order).
Abstract This manuscript is dedicated to prove a new inequality that involves an important case of Leibniz rule regarding RiemannâLiouville and Caputo fractional derivatives of order . In the context âŠ
Abstract This manuscript is dedicated to prove a new inequality that involves an important case of Leibniz rule regarding RiemannâLiouville and Caputo fractional derivatives of order . In the context of partial differential equations, the aforesaid inequality allows us to address the FaedoâGalerkin method to study several kinds of partial differential equations with fractional derivative in the time variable; particularly, we apply these ideas to prove the existence and uniqueness of solution to the fractional version of the 2D unsteady Stokes equations in bounded domains.
The purpose of this work is to study the memory effect analysis of CaputoâFabrizio time fractional diffusion equation by means of cubic B-spline functions. The CaputoâFabrizio interpretation of fractional derivative âŠ
The purpose of this work is to study the memory effect analysis of CaputoâFabrizio time fractional diffusion equation by means of cubic B-spline functions. The CaputoâFabrizio interpretation of fractional derivative involves a non-singular kernel that permits to describe some class of material heterogeneities and the effect of memory more effectively. The proposed numerical technique relies on finite difference approach and cubic B-spline functions for discretization along temporal and spatial grids, respectively. To ensure that the error does not amplify during computational process, stability analysis is performed. The described algorithm is second-order convergent along time and space directions. The computational competence of the scheme is tested through some numerical examples. The results reveal that the current scheme is reasonably efficient and reliable to be used for solving the subject problem.
Inequalities play important roles not only in mathematics but also in other fields, such as economics and engineering. Even though many results are published as HermiteâHadamard (H-H)-type inequalities, new researchers âŠ
Inequalities play important roles not only in mathematics but also in other fields, such as economics and engineering. Even though many results are published as HermiteâHadamard (H-H)-type inequalities, new researchers to these fields often find it difficult to understand them. Thus, some important discoverers, such as the formulations of H-H-type inequalities of α-type real-valued convex functions, along with various classes of convexity through differentiable mappings and for fractional integrals, are presented. Some well-known examples from the previous literature are used as illustrations. In the many above-mentioned inequalities, the symmetrical behavior arises spontaneously.
Abstract The purpose of this paper is to introduce a new time-fractional heat conduction model with three-phase-lags and three distinct fractional-order derivatives. We investigate the introduced model in the situation âŠ
Abstract The purpose of this paper is to introduce a new time-fractional heat conduction model with three-phase-lags and three distinct fractional-order derivatives. We investigate the introduced model in the situation of an isotropic and homogeneous solid sphere. The exterior of the sphere is exposed to a thermal shock and a decaying heat generation rate. We recuperate some earlier thermoelasticity models as particular cases from the proposed model. Moreover, the effects of different fractional thermoelastic models and the effect of instant time on the physical variables of the medium are studied. We obtain the numerical solutions for the various physical fields using a numerical Laplace inversion technique. We represent the obtained results graphically and discuss them. Physical views presented in this article may be useful for the design of new materials, bio-heat transfer mechanisms between tissues and other scientific domains.
In this paper, approximation of space fractional order diffusion equation are considered using compact finite difference technique to discretize the time derivative, which was then approximated via shifted Gegenbauer polynomials âŠ
In this paper, approximation of space fractional order diffusion equation are considered using compact finite difference technique to discretize the time derivative, which was then approximated via shifted Gegenbauer polynomials using zeros of (N - 1) degree shifted Gegenbauer polynomial as collocation points. The important feature in this approach is that it reduces the problems to algebraic linear system of equations together with the boundary conditions gives (N + 1) linear equations. Some theorems are given to establish the convergence and the stability of the proposed method. To validate the efficiency and the accuracy of the method, obtained results are compared with the existing results in the literature. The graphical representation are also displayed for various values of \beta Gegenbauer polynomials. It can be observe in the tables of the results and figures that the proposed method performs better than the existing one in the literature.
