Polynomial spectral collocation method for space fractional advection–diffusion equation

This article discusses the spectral collocation method for numerically solving nonlocal problems: one‐dimensional space fractional advection–diffusion equation; and two‐dimensional linear/nonlinear space fractional advection–diffusion equation. The differentiation matrixes of the left and right Riemann–Liouville and Caputo fractional derivatives are derived for any collocation points within any given bounded interval. Several numerical examples with different boundary conditions are computed to verify the efficiency of the numerical schemes and confirm the exponential convergence; the physical simulations for Lévy–Feller advection–diffusion equation and space fractional Fokker–Planck equation with initial δ‐peak and reflecting boundary conditions are performed; and the eigenvalue distributions of the iterative matrix for a variety of systems are displayed to illustrate the stabilities of the numerical schemes in more general cases. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 514–535, 2014     

Numerical methods for fractional partial differential equations with Riesz space fractional derivatives

In this paper, we consider the numerical solution of a fractional partial differential equation with Riesz space fractional derivatives (FPDE-RSFD) on a finite domain. Two types of FPDE-RSFD are considered: the Riesz fractional diffusion equation (RFDE) and the Riesz fractional advection–dispersion equation (RFADE). The RFDE is obtained from the standard diffusion equation by replacing the second-order space derivative with the Riesz fractional derivative of order α(1,2]. The RFADE is obtained from the standard advection–dispersion equation by replacing the first-order and second-order space derivatives with the Riesz fractional derivatives of order β(0,1) and of order α(1,2], respectively. Firstly, analytic solutions of both the RFDE and RFADE are derived. Secondly, three numerical methods are provided to deal with the Riesz space fractional derivatives, namely, the L1/L2-approximation method, the standard/shifted Grünwald method, and the matrix transform method (MTM). Thirdly, the RFDE and RFADE are transformed into a system of ordinary differential equations, which is then solved by the method of lines. Finally, numerical results are given, which demonstrate the effectiveness and convergence of the three numerical methods.     

Exact solutions for fractional DDEs via auxiliary equation method coupled with the fractional complex transform

Dynamical behavior of many nonlinear systems can be described by fractional‐order equations. This study is devoted to fractional differential–difference equations of rational type. Our focus is on the construction of exact solutions by means of the (G'/G)‐expansion method coupled with the so‐called fractional complex transform. The solution procedure is elucidated through two generalized time‐fractional differential–difference equations of rational type. As a result, three types of discrete solutions emerged: hyperbolic, trigonometric, and rational. Copyright © 2016 John Wiley & Sons, Ltd.     

A new method for exact solutions of variant types of time‐fractional Korteweg‐de Vries equations in shallow water waves

The current article devoted on the new method for finding the exact solutions of some time‐fractional Korteweg–de Vries (KdV) type equations appearing in shallow water waves. We employ the new method here for time‐fractional equations viz. time‐fractional KdV‐Burgers and KdV‐mKdV equations for finding the exact solutions. We use here the fractional complex transform accompanied by properties of local fractional calculus for reduction of fractional partial differential equations to ordinary differential equations. The obtained results are demonstrated by graphs for the new solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

High‐order characteristic‐tracking strategy for simulation of a nonlinear advection–diffusion equation

In this study, new high‐order backward semi‐Lagrangian methods are developed to solve nonlinear advection–diffusion type problems, which are realized using high‐order characteristic‐tracking strategies. The proposed characteristic‐tracking strategies are second‐order L‐stable and third‐order L(α)‐stable methods, which are based on a classical implicit multistep method combined with a error‐correction method. We also use backward differentiation formulas and the fourth‐order finite‐difference scheme for diffusion problem discretization in the temporal and spatial domains, respectively. To demonstrate the adaptability and efficiency of these time‐discretization strategies, we apply these methods to nonlinear advection–diffusion type problems such as the viscous Burgers' equation. Through simulations, not only the temporal and spatial accuracies are numerically evaluated but also the proposed methods are shown to be superior to the compared existing characteristic‐tracking methods under the same rates of convergence in terms of accuracy and efficiency. Finally, we have shown that the proposed method well preserves the energy and mass when the viscosity coefficient becomes zero.     

