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 …
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.
The concept of fractional calculus (fractional derivative and fractional integral) is not new. In recent years, fractional calculus has found use in studies of viscoelastic materials, as well as in …
The concept of fractional calculus (fractional derivative and fractional integral) is not new. In recent years, fractional calculus has found use in studies of viscoelastic materials, as well as in many fields of science and engineering including fluid flow, rheology, diffusive transport, electrical
networks, electromagnetic theory and probability. The definitions of Fractional Derivatives and Integrals, that I chose to consider: Grunwald-Letnikov Fractional Derivatives; Riemann-Liouville Fractional Integrals; Riemann-Liouville Fractional Derivatives; Caputo Fractional Derivatives. We take the phenomenological diffusion equation and study the different cases leading to anomalous
diffusion. Modelling the memory effects of anomalous diffusion, we get the fractional time form of
Cattaneo equation, with 0 < ɑ < 1 and D is the diffusion constant with τ<< 1. If ɑ= 1, the normal Cattaneo diffusion equation is recovered.
We try to model the fractional diffusion equations for anomalous diffusion cases and simulate the
diffusion phenomena.
Fractional calculus (FC) is a new mathematical concept that has a wide range of applicability in science and engineering. A few of the findings have been published in publications or …
Fractional calculus (FC) is a new mathematical concept that has a wide range of applicability in science and engineering. A few of the findings have been published in publications or relevant research papers. Nevertheless, we are only at the beginning of using this extremely powerful instrument in a variety of fields of study. At this time, proportional calculus has spread its wings even wider to include the dynamics of the actual world's complexities, and novel concepts are being applied and evaluated on real data.
The subject of fractional calculus has emerged as a powerful and efficient mathematical instrument during the past six decades, mainly due to its demonstrated applications in numerous seemingly diverse and …
The subject of fractional calculus has emerged as a powerful and efficient mathematical instrument during the past six decades, mainly due to its demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering. Although researchers have already reported many excellent results inseveral seminal monographs and review articles, there arestill alarge number of non-local phenomena unexplored and waiting to be discovered. In this perspective, this paper investigates the use of Fractional Calculus in the fields of Physics, Mechanics, Biology, Engineering and Signal Processing. We hope this incomplete, but significant, details will guide young researchers and help new comers to see some of the important applications. We expect this collection of review will also benefit our society.
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 …
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.
In this paper we present a new approach leads to innovative methods for estimating the parameters of distributions defined on R + by ap- plying Fourier transform on Riemann-Liouville fractional …
In this paper we present a new approach leads to innovative methods for estimating the parameters of distributions defined on R + by ap- plying Fourier transform on Riemann-Liouville fractional differential and inte- gral operator and by Mellin transform of the characterization function. These methods are useful specially when the density of the random variable has power-law tails. Estimating the class of complex moments using characteristic function, which include both integer and fractional moments, we show that random variable with power law distribution can be represented within this approach, even if its integer moments diverge.
Preface ix Part 1. Mathematical Preliminaries, Definitions and Properties of Fractional Integrals and Derivatives 1 Chapter 1. Mathematical Preliminaries 3 Chapter 2. Basic Definitions and Properties of Fractional Integrals and …
Preface ix Part 1. Mathematical Preliminaries, Definitions and Properties of Fractional Integrals and Derivatives 1 Chapter 1. Mathematical Preliminaries 3 Chapter 2. Basic Definitions and Properties of Fractional Integrals and Derivatives 17 Part 2. Mechanical Systems 49 Chapter 3. Restrictions Following from the Thermodynamics for Fractional Derivative Models of a Viscoelastic Body 51 Chapter 4. Vibrations with Fractional Dissipation 83 Chapter 5. Lateral Vibrations and Stability of Viscoelastic Rods 123 Chapter 6. Fractional Diffusion-Wave Equations 185 Chapter 7. Fractional Heat Conduction Equations 257 Bibliography 289 Index 311
The fractional calculus (FC) goes back to the beginning of the theory of differential calculus. Nevertheless, the application of FC just emerged in the last two decades, due to the …
The fractional calculus (FC) goes back to the beginning of the theory of differential calculus. Nevertheless, the application of FC just emerged in the last two decades, due to the progress in the area of chaos and nonlinear dynamics that revealed subtle relationships with the FC concepts. In the field of dynamical systems theory some work has been carried out but the proposed models and algorithms are still in a preliminary stage of establishment. Having these ideas in mind, the paper discusses the application of FC in engineering sciences.