Orthogonal polynomials are the natural way to express the elements of the inner product spaces as an infinite sum of orthonormal basis sets. The construction and development of the many âŠ
Orthogonal polynomials are the natural way to express the elements of the inner product spaces as an infinite sum of orthonormal basis sets. The construction and development of the many important numerical algorithms are based on the operational matrices of orthogonal polynomials including spectral tau, spectral collocation, and operational matrices approach are few of them. The widely used orthogonal polynomials are Legendre, Jacobi, and Chebyshev. However, only a few papers are available where the Hermite polynomials (HPs) were exploited to solve numerically the differential equations. The notable characteristic of the HPs is its ability to approximate the square-integrable functions on the entire real line. The prime objective of this chapter is to introduce the two new generalized operational matrices of HPs which are developed in the sense of the Riemann-Liouville fractional-order integral operator and Hilfer fractional-order derivative operator. The newly derived operational matrices are further used to construct a numerical algorithm for solving the Bagley--Trovik types fractional derivative differential equations (FDDE). Moreover, the results obtained by using the proposed algorithm are compared with the results obtained otherwise to demonstrate the efficiency and accuracy of the proposed numerical algorithm. Some examples are solved for application purposes.
<abstract><p>In this work, an epidemic model of a susceptible, exposed, infected and recovered SEIR-type is established for the distinctive dynamic compartments and epidemic characteristics of COVID-19 as it spreads across âŠ
<abstract><p>In this work, an epidemic model of a susceptible, exposed, infected and recovered SEIR-type is established for the distinctive dynamic compartments and epidemic characteristics of COVID-19 as it spreads across a population with a heterogeneous rate. The proposed model is investigated using a novel approach of fractional calculus known as piecewise derivatives. The existence theory is demonstrated through the establishment of sufficient conditions. In addition, result related to Hyers-Ulam stability is also derived for the considered model. A numerical method based on modified Euler procedure is also constructed to simulate the approximate solutions of the proposed model by employing various values of fractional orders. We testified the numerical results by using real available data of Japan. In addition, some results for the SEIR-type model are also presented graphically using the stochastic process, and the obtained results are discussed.</p></abstract>
Fractional-order differential equations are increasingly used to model systems in engineering for purposes such as control and health-monitoring. Because of the nature of a fractional derivative, mechanistically fractional-order dynamics will âŠ
Fractional-order differential equations are increasingly used to model systems in engineering for purposes such as control and health-monitoring. Because of the nature of a fractional derivative, mechanistically fractional-order dynamics will most naturally arise when there are non-local features in the dynamics. Even if there are no non-local effects, however, when searching for an approximate model for a very high order system, it is worth considering whether a fractional-order model is better than an integer-order model. This work is motivated by the challenges presented by very large scale systems, which will be increasingly common as integration of the control of formerly decoupled systems occurs such as in cyber-physical systems. Because fractional-order differential equations are more difficult to numerically compute, justifying the use of a fractional-order model is a balance between accuracy of the approximation and ease of computation. This paper constructs large, random networks and compares the accuracy of integer-order and fractional-order models for their dynamics. Over the range of parameter values considered, fractional-order models generally provide a more accurate approximation to the response of the system than integer order models. To ensure a fair comparison, both the fractional-order and integer-order models considered had two parameters.
This paper mainly focuses on the LMS algorithm based on fractional order gradient information, which extends the first order gradient of traditional LMS algorithm to fractional order a (0 < âŠ
This paper mainly focuses on the LMS algorithm based on fractional order gradient information, which extends the first order gradient of traditional LMS algorithm to fractional order a (0 < a †1). Since the LMS algorithm generally does not converge to the true extreme value point of the objective function when fractional order gradient is used. To ensure convergence, the strategy of truncating the second order term of the fractional order gradient expansion of the objective function is adopted. The step size condition for convergence is given and the performance of the algorithm with different fractional orders a is analyzed. It is shown that under certain conditions, a larger fractional order a will lead a faster convergence speed. Finally, the effectiveness of the proposed algorithm is illustrated by four simulation examples.
