Matrix approach to discrete fractional calculus II: Partial fractional differential equations

A new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order (so-called fractional derivatives) and delays is presented and illustrated on numerical solution of various types of fractional diffusion equation. The suggested method is the development of Podlubny’s matrix approach [I. Podlubny, Matrix approach to discrete fractional calculus, Fractional Calculus and Applied Analysis 3 (4) (2000) 359–386]. Four examples of numerical solution of fractional diffusion equation with various combinations of time-/space-fractional derivatives (integer/integer, fractional/integer, integer/fractional, and fractional/fractional) with respect to time and to the spatial variable are provided in order to illustrate how simple and general is the suggested approach. The fifth example illustrates that the method can be equally simply used for fractional differential equations with delays. A set of MATLAB routines for the implementation of the method as well as sample code used to solve the examples have been developed.     

Fractional vector calculus and fractional Maxwell’s equations

The theory of derivatives and integrals of non-integer order goes back to Leibniz, Liouville, Grunwald, Letnikov and Riemann. The history of fractional vector calculus (FVC) has only 10 years. The main approaches to formulate a FVC, which are used in the physics during the past few years, will be briefly described in this paper. We solve some problems of consistent formulations of FVC by using a fractional generalization of the Fundamental Theorem of Calculus. We define the differential and integral vector operations. The fractional Green’s, Stokes’ and Gauss’s theorems are formulated. The proofs of these theorems are realized for simplest regions. A fractional generalization of exterior differential calculus of differential forms is discussed. Fractional nonlocal Maxwell’s equations and the corresponding fractional wave equations are considered.     

Analytical approximate solution for nonlinear space-time fractional Klein Gordon equation

The fractional derivatives in the sense of Caputo and the homotopy analysis method are used to construct an approximate solution for the nonlinear space-time fractional derivatives Klein-Gordon equation. The numerical results show that the approaches are easy to implement and accurate when applied to the nonlinear space-time fractional derivatives KleinGordon equation. This method introduces a promising tool for solving many space-time fractional partial differential equations. This method is efficient and powerful in solving wide classes of nonlinear evolution fractional order equations.     

A survey on the stability of fractional differential equations

Recently, fractional calculus has attracted much attention since it plays an important role in many fields of science and engineering. Especially, the study on stability of fractional differential equations appears to be very important. In this paper, a brief overview on the recent stability results of fractional differential equations and the analytical methods used are provided. These equations include linear fractional differential equations, nonlinear fractional differential equations, fractional differential equations with time-delay. Some conclusions for stability are similar to that of classical integer-order differential equations. However, not all of the stability conditions are parallel to the corresponding classical integer-order differential equations because of non-locality and weak singularities of fractional calculus. Some results and remarks are also included.     

A matrix approach for partial differential equations with Riesz space fractional derivatives

Fractional partial differential equations are emerging in many scientific fields and their numerical solution is becoming a fundamental topic. In this paper we consider the Riesz fractional derivative operator and its discretization by fractional centered differences. The resulting matrix is studied, with an interesting result on a connection between the decay behavior of its entries and the short memory principle from fractional calculus. The Shift-and-Invert method is then applied to approximate the solution of the partial differential equation as the action of the matrix exponential on a suitable vector which mimics the given initial conditions. The numerical results confirm the good approximation quality and encourage the use of the proposed approach.     

Explicit solutions of a certain class of differential equations by means of fractional calculus

Judging by the remarkably large number of recent publications on Fractional Calculus and Its Applications in several widely diverse areas of mathematical, physical, and engineering sciences, the current popularity and importance of the subject of fractional calculus cannot be overemphasized. Motivated by some of these interesting developments, many authors have recently demonstrated the usefulness of fractional calculus in the derivation of explicit particular solutions of a number of linear ordinary and partial differential equations of the second and higher orders. The main object of the present paper is to show how several recent contributions on this subject, involving a certain class of ordinary differential equations, can be obtained (in a unified manner) by suitably applying some general theorems on explicit particular solutions of a family of linear ordinary fractional differintegral equations.     