In this paper a new function called as K-function, which is an extension of the generalization of the Mittag-Leffler function[10,11 ] and its generalized form introduced by Prabhakar[20], is introduced …
In this paper a new function called as K-function, which is an extension of the generalization of the Mittag-Leffler function[10,11 ] and its generalized form introduced by Prabhakar[20], is introduced and stud ied by the author in terms of some special functions and derived the relations t hat exists between the Kfunction and the operators of Riemann-Liouville fra ctional integrals and derivatives.
We study fractional variational problems in terms of a generalized fractional integral with Lagrangians depending on classical derivatives, generalized fractional integrals and derivatives. We obtain necessary optimality conditions for the …
We study fractional variational problems in terms of a generalized fractional integral with Lagrangians depending on classical derivatives, generalized fractional integrals and derivatives. We obtain necessary optimality conditions for the basic and isoperimetric problems, as well as natural boundary conditions for free‐boundary value problems. The fractional action‐like variational approach (FALVA) is extended and some applications to physics discussed.
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 …
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.
We investigate some basic applications of Fractional Calculus (FC) to Newtonian mechanics. After a brief review of FC, we consider a possible generalization of Newton's second law of motion and …
We investigate some basic applications of Fractional Calculus (FC) to Newtonian mechanics. After a brief review of FC, we consider a possible generalization of Newton's second law of motion and apply it to the case of a body subject to a constant force. In our second application of FC to Newtonian gravity, we consider a generalized fractional gravitational potential and derive the related circular orbital velocities. This analysis might be used as a tool to model galactic rotation curves, in view of the dark matter problem. Both applications have a pedagogical value in connecting fractional calculus to standard mechanics and can be used as a starting point for a more advanced treatment of fractional mechanics.
Abstract Complex fractional moment (CFM), which is defined as the Mellin transform of a probability density function (PDF), has been successfully employed to find the response PDF of a wide …
Abstract Complex fractional moment (CFM), which is defined as the Mellin transform of a probability density function (PDF), has been successfully employed to find the response PDF of a wide variety of integer-order nonlinear oscillators. In this paper, a CFM-based analysis is performed to determine the transient response PDF of nonlinear oscillators with fractional derivative elements under Gaussian white noise. First, an equivalent linear system is introduced for the purpose of deriving the Fokker–Planck (FP) equation for response amplitude. The equivalent natural frequency and equivalent damping coefficient of the system need to be determined, taking into account both the nonlinear and fractional derivative elements of the original oscillator. Moreover, to convert the FP equation into the governing equation of CFMs, these equivalent coefficients must be given in polynomial form of amplitude. This paper proposes formulas for appropriately determining the equivalent coefficients, based on an equivalent linearization technique. Then, applying stochastic averaging, the FP equation is derived from the equivalent linear system. Next, the Mellin transform converts the FP equation into coupled linear ordinary differential equations for amplitude CFMs, which are solved with a constraint corresponding to the normalization condition for a PDF. Finally, the inverse Mellin transform of the CFMs yields the amplitude PDF. The joint PDF of displacement and velocity is also obtained from the amplitude PDF. Three linear and nonlinear fractional oscillators are considered in numerical examples. For all cases, the analytical results are in good agreement with the pertinent Monte Carlo simulation results.
In the present paper, we are concerned with an inverse source Cauchy weighted problem involving a one-dimensional diffusion equation with a time-fractional Riemann-Liouville derivative with $0&lt;\alpha &lt;1$. We start with …
In the present paper, we are concerned with an inverse source Cauchy weighted problem involving a one-dimensional diffusion equation with a time-fractional Riemann-Liouville derivative with $0&lt;\alpha &lt;1$. We start with results on the existence and regularity of the weak solution of the direct problem. Then, we investigate the invertibility of the input-output mapping defined by the additional over-determination integral data in order to the determination of the unknown time-dependent source coefficient.