Abstract In this paper, the new exact solutions of nonlinear conformable fractional partial differential equations(CFPDEs) are achieved by using auxiliary equation method for the nonlinear space-time fractional Klein-Gordon equation and âŠ
Abstract In this paper, the new exact solutions of nonlinear conformable fractional partial differential equations(CFPDEs) are achieved by using auxiliary equation method for the nonlinear space-time fractional Klein-Gordon equation and the (2+1)-dimensional time-fractional Zoomeron equation. The technique is easily applicable which can be applied successfully to get the solutions for different types of nonlinear CFPDEs. The conformable fractional derivative(CFD) definitions are used to cope with the fractional derivatives.
Abstract In this paper, we study a type of nonlinear fractional differential equations multi-point boundary value problem with fractional derivative in the boundary conditions. By using the upper and lower âŠ
Abstract In this paper, we study a type of nonlinear fractional differential equations multi-point boundary value problem with fractional derivative in the boundary conditions. By using the upper and lower solutions method and fixed point theorems, some results for the existence of positive solutions for the boundary value problem are established. Some examples are also given to illustrate our results.
A mathematical model which is non-linear in nature with non-integer order Ï, $0 < \phi \leq 1$ is presented for exploring the SIRV model with the rate of vaccination $\mu âŠ
A mathematical model which is non-linear in nature with non-integer order Ï, $0 < \phi \leq 1$ is presented for exploring the SIRV model with the rate of vaccination $\mu _{1}$ and rate of treatment $\mu _{2}$ to describe a measles model. Both the disease free $\mathcal{F}_{0}$ and the endemic $\mathcal{F}^{*}$ points have been calculated. The stability has also been argued for using the theorem of stability of non-integer order differential equations. $\mathcal{R} _{0}$ , the basic reproduction number exhibits an imperative role in the stability of the model. The disease free equilibrium point $\mathcal{F}_{0}$ is an attractor when $\mathcal{R}_{0} < 1$ . For $\mathcal{R}_{0} > 1$ , $\mathcal{F}_{0}$ is unstable, the endemic equilibrium $\mathcal{F}^{*}$ subsists and it is an attractor. Numerical simulations of considerable model are also supported to study the behavior of the system.
Recently, various techniques and methods have been employed by mathematicians to solve specific types of fractional differential equations (FDEs) with symmetric properties. The study focuses on Navier-Stokes equations (NSEs) that âŠ
Recently, various techniques and methods have been employed by mathematicians to solve specific types of fractional differential equations (FDEs) with symmetric properties. The study focuses on Navier-Stokes equations (NSEs) that involve MHD effects with time-fractional derivatives (FDs). The (NSEs) with time-FDs of order ÎČâ(0,1) are investigated. To facilitate anomalous diffusion in fractal media, mild solutions and Mittag-Leffler functions are used. In HÎŽ,r, the existence, and uniqueness of local and global mild solutions are proved, as well as the symmetric structure created. Moderate local solutions are provided in Jr. Moreover, the regularity and existence of classical solutions to the equations in Jr. are established and presented.
This manuscript is devoted to presenting an approximation method based upon a new set of fractional functions named fractional-order MittagâLeffler functions (FM-LFs). This scheme is implemented to approximate the solution âŠ
This manuscript is devoted to presenting an approximation method based upon a new set of fractional functions named fractional-order MittagâLeffler functions (FM-LFs). This scheme is implemented to approximate the solution of a multi-dimensional fractional pantograph differential equation. To this approach, FM-LFs are introduced. Then, we employ FM-LFs to construct the pseudo-operational matrix of fractional integration and pantograph operational matrix (P-OM). We reduce the considered problems to systems of algebraic equations with the help of the mentioned matrices, and the collocation technique, respectively. Error analysis is proposed. Moreover, several numerical experiments have been considered to confirm the efficiency and applicability of the suggested scheme.
A coupled system under Caputo-Fabrizio fractional order derivative (CFFOD) with antiperiodic boundary condition is considered. We use piecewise version of CFFOD. Sufficient conditions for the existence and uniqueness of solution âŠ
A coupled system under Caputo-Fabrizio fractional order derivative (CFFOD) with antiperiodic boundary condition is considered. We use piecewise version of CFFOD. Sufficient conditions for the existence and uniqueness of solution by ap?plying the Banach, Krasnoselskii?s fixed point theorems. Also some appropriate results for Hyers-Ulam (H-U) stability analysis is established. Proper example is given to verify the results.