Numerical solution of fractional differential equations by using fractional B-spline

In this paper, we present fractional B-spline collocation method for the numerical solution of fractional differential equations. We consider this method for solving linear fractional differential equations which involve Caputo-type fractional derivatives. The numerical results demonstrate that the method is efficient and quite accurate and it requires relatively less computational work. For this reason one can conclude that this method has advantage on other methods and hence demonstrates the importance of this work.     

Finite difference methods for two-dimensional fractional dispersion equation

Fractional order partial differential equations, as generalizations of classical integer order partial differential equations, are increasingly used to model problems in fluid flow, finance and other areas of application. In this paper we discuss a practical alternating directions implicit method to solve a class of two-dimensional initial-boundary value fractional partial differential equations with variable coefficients on a finite domain. First-order consistency, unconditional stability, and (therefore) first-order convergence of the method are proven using a novel shifted version of the classical Grünwald finite difference approximation for the fractional derivatives. A numerical example with known exact solution is also presented, and the behavior of the error is examined to verify the order of convergence.     

分数阶Oldroyd-B黏弹性Poiseuille流的Laplace数值反演分析

采用Laplace数值反演的Stehfest算法研究了分数阶Oldroyd-B粘弹性流体在两平板间非定常的Poiseuille流动问题.首先,通过数值解与近似解析解的比较验证了Stehfest算法的有效性.其次,运用Stehfest算法对平板Poiseuille流动进行了研究,揭示了分数阶黏弹性平板流的速度过冲和应力过冲现象,指出这些现象对分数导数的阶数存在明显的依赖性.同时,数值结果表明,整数阶本构方程仅仅是分数阶本构方程的特例,分数阶本构方程较整数阶本构方程具有更广泛的适用性。     

Numerical solution of fractional differential equations via a Volterra integral equation approach

The main focus of this paper is to present a numerical method for the solution of fractional differential equations. In this method, the properties of the Caputo derivative are used to reduce the given fractional differential equation into a Volterra integral equation. The entire domain is divided into several small domains, and by collocating the integral equation at two adjacent points a system of two algebraic equations in two unknowns is obtained. The method is applied to solve linear and nonlinear fractional differential equations. Also the error analysis is presented. Some examples are given and the numerical simulations are also provided to illustrate the effectiveness of the new method.     

Synchronization of chaotic fractional-order systems via active sliding mode controller

In this paper, we propose a controller based on active sliding mode theory to synchronize chaotic fractional-order systems in master-slave structure. Master and slave systems may be identical or different. Based on stability theorems in the fractional calculus, analysis of stability is performed for the proposed method. Finally, three numerical simulations (synchronizing fractional-order Lü-Lü systems, synchronizing fractional order Chen-Chen systems and synchronizing fractional-order Lü-Chen systems) are presented to show the effectiveness of the proposed controller. The simulations are implemented using two different numerical methods to solve the fractional differential equations.     

A fractional calculus approach to nonlocal elasticity

If the attenuation function of strain is expressed as a power law, the formalism of fractional calculus may be used to handle Eringen nonlocal elastic model. Aim of the present paper is to provide a mechanical interpretation to this nonlocal fractional elastic model by showing that it is equivalent to a discrete, point-spring model. A one-dimensional geometry is considered; the static, kinematic and constitutive equations are presented and the governing fractional differential equation highlighted. Two numerical procedures to solve the fractional equation are finally implemented and applied to study the strain field in a finite bar under given edge displacements.     