We study a Dirichlet boundary value problem for Langevin equation involving two fractional orders. Langevin equation has been widely used to describe the evolution of physical phenomena in fluctuating environments. …
We study a Dirichlet boundary value problem for Langevin equation involving two fractional orders. Langevin equation has been widely used to describe the evolution of physical phenomena in fluctuating environments. However, ordinary Langevin equation does not provide the correct description of the dynamics for systems in complex media. In order to overcome this problem and describe dynamical processes in a fractal medium, numerous generalizations of Langevin equation have been proposed. One such generalization replaces the ordinary derivative by a fractional derivative in the Langevin equation. This gives rise to the fractional Langevin equation with a single index. Recently, a new type of Langevin equation with two different fractional orders has been introduced which provides a more flexible model for fractal processes as compared with the usual one characterized by a single index. The contraction mapping principle and Krasnoselskii′s fixed point theorem are applied to prove the existence of solutions of the problem in a Banach space.
We introduce fractional order into an HIV model. We consider the effect of viral diversity on the human immune system with frequency dependent rate of proliferation of cytotoxic T‐lymphocytes (CTLs) …
We introduce fractional order into an HIV model. We consider the effect of viral diversity on the human immune system with frequency dependent rate of proliferation of cytotoxic T‐lymphocytes (CTLs) and rate of elimination of infected cells by CTLs, based on a fractional‐order differential equation model. For the one‐virus model, our analysis shows that the interior equilibrium which is unstable in the classical integer‐order model can become asymptotically stable in our fractional‐order model and numerical simulations confirm this. We also present simulation results of the chaotic behaviors produced from the fractional‐order HIV model with viral diversity by using an Adams‐type predictor‐corrector method.
A new direct operational inversion method is introduced for solving coupled linear systems of ordinary fractional differential equations. The solutions so‐obtained can be expressed explicitly in terms of multivariate Mittag‐Leffler …
A new direct operational inversion method is introduced for solving coupled linear systems of ordinary fractional differential equations. The solutions so‐obtained can be expressed explicitly in terms of multivariate Mittag‐Leffler functions. In the case where the multiorders are multiples of a common real positive number, the solutions can be reduced to linear combinations of Mittag‐Leffler functions of a single variable. The solutions can be shown to be asymptotically oscillatory under certain conditions. This technique is illustrated in detail by two concrete examples, namely, the coupled harmonic oscillator and the fractional Wien bridge circuit. Stability conditions and simulations of the corresponding solutions are given.
The applicability of the factorization method is extended to the case of quantum fractional-differential Hamiltonians. In contrast with the conventional factorization, it is shown that the 'factorization energy' is now …
The applicability of the factorization method is extended to the case of quantum fractional-differential Hamiltonians. In contrast with the conventional factorization, it is shown that the 'factorization energy' is now a fractional-differential operator rather than a constant. As a first example, the energies and wave-functions of a fractional version of the quantum oscillator are determined. Interestingly, the energy eigenvalues are expressed as power-laws of the momentum in terms of the non-integer differential order of the eigenvalue equation.
A Ffactional element model describes a special kind of viscoelastic material. Its stress is proportional to the fractional-order derivative of strain. Physically the mechanical analogies of fractional elements can be …
A Ffactional element model describes a special kind of viscoelastic material. Its stress is proportional to the fractional-order derivative of strain. Physically the mechanical analogies of fractional elements can be represented by spring-dashpot fractal networks. We introduce a constitutive operator in the constitutive equations of viscoelastic materials. To derive constitutive operators for spring-dashpot fractal networks, we use Heaviside operational calculus, which provides explicit answers not otherwise obtainable simply. Then the series-parallel formulas for the constitutive operator are derived. Using these formulas, a constitutive equation of fractional element with 1/2-order derivative is obtained. Finally we find the way to derive the constitutive equations with other fractional-order derivatives and their mechanical analogies.