This paper solves a generalized class of first-order fractional ordinary differential equations (1st-order FODEs) by means of RiemannâLiouville fractional derivative (RLFD). The principal incentive of this paper is to generalize âŠ
This paper solves a generalized class of first-order fractional ordinary differential equations (1st-order FODEs) by means of RiemannâLiouville fractional derivative (RLFD). The principal incentive of this paper is to generalize some existing results in the literature. An effective approach is applied to solve non-homogeneous fractional differential systems containing 2n periodic terms. The exact solutions are determined explicitly in a straightforward manner. The solutions are expressed in terms of entire functions with fractional order arguments. Features of the current solutions are discussed and analyzed. In addition, the existing solutions in the literature are recovered as special cases of our results.
Historical Survey The Modern Approach The Riemann-Liouville Fractional Integral The Riemann-Liouville Fractional Calculus Fractional Differential Equations Further Results Associated with Fractional Differential Equations The Weyl Fractional Calculus Some Historical Arguments.
Historical Survey The Modern Approach The Riemann-Liouville Fractional Integral The Riemann-Liouville Fractional Calculus Fractional Differential Equations Further Results Associated with Fractional Differential Equations The Weyl Fractional Calculus Some Historical Arguments.
On the H-Function With Applications.- H-Function in Science and Engineering.- Fractional Calculus.- Applications in Statistics.- Functions of Matrix Argument.- Applications in Astrophysics Problems.
On the H-Function With Applications.- H-Function in Science and Engineering.- Fractional Calculus.- Applications in Statistics.- Functions of Matrix Argument.- Applications in Astrophysics Problems.
Introduction Generalized operatorsz of fractional integration and differentiation Recent aspects of classical Erdelyi-Kober operators Hyper-Bessel differential and integral operators and equations Applications to the generalized hypergeometric functions Further generalizations and âŠ
Introduction Generalized operatorsz of fractional integration and differentiation Recent aspects of classical Erdelyi-Kober operators Hyper-Bessel differential and integral operators and equations Applications to the generalized hypergeometric functions Further generalizations and applications Appendis: Definitions, examples and properties of the special functions used in this book References Citation index
In the last two decades, fractional (or non integer) differentiation has played a very important role in various fields such as mechanics, electricity, chemistry, biology, economics, control theory and signal âŠ
In the last two decades, fractional (or non integer) differentiation has played a very important role in various fields such as mechanics, electricity, chemistry, biology, economics, control theory and signal and image processing. For example, in the last three fields, some important considerations such as modelling, curve fitting, filtering, pattern recognition, edge detection, identification, stability, controllability, observability and robustness are now linked to long-range dependence phenomena. Similar progress has been made in other fields listed here. The scope of the book is thus to present the state of the art in the study of fractional systems and the application of fractional differentiation. As this volume covers recent applications of fractional calculus, it will be of interest to engineers, scientists, and applied mathematicians.
We present a numerical method for the Monte Carlo simulation of uncoupled continuous-time random walks with a Levy alpha-stable distribution of jumps in space and a Mittag-Leffler distribution of waiting âŠ
We present a numerical method for the Monte Carlo simulation of uncoupled continuous-time random walks with a Levy alpha-stable distribution of jumps in space and a Mittag-Leffler distribution of waiting times, and apply it to the stochastic solution of the Cauchy problem for a partial differential equation with fractional derivatives both in space and in time. The one-parameter Mittag-Leffler function is the natural survival probability leading to time-fractional diffusion equations. Transformation methods for Mittag-Leffler random variables were found later than the well-known transformation method by Chambers, Mallows, and Stuck for Levy alpha-stable random variables and so far have not received as much attention; nor have they been used together with the latter in spite of their mathematical relationship due to the geometric stability of the Mittag-Leffler distribution. Combining the two methods, we obtain an accurate approximation of space- and time-fractional diffusion processes almost as easy and fast to compute as for standard diffusion processes.