A novel method for nonlinear fractional differential equations using symbolic computation

In this paper, a new approach, namely an ansatz method is applied to find exact solutions for nonlinear fractional differential equations in the sense of modified Riemann–Liouville derivative. Based on a nonlinear fractional complex transformation, a certain fractional partial differential equation can be turned into another ordinary differential equation of integer order. For illustrating the validity of this method, we apply it to solve the fractional-order biological population model and the space–time fractional modified equal width equation, and as a result, some dark soliton solutions for them are established.     

Exact solutions for nonlinear partial fractional differential equations

In this article,we use the fractional complex transformation to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations.We use the improved(G’/G)-expansion function method to calculate the exact solutions to the time-and space-fractional derivative foam drainage equation and the time-and space-fractional derivative nonlinear KdV equation.This method is efficient and powerful for solving wide classes of nonlinear evolution fractional order equations.     

Fractional derivative order determination from harmonic oscillator damping factor

This article analysis differential equations which represents damped and fractional oscillators. First, it is shown that prior to using physical quantities in fractional calculus, it is imperative that they are turned dimensionless. Approximated expressions that relate the two equations parameters, when fractional order is close to an integer, are presented. Following, a numerical regression is made using power series expansion, and, also from fractional calculus, the fact that both equations cannot be equivalent is concluded. In the end, from the numerical regression data, the analytical approximated expressions that relate the two equations’ parameters are refined.     

Existence and approximation of solutions of fractional order iterative differential equations

In this paper, we investigate existence and approximation of solutions of fractional order iterative differential equations by virtue of nonexpansive mappings, fractional calculus and fixed point methods. Three existence theorems as well as convergence theorems for a fixed point iterative method designed to approximate these solutions are obtained in two different work spaces via Chebyshev’s norm, Bielecki’s norm and β norm. Finally, an example is given to illustrate the obtained results.     

A Tutorial Review on Fractal Spacetime and Fractional Calculus

This tutorial review of fractal-Cantorian spacetime and fractional calculus begins with Leibniz’s notation for derivative without limits which can be generalized to discontinuous media like fractal derivative and q-derivative of quantum calculus. Fractal spacetime is used to elucidate some basic properties of fractal which is the foundation of fractional calculus, and El Naschie’s mass-energy equation for the dark energy. The variational iteration method is used to introduce the definition of fractional derivatives. Fractal derivative is explained geometrically and q-derivative is motivated by quantum mechanics. Some effective analytical approaches to fractional differential equations, e.g., the variational iteration method, the homotopy perturbation method, the exp-function method, the fractional complex transform, and Yang-Laplace transform, are outlined and the main solution processes are given.     

Phase synchronization in fractional differential chaotic systems

We propose an analytical justification for phase synchronization of fractional differential equations. This justification is based upon a linear stability criterion for fractional differential equations. We then investigate the existence of phase synchronization in chaotic forced Duffing and Sprott-L fractional differential systems of equations. Our numerical results agree with those analytical justifications.     

Analysis of a time fractional wave-like equation with the homotopy analysis method

The time fractional wave-like differential equation with a variable coefficient is studied analytically. By using a simple transformation, the governing equation is reduced to two fractional ordinary differential equations. Then the homotopy analysis method is employed to derive the solutions of these equations. The accurate series solutions are obtained. Especially, when ?f=?g=−1, these solutions are exactly the same as those results given by the Adomian decomposition method. The present work shows the validity and great potential of the homotopy analysis method for solving nonlinear fractional differential equations. The basic idea described in this Letter is expected to be further employed to solve other similar nonlinear problems in fractional calculus.     

Numerical solution of the conformable space-time fractional wave equation

In this paper, an efficient numerical method is considered for solving space-time fractional wave equation. The fractional derivatives are described in the conformable sense. The method is based on shifted Chebyshev polynomials of the second kind. Unknown function is written as Chebyshev series with the N term. The space-time fractional wave equation is reduced to a system of ordinary differential equations by using the properties of Chebyshev polynomials. The finite difference method is applied to solve this system of equations. Numerical results are provided to verify the accuracy and efficiency of the proposed approach.