An extended fractional subequation method is proposed for solving fractional differential equations by introducing a new general ansätz and Bäcklund transformation of the fractional Riccati equation with known solutions. Being …
An extended fractional subequation method is proposed for solving fractional differential equations by introducing a new general ansätz and Bäcklund transformation of the fractional Riccati equation with known solutions. Being concise and straightforward, this method is applied to the space‐time fractional coupled Burgers’ equations and coupled MKdV equations. As a result, many exact solutions are obtained. It is shown that the considered method provides a very effective, convenient, and powerful mathematical tool for solving fractional differential equations.
The closed-form wave solutions to the time-fractional Burgers’ equation have been investigated by the use of the two variables <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <mfenced open="(" close=")" separators="|"> <mrow> <mfenced open="(" close=")" …
The closed-form wave solutions to the time-fractional Burgers’ equation have been investigated by the use of the two variables <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <mfenced open="(" close=")" separators="|"> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mrow> <mrow> <msup> <mrow> <mi>G</mi> </mrow> <mrow> <mo>′</mo> </mrow> </msup> </mrow> <mo>/</mo> <mi>G</mi> </mrow> </mrow> </mfenced> <mo>,</mo> <mfenced open="(" close=")" separators="|"> <mrow> <mrow> <mn>1</mn> <mo>/</mo> <mi>G</mi> </mrow> </mrow> </mfenced> </mrow> </mfenced> </math> -expansion, the extended tanh function, and the exp-function methods translating the nonlinear fractional differential equations (NLFDEs) into ordinary differential equations. In this article, we ascertain the solutions in terms of <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"> <mtext>tanh</mtext> </math> , <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3"> <mtext>sech</mtext> </math> , <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4"> <mtext>sinh</mtext> </math> , rational function, hyperbolic rational function, exponential function, and their integration with parameters. Advanced and standard solutions can be found by setting definite values of the parameters in the general solutions. Mathematical analysis of the solutions confirms the existence of different soliton forms, namely, kink, single soliton, periodic soliton, singular kink soliton, and some other types of solitons which are shown in three-dimensional plots. The attained solutions may be functional to examine unidirectional propagation of weakly nonlinear acoustic waves, the memory effect of the wall friction through the boundary layer, bubbly liquids, etc. The methods suggested are direct, compatible, and speedy to simulate using algebraic computation schemes, such as Maple, and can be used to verify the accuracy of results.
The fractional diffusion equation is one of the important recent models that can efficiently characterize various complex diffusion processes, such as in inhomogeneous or heterogeneous media or in porous media. …
The fractional diffusion equation is one of the important recent models that can efficiently characterize various complex diffusion processes, such as in inhomogeneous or heterogeneous media or in porous media. This article provides a method for the numerical simulation of time-fractional diffusion equations. The proposed scheme combines the local meshless method based on a radial basis function (RBF) with Laplace transform. This scheme first implements the Laplace transform to reduce the given problem to a time-independent inhomogeneous problem in the Laplace domain, and then the RBF-based local meshless method is utilized to obtain the solution of the reduced problem in the Laplace domain. Finally, Stehfest’s method is utilized to convert the solution from the Laplace domain into the real domain. The proposed method uses Laplace transform to handle the fractional order derivative, which avoids the computation of a convolution integral in a fractional order derivative and overcomes the effect of time-stepping on stability and accuracy. The method is tested using four numerical examples. All the results demonstrate that the proposed method is easy to implement, accurate, efficient and has low computational costs.
This study develops the governing equations of unsteady multi-dimensional incompressible and compressible flow in fractional time and multi-fractional space. When their fractional powers in time and in multi-fractional space are …
This study develops the governing equations of unsteady multi-dimensional incompressible and compressible flow in fractional time and multi-fractional space. When their fractional powers in time and in multi-fractional space are specified to unit integer values, the developed fractional equations of continuity and momentum for incompressible and compressible fluid flow reduce to the classical Navier-Stokes equations. As such, these fractional governing equations for fluid flow may be interpreted as generalizations of the classical Navier-Stokes equations. The derived governing equations of fluid flow in fractional differentiation framework herein are nonlocal in time and space. Therefore, they can quantify the effects of initial and boundary conditions better than the classical Navier-Stokes equations. For the frictionless flow conditions, the corresponding fractional governing equations were also developed as a special case of the fractional governing equations of incompressible flow. When their derivative fractional powers are specified to unit integers, these equations are shown to reduce to the classical Euler equations. The numerical simulations are also performed to investigate the merits of the proposed fractional governing equations. It is shown that the developed equations are capable of simulating anomalous sub- and super-diffusion due to their nonlocal behavior in time and space.