The generalized Mittag-Leffler function Eα, ÎČ(z) has been studied for arbitrary complex argument zââ and parameters αââ+ and ÎČââ. This function plays a fundamental role in the theory of fractional âŠ
The generalized Mittag-Leffler function Eα, ÎČ(z) has been studied for arbitrary complex argument zââ and parameters αââ+ and ÎČââ. This function plays a fundamental role in the theory of fractional differential equations and numerous applications in physics. The Mittag-Leffler function interpolates smoothly between exponential and algebraic functional behaviour. A numerical algorithm for its evaluation has been developed. The algorithm is based on integral representations and exponential asymptotics. Results of extensive numerical calculations for Eα, ÎČ(z) in the complex z-plane are reported here. We find that all complex zeros emerge from the point z=1 for small α. They diverge towards ââ+(2kâ1)Ïi for αâ1â and towards ââ+2kÏi for αâ1+ (kââ€). All the complex zeros collapse pairwise onto the negative real axis for αâ2. We introduce and study also the inverse generalized Mittag-Leffler function Lα, ÎČ(z) defined as the solution of the equation Lα, ÎČ(Eα, ÎČ(z))=z. We determine its principal branch numerically.
We review the function theoretical properties of the Mittag-Leffler function $E_{a,b}\left( z\right) $ in a self-contained manner, but also add new results; more than half is new!
We review the function theoretical properties of the Mittag-Leffler function $E_{a,b}\left( z\right) $ in a self-contained manner, but also add new results; more than half is new!
The subject of fractional calculus (that is, calculus of integrals and derivatives of any arbitrary real or complex order) has gained importance and popularity during the past three decades or âŠ
The subject of fractional calculus (that is, calculus of integrals and derivatives of any arbitrary real or complex order) has gained importance and popularity during the past three decades or so, due mainly to its demonstrated applications in numerous seemingly diverse fields of science and engineering. Indeed it provides several potentially useful tools for solving differential, integral, and integro-differential 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 purpose of this expository article is to provide a rather brief introduction to the theory and applications of fractional calculus.
The Derivatives. Limits. Indefinite Integrals. Definite Integrals. Infinite Series. The Connection Formulas. Representations of Hypergeometric Functions and the Meijer G Function.
The Derivatives. Limits. Indefinite Integrals. Definite Integrals. Infinite Series. The Connection Formulas. Representations of Hypergeometric Functions and the Meijer G Function.
An introduction to fractional calculus, P.L. Butzer & U. Westphal fractional time evolution, R. Hilfer fractional powers of infinitesimal generators of semigroups, U. Westphal fractional differences, derivatives and fractal time âŠ
An introduction to fractional calculus, P.L. Butzer & U. Westphal fractional time evolution, R. Hilfer fractional powers of infinitesimal generators of semigroups, U. Westphal fractional differences, derivatives and fractal time series, B.J. West and P. Grigolini fractional kinetics of Hamiltonian chaotic systems, G.M. Zaslavsky polymer science applications of path integration, integral equations, and fractional calculus, J.F. Douglas applications to problems in polymer physics and rheology, H. Schiessel et al applications of fractional calculus and regular variation in thermodynamics, R. Hilfer.