A novel method for travelling wave solutions of fractional Whitham–Broer–Kaup,fractional modified Boussinesq and fractional approximate long wave equations in shallow water

In this paper, the analytical approximate traveling wave solutions of Whitham–Broer–Kaup (WBK) equations, which contain blow‐up solutions and periodic solutions, have been obtained by using the coupled fractional reduced differential transform method. By using this method, the solutions were calculated in the form of a generalized Taylor series with easily computable components. The convergence of the method as applied to the WBK equations is illustrated numerically as well as analytically. By using the present method, we can solve many linear and nonlinear coupled fractional differential equations. The results justify that the proposed method is also very efficient, effective and simple for obtaining approximate solutions of fractional coupled modified Boussinesq and fractional approximate long wave equations. Numerical solutions are presented graphically to show the reliability and efficiency of the method. Moreover, the results are compared with those obtained by the Adomian decomposition method (ADM) and variational iteration method (VIM), revealing that the present method is superior to others. Copyright © 2014 John Wiley & Sons, Ltd.     

Analytical solution of time‐fractional Navier–Stokes equation in polar coordinate by homotopy perturbation method

In this letter, we implement a relatively new analytical technique, the homotopy perturbation method (HPM), for solving linear partial differential equations of fractional order arising in fluid mechanics. The fractional derivatives are described in Caputo derivatives. This method can be used as an alternative to obtain analytic and approximate solutions of different types of fractional differential equations applied in engineering mathematics. The corresponding solutions of the integer order equations are found to follow as special cases of those of fractional order equations. Some numerical examples are presented to illustrate the efficiency and reliability of HPM. He's HPM, which does not need small parameter is implemented for solving the differential equations. In this method, a homotopy is introduced to be constructed for the equation. The initial approximations can be freely chosen with possible unknown constants that can be determined by imposing the boundary and initial conditions. It is predicted that HPM can be found widely applicable in engineering. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Analytical and approximate solutions of fractional‐order susceptible‐infected‐recovered epidemic model of childhood disease

In this work, we make use of the conformable fractional differential transform method (CFDTM) in order to compute an approximate solution of the fractional‐order susceptible‐infected‐recovered (SIR) epidemic model of childhood disease. The method provides the solution in the form of a rapidly convergent series. Two explanatory and illustrative examples are given to represent the efficacy of the obtained results.     

On the exact solutions,lie symmetry analysis,and conservation laws of Schamel–Korteweg–de Vries equation

In this work, we study the integrability aspects of the Schamel–Korteweg–de Vries equation that play an important role in studying the effect of electron trapping on the nonlinear interaction of ion‐acoustic waves by including a quasi‐potential. Lie symmetry analysis together with the simplest equation method and Kudryashov method is used to obtain exact traveling wave solutions for this equation. In addition, conservation laws are constructed using two different techniques, namely, the multiplier method and the new conservation theorem. Using the conservation laws and symmetries of the underlying equation, double reduction and exact solution were also constructed. Copyright © 2017 John Wiley & Sons, Ltd.     

The dual reciprocity boundary integral equation technique to solve a class of the linear and nonlinear fractional partial differential equations

In this paper, we apply the boundary integral equation technique and the dual reciprocity boundary elements method (DRBEM) for the numerical solution of linear and nonlinear time‐fractional partial differential equations (TFPDEs). The main aim of the present paper is to examine the applicability and efficiency of DRBEM for solving TFPDEs. We employ the time‐stepping scheme to approximate the time derivative, and the method of linear radial basis functions is also used in the DRBEM technique. This method is improved by using a predictor–corrector scheme to overcome the nonlinearity that appears in the nonlinear problems under consideration. To confirm the accuracy of the new approach, several examples are presented. The convergence of the DRBEM is studied numerically by comparing the exact solutions of the problems under investigation. Copyright © 2015 John Wiley & Sons, Ltd.