This paper introduces a new fractional operator by using the concepts of fractional q-calculus and q-Mittag-Leffler functions. With this fractional operator, Janowski functions are generalized and studied regarding their certain …
This paper introduces a new fractional operator by using the concepts of fractional q-calculus and q-Mittag-Leffler functions. With this fractional operator, Janowski functions are generalized and studied regarding their certain geometric characteristics. It also establishes the solution of the complex Briot–Bouquet differential equation by using the newly defined operator.
This paper focuses on the exploration of fractional Birkhoffian mechanics and its fractional Noether theorems under quasi-fractional dynamics models. The quasi-fractional dynamics models under study are nonconservative dynamics models proposed …
This paper focuses on the exploration of fractional Birkhoffian mechanics and its fractional Noether theorems under quasi-fractional dynamics models. The quasi-fractional dynamics models under study are nonconservative dynamics models proposed by El-Nabulsi, including three cases: extended by Riemann–Liouville fractional integral (abbreviated as ERLFI), extended by exponential fractional integral (abbreviated as EEFI), and extended by periodic fractional integral (abbreviated as EPFI). First, the fractional Pfaff–Birkhoff principles based on quasi-fractional dynamics models are proposed, in which the Pfaff action contains the fractional-order derivative terms, and the corresponding fractional Birkhoff’s equations are obtained. Second, the Noether symmetries and conservation laws of the systems are studied. Finally, three concrete examples are given to demonstrate the validity of the results.
Abstract We develop walk‐on‐sphere method for fractional Poisson equations with Dirichilet boundary conditions in high dimensions. The walk‐on‐sphere method is based on probabilistic representation of the fractional Poisson equation. We …
Abstract We develop walk‐on‐sphere method for fractional Poisson equations with Dirichilet boundary conditions in high dimensions. The walk‐on‐sphere method is based on probabilistic representation of the fractional Poisson equation. We propose efficient quadrature rules to evaluate integral representation in the ball and apply rejection sampling method to drawing from the computed probabilities in general domains. Moreover, we provide an estimate of the number of walks in the mean value for the method when the domain is a ball. We show that the number of walks is increasing in the fractional order and the distance of the starting point to the origin. We also give the relationship between the Green function of fractional Laplace equation and that of the classical Laplace equation. Numerical results for problems in 2–10 dimensions verify our theory and the efficiency of the modified walk‐on‐sphere method.
In this paper, the fractional-order chaotic system form of a four-dimensional system with cross-product nonlinearities is introduced. The stability of the equilibrium points of the system and then the feedback …
In this paper, the fractional-order chaotic system form of a four-dimensional system with cross-product nonlinearities is introduced. The stability of the equilibrium points of the system and then the feedback control design to achieve this goal have been analyzed. Furthermore, further dynamical behaviors including, phase portraits, bifurcation diagrams, and the largest Lyapunov exponent are presented. Finally, the global Mittag–Leffler attractive sets (MLASs) and Mittag–Leffler positive invariant sets (MLPISs) of the considered fractional order system are presented. Numerical simulations are provided to show the effectiveness of the results.
Abstract In a recent paper (Filomat 32:4577–4586, 2018) the authors have investigated the existence and uniqueness of a solution for a nonlinear sequential fractional differential equation. To present an analytical …
Abstract In a recent paper (Filomat 32:4577–4586, 2018) the authors have investigated the existence and uniqueness of a solution for a nonlinear sequential fractional differential equation. To present an analytical improvement for Fazli–Nieto’s results with some conditions removed based on a new technique is the main objective of this paper. In addition, we introduce an infinite system of nonlinear sequential fractional differential equations and discuss the existence of a solution for them in the classical Banach sequence spaces $c_{0}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>c</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> and $\ell_{p}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>ℓ</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:math> by applying the Darbo fixed point theorem. Moreover, the proposed method is applied to several examples to show the clarity and effectiveness.