In Memoriam: Radu Balescu Part I FRACTIONAL CALCULUS AND STOCHASTIC THEORY Threefold Introduction to Fractional Derivatives (Rudolf Hilfer) Random Processes with Infinite Moments (Michael F. Shlesinger) Continuous Time Random Walk, âŠ
In Memoriam: Radu Balescu Part I FRACTIONAL CALCULUS AND STOCHASTIC THEORY Threefold Introduction to Fractional Derivatives (Rudolf Hilfer) Random Processes with Infinite Moments (Michael F. Shlesinger) Continuous Time Random Walk, Mittag-Leffler Waiting Time and Fractional Diffusion: Mathematical Aspects (Rudolf Gorenflo and Francesco Mainardi) Introduction to the Theory of Levy Flights (Alexei V. Chechkin, Ralf Metzler, Joseph Klafter, and Vsevolod Yu. Gonchar) Fractional Diffusion Models of Anomalous Transport (Diego del-Castillo-Negrete) Anomalous Kinetics Leads to Weak Ergodicity Breaking (Eli Barkai) Part II DYNAMICAL SYSTEMS AND DETERMINISTIC TRANSPORT Deterministic (Anomalous) Transport (Roberto Artuso and Giampaolo Cristadoro) Anomalous Transport in Hamiltonian Systems (Eduardo G. Altmann and Holger Kantz) Anomalous Heat Conduction (Stefano Lepri, Roberto Livi, and Antonio Politi) Part III ANOMALOUS TRANSPORT IN DISORDERED SYSTEMS Anomalous Relaxation in Complex Systems: From Stretched to Compressed Exponentials (Jean-Philippe Bouchaud) Anomalous Transport in Glass-Forming Liquids (Walter Kob, Gustavo A. Appignanesi, J. Ariel Rodriguez Fris, and Ruben A. Montani) Subdiffusion-Limited Reactions (Santos Bravo Yuste, Katja Lindenberg, and Juan Jesus Ruiz-Lorenzo) Anomalous Transport on Disordered Fractals (Karl Heinz Hoffmann and Janett Prehl) Part IV APPLICATIONS TO COMPLEX SYSTEMS AND EXPERIMENTAL RESULTS Superstatistics: Theoretical Concepts and Physical Applications (Christian Beck) Money Circulation Science - Fractional Dynamics in Human Mobility (Dirk Brockmann) Anomalous Molecular Displacement Laws in Porous Media and Polymers Probed by Nuclear Magnetic Resonance Techniques (Rainer Kimmich, Nail Fatkullin, Markus Kehr, and Yujie Li) Anomalous Molecular Dynamics in Confined Spaces (Rustem Valiullin and Jorg Karger) Paradigm Shift of the Molecular Dynamics Concept in the Cell Membrane: High-Speed Single-Molecule Tracking Revealed the Partitioning of the Cell Membrane (A. Kusumi, Y. Umemura, N. Morone, and T. Fujiwara)
Definition, Representations and Expansions of the H-Function. Properties of the H-Function H-Transform on the Space Ln,2. H-Transform on the Space L n,t Modified H-Transforms on the Space L n,t. G-Transform âŠ
Definition, Representations and Expansions of the H-Function. Properties of the H-Function H-Transform on the Space Ln,2. H-Transform on the Space L n,t Modified H-Transforms on the Space L n,t. G-Transform and Modified G-Transforms on the Space L n,t. Hypergeometric Type Integral Transforms on the Space L n,t. Bessel Type Integral Transforms on the Space L n,t. Bibliography Subject Index Author Index Symbol Index.
Preface. Acknowledgments. Introduction. Signals, Systems, and Transformations. Wigner Distributions and Linear Canonical Transforms. The Fractional Fourier Transform. Time-Order and Space-Order Representations. The Discrete Fractional Fourier Transform. Optical Signals and Systems. âŠ
Preface. Acknowledgments. Introduction. Signals, Systems, and Transformations. Wigner Distributions and Linear Canonical Transforms. The Fractional Fourier Transform. Time-Order and Space-Order Representations. The Discrete Fractional Fourier Transform. Optical Signals and Systems. Phase-Space Optics. The Fractional Fourier Transform in Optics. Applications of the Fractional Fourier Transform to Filtering, Estimation, and Signal Recovery. Applications of the Fractional Fourier Transform to Matched Filtering, Detection, and Pattern Recognition. Bibliography on the Fractional Fourier Transform. Other Cited Works. Credits. Index.
Basic theory and representation formulas.- Applications of Abel's original integral equation: Determination of potentials.- Applications of a transformed abel integral equation.- Smoothing properties of the abel operators.- Existence and uniqueness âŠ
Basic theory and representation formulas.- Applications of Abel's original integral equation: Determination of potentials.- Applications of a transformed abel integral equation.- Smoothing properties of the abel operators.- Existence and uniqueness theorems.- Relations between abel transform and other integral transforms.- Nonlinear abel integral equations of second kind.- Illposedness and stabilization of linear abel integral equations of first kind.- On numerical treatment of first kind abel integral equations.