The transport dynamics in complex systems governing by anomalous diffusion modelled with Riesz fractional partial differential equations

In this paper, numerical solutions of fractional Fokker–Planck equations with Riesz space fractional derivatives have been developed. Here, the fractional Fokker–Planck equations have been considered in a finite domain. In order to deal with the Riesz fractional derivative operator, shifted Grünwald approximation and fractional centred difference approaches have been used. The explicit finite difference method and Crank–Nicolson implicit method have been applied to obtain the numerical solutions of fractional diffusion equation and fractional Fokker–Planck equations, respectively. Numerical results are presented to demonstrate the accuracy and effectiveness of the proposed numerical solution techniques. Copyright © 2016 John Wiley & Sons, Ltd.     

Analytical approximate solutions of Riesz fractional diffusion equation and Riesz fractional advection–dispersion equation involving nonlocal space fractional derivatives

In this paper, we consider the analytical solutions of fractional partial differential equations (PDEs) with Riesz space fractional derivatives on a finite domain. Here we considered two types of fractional PDEs with Riesz space fractional derivatives such as Riesz fractional diffusion equation (RFDE) and 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. Here the analytic solutions of both the RFDE and RFADE are derived by using modified homotopy analysis method with Fourier transform. Then, we analyze the results by numerical simulations, which demonstrate the simplicity and effectiveness of the present method. Here the space fractional derivatives are defined as Riesz fractional derivatives. Copyright © 2015 John Wiley & Sons, Ltd.     

Numerical methods for nonlinear partial differential equations of fractional order

In this article, we implement relatively new analytical techniques, the variational iteration method and the Adomian decomposition method, for solving nonlinear partial differential equations of fractional order. The fractional derivatives are described in the Caputo sense. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of fractional differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. Numerical results show that the two approaches are easy to implement and accurate when applied to partial differential equations of fractional order.     

On solvability of differential equations with the Riesz fractional derivative

We consider the solvability of fractional differential equations involving the Riesz fractional derivative. Our approach basically relies on the reduction of the problem considered to the equivalent nonlinear mixed Volterra and Cauchy-type singular integral equation and on the theory of fractional calculus. By establishing a compactness property of the Riemann–Liouville fractional integral operator on Lebesgue spaces and using the well-known Krasnoselskii's fixed point theorem, an existence of at least one solution is gleaned. An example is finally included to show the applicability of the theory.     

An efficient difference scheme for the coupled nonlinear fractional Ginzburg–Landau equations with the fractional Laplacian

In this article, an efficient difference scheme for the coupled fractional Ginzburg–Landau equations with the fractional Laplacian is studied. We construct the discrete scheme based on the implicit midpoint method in time and a weighted and shifted Grünwald difference method in space. Then, we prove that the scheme is uniquely solvable, and the numerical solutions are bounded and unconditionally convergent in the norm. Finally, numerical tests are given to confirm the theoretical results and show the effectiveness of the scheme.     

Positive solutions of fractional differential equations with the Riesz space derivative

A new class of fractional differential equations with the Riesz–Caputo derivative is proposed and the physical meaning is introduced in this paper. The boundary value problem is investigated under some conditions. Leray–Schauder and Krasnoselskii’s fixed point theorems in a cone are adopted. Existence of positive solutions is provided. Finally, two examples with numerical solutions are given to support theoretical results.     

A new numerical approach for single rational soliton solution of Chen‐Lee‐Liu equation with Riesz fractional derivative in optical fibers

In this paper, time‐splitting spectral approximation technique has been proposed for Chen‐Lee‐Liu (CLL) equation involving Riesz fractional derivative. The proposed numerical technique is efficient, unconditionally stable, and of second‐order accuracy in time and of spectral accuracy in space. Moreover, it conserves the total density in the discretized level. In order to examine the results, with the aid of weighted shifted Grünwald‐Letnikov formula for approximating Riesz fractional derivative, Crank‐Nicolson weighted and shifted Grünwald difference (CN‐WSGD) method has been applied for Riesz fractional CLL equation. The comparison of results reveals that the proposed time‐splitting spectral method is very effective and simple for obtaining single soliton numerical solution of Riesz fractional CLL equation.     

Existence of solutions to some classes of partial fractional differential equations

In this paper we obtain the existence of solutions to some classes of partial fractional differential equations. Applications include the existence of solutions to a fractional heat-like equation.     