This paper considers nonlinear fractional mixed Volterra-Fredholm integro-differential equation with a nonlocal initial condition. We propose a fixed-point approach to investigate the existence, uniqueness, and Hyers-Ulam-Rassias stability of solutions. Results …
This paper considers nonlinear fractional mixed Volterra-Fredholm integro-differential equation with a nonlocal initial condition. We propose a fixed-point approach to investigate the existence, uniqueness, and Hyers-Ulam-Rassias stability of solutions. Results of this paper are based on nonstandard assumptions and hypothesis and provide a supplementary result concerning the regularity of solutions. We show and illustrate the wide validity field of our findings by an example of problem with nonlocal neutral pantograph equation, involving functional derivative and <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mi>ψ</a:mi> </a:math> -Caputo fractional derivative.
Abstract In this article, a novel fourth‐order accurate difference method is derived for the distributed‐order Riesz space fractional diffusion equation in one‐dimensional (1D) and two‐dimensional (2D) cases, respectively. First, the …
Abstract In this article, a novel fourth‐order accurate difference method is derived for the distributed‐order Riesz space fractional diffusion equation in one‐dimensional (1D) and two‐dimensional (2D) cases, respectively. First, the distributed integral terms are discretized by using the Simpson quadrature rule into the multi‐term Riesz space fractional diffusion equations. Then, a fourth‐order accurate difference scheme is presented to approximate the multi‐term Riesz fractional diffusion equations. Moreover, the proposed difference schemes are proved to be unconditionally stable and convergent in norm for both 1D and 2D cases. Finally, numerical experiments are given to verify the efficiency of the schemes.
Abstract In the present study, we deal with the space–time fractional KdV–MKdV equation and the space–time fractional Konopelchenko–Dubrovsky equation in the sense of the conformable fractional derivative. By means of …
Abstract In the present study, we deal with the space–time fractional KdV–MKdV equation and the space–time fractional Konopelchenko–Dubrovsky equation in the sense of the conformable fractional derivative. By means of the extend \left(\tfrac{G^{\prime} }{G}\right) -expansion method, many exact solutions are obtained, which include hyperbolic function solutions, trigonometric function solutions and rational solutions. The results show that the extend \left(\tfrac{G^{\prime} }{G}\right) -expansion method is an efficient technique for solving nonlinear fractional partial equations. We also provide some graphical representations to demonstrate the physical features of the obtained solutions.
Abstract The numerical solution of the time-fractional sub-diffusion equation on an unbounded domain in two-dimensional space is considered, where a circular artificial boundary is introduced to divide the unbounded domain …
Abstract The numerical solution of the time-fractional sub-diffusion equation on an unbounded domain in two-dimensional space is considered, where a circular artificial boundary is introduced to divide the unbounded domain into a bounded computational domain and an unbounded exterior domain. The local artificial boundary conditions for the fractional sub-diffusion equation are designed on the circular artificial boundary by a joint Laplace transform and Fourier series expansion, and some auxiliary variables are introduced to circumvent high-order derivatives in the artificial boundary conditions. The original problem defined on the unbounded domain is thus reduced to an initial boundary value problem on a bounded computational domain. A finite difference and L1 approximation are applied for the space variables and the Caputo time-fractional derivative, respectively. Two numerical examples demonstrate the performance of the proposed method.
In this paper, we investigate the existence and uniqueness of S-asymptotically ω-periodic solutions to fractional differential equations of order $q\in(0, 1)$ with finite delay in a Banach space X. Existence …
In this paper, we investigate the existence and uniqueness of S-asymptotically ω-periodic solutions to fractional differential equations of order $q\in(0, 1)$ with finite delay in a Banach space X. Existence and uniqueness theorems, which are new even in the case of $X=\mathbf{R}^{n}$ or $A=0$ , are established. As examples of applications of our existence and uniqueness results, we obtain the S-asymptotically ω-periodic solutions for the fractional-order autonomous neural networks with delay.
The one-parameter fractional linear prediction (FLP) is presented and the closed-form expressions for the evaluation of FLP coefficients are derived. Contrary to the classical first-order linear prediction (LP) that uses …
The one-parameter fractional linear prediction (FLP) is presented and the closed-form expressions for the evaluation of FLP coefficients are derived. Contrary to the classical first-order linear prediction (LP) that uses one previous sample and one predictor coefficient, the one-parameter FLP model is derived using the memory of two, three or four samples, while not increasing the number of predictor coefficients. The first-order LP is only a special case of the proposed one-parameter FLP when the order of fractional derivative tends to zero. Based on the numerical experiments using test signals (sine test waves), and real-data signals (speech and electrocardiogram), the hypothesis for estimating the fractional derivative order used in the model is given. The one-parameter FLP outperforms the classical first-order LP in terms of the prediction gain, having comparable performance with the second-order LP, although using one predictor coefficient less.
Abstract The aim of this manuscript is to handle the nonlocal boundary value problem for a specific kind of nonlinear fractional differential equations involving a ξ -Hilfer derivative. The used …
Abstract The aim of this manuscript is to handle the nonlocal boundary value problem for a specific kind of nonlinear fractional differential equations involving a ξ -Hilfer derivative. The used fractional operator is generated by the kernel of the kind $k(\vartheta,s)=\xi (\vartheta )-\xi (s)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi><mml:mo>(</mml:mo><mml:mi>ϑ</mml:mi><mml:mo>,</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mi>ξ</mml:mi><mml:mo>(</mml:mo><mml:mi>ϑ</mml:mi><mml:mo>)</mml:mo><mml:mo>−</mml:mo><mml:mi>ξ</mml:mi><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:math> and the operator of differentiation ${ D}_{\xi } = ( \frac{1}{\xi ^{\prime }(\vartheta )}\frac{d}{d\vartheta } ) $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>D</mml:mi><mml:mi>ξ</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mo>(</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:msup><mml:mi>ξ</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>(</mml:mo><mml:mi>ϑ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mfrac><mml:mfrac><mml:mi>d</mml:mi><mml:mrow><mml:mi>d</mml:mi><mml:mi>ϑ</mml:mi></mml:mrow></mml:mfrac><mml:mo>)</mml:mo></mml:math> . The existence and uniqueness of solutions are established for the considered system. Our perspective relies on the properties of the generalized Hilfer derivative and the implementation of Krasnoselskii’s fixed point approach and Banach’s contraction principle with respect to the Bielecki norm to obtain the uniqueness of solution on a bounded domain in a Banach space. Besides, we discuss the Ulam–Hyers stability criteria for the main fractional system. Finally, some examples are given to illustrate the viability of the main theories.
In this paper, the WKB method is extended to be applicable for conformable Hamiltonian systems where the concept of conformable operator with fractional order $\alpha$ is used. The WKB approximation …
In this paper, the WKB method is extended to be applicable for conformable Hamiltonian systems where the concept of conformable operator with fractional order $\alpha$ is used. The WKB approximation for the $\alpha$-wavefunction is derived when the potential is slowly varying in space. The paper is furnished with some illustrative examples to demonstrate the method. The quantities of the conformable form are found to be inexact agreement with traditional quantities when $\alpha=1$.
This paper focuses on the bounds of the Lyapunov exponents for fractional differential systems, where the fractional derivatives are Riemann–Liouville and Caputo fractional derivatives with the exponential kernel. First, the …
This paper focuses on the bounds of the Lyapunov exponents for fractional differential systems, where the fractional derivatives are Riemann–Liouville and Caputo fractional derivatives with the exponential kernel. First, the essential properties of fractional integral and derivatives with the exponential kernel are given. Then the continuous dependence of solutions on the initial value problems of some particular parameters is studied. On these bases, the bounds of Lyapunov exponents are estimated. Finally, the theoretical results are illustrated by numerical simulations.