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     

A novel attempt for finding comparatively accurate solution for sine‐Gordon equation comprising Riesz space fractional derivative

In this paper, a numerical procedure involving Chebyshev wavelet method has been implemented for computing the approximate solution of Riesz space fractional sine‐Gordon equation (SGE). Two‐dimensional Chebyshev wavelet method is implemented to calculate the numerical solution of space fractional SGE. The fractional SGE is considered as an interpolation between the classical SGE (corresponding to α = 2) and nonlocal SGE (corresponding to α = 1). As a consequence, the approximate solutions of fractional SGE obtained by using Chebyshev wavelet approach were compared with those derived by using modified homotopy analysis method with Fourier transform. Copyright © 2015 John Wiley & Sons, Ltd.     

A novel method for nonlinear fractional partial differential equations: Combination of DTM and generalized Taylor's formula

In this article, a novel numerical method is proposed for nonlinear partial differential equations with space- and time-fractional derivatives. This method is based on the two-dimensional differential transform method (DTM) and generalized Taylor's formula. The fractional derivatives are considered in the Caputo sense. Several illustrative examples are given to demonstrate the effectiveness of the present method. Results obtained using the scheme presented here agree well with the analytical solutions and the numerical results presented elsewhere. Results also show that the numerical scheme is very effective and convenient for solving nonlinear partial differential equations of fractional order.     

A reliable algorithm of homotopy analysis method for solving nonlinear fractional differential equations

In this paper, based on the homotopy analysis method (HAM), a powerful algorithm is developed for the solution of nonlinear ordinary differential equations of fractional order. The proposed algorithm presents the procedure of constructing the set of base functions and gives the high-order deformation equation in a simple form. Different from all other analytic methods, it provides us with a simple way to adjust and control the convergence region of solution series by introducing an auxiliary parameter ?$?$. The analysis is accompanied by numerical examples. The algorithm described in this paper is expected to be further employed to solve similar nonlinear problems in fractional calculus.     

Numerical methods for multi-term fractional (arbitrary) orders differential equations

Our main concern here is to give a numerical scheme to solve a nonlinear multi-term fractional (arbitrary) orders differential equation.     

Maximal ℓp‐regularity of multiterm fractional equations with delay

We provide a characterization for the existence and uniqueness of solutions in the space of vector‐valued sequences ${?}_p(?,X)$ for the multiterm fractional delayed model in the form $${\mathrm{\Delta }}^{\alpha }u(n)+\lambda {\mathrm{\Delta }}^{\beta }u(n)=Au(n)+u(n?\tau )+f(n),n\in ?,\alpha ,\beta \in {?}_{+},\tau \in ?,\lambda \in ?,$$ where X is a Banach space, A is a closed linear operator with domain D(A) defined on X, $f\in {?}_p(?,X)$ and Δ^Γ denotes the Grünwald–Letkinov fractional derivative of order Γ > 0. We also give some conditions to ensure the existence of solutions when adding nonlinearities. Finally, we illustrate our results with an example given by a general abstract nonlinear model that includes the fractional Fisher equation with delay.     

Darboux problem for perturbed partial differential equations of fractional order with finite delay

In this paper we investigate the existence of solutions for functional partial perturbed hyperbolic differential equations with fractional order. These results are based upon a fixed point theorem for the sum of contraction and compact operators.     

A finite difference method for fractional partial differential equation

An implicit unconditional stable difference scheme is presented for a kind of linear space–time fractional convection–diffusion equation. The equation is obtained from the classical integer order convection–diffusion equations with fractional order derivatives for both space and time. First-order consistency, unconditional stability, and 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.     

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.     

On the convergence of spline collocation methods for solving fractional differential equations

In the first part of this paper we study the regularity properties of solutions of initial value problems of linear multi-term fractional differential equations. We then use these results in the convergence analysis of a polynomial spline collocation method for solving such problems numerically. Using an integral equation reformulation and special non-uniform grids, global convergence estimates are derived. From these estimates it follows that the method has a rapid convergence if we use suitable nonuniform grids and the nodes of the composite Gaussian quadrature formulas as collocation points. Theoretical results are verified by some numerical examples.     

A new approximation for the generalized fractional derivative and its application to generalized fractional diffusion equation

In this paper, we derive a fourth order approximation for the generalized fractional derivative that is characterized by a scale function z(t) and a weight function w(t) . Combining the new approximation with compact finite difference method, we develop a numerical scheme for a generalized fractional diffusion problem. The stability and convergence of the numerical scheme are proved by the energy method, and it is shown that the temporal and spatial convergence orders are both 4. Several numerical experiments are provided to illustrate the efficiency of our scheme.     

Initial value problems for arbitrary order fractional differential equations with delay

By fixed point theory the nonlinear alternative of Leray–Schauder type, and the properties of absolutely continuous functions space, we study the existence and uniqueness of initial value problems for nonlinear higher fractional equations with delay, and obtain some new results involving local and global solutions.