Riesz Fractional Derivative Of Order Research Articles

Numerical solution of two‐dimensional nonlinear Riesz space‐fractional reaction–advection–diffusion equation using fast compact implicit integration factor method

AbstractIn the present article, a finite domain is considered to find the numerical solution of a two‐dimensional nonlinear fractional‐order partial differential equation (FPDE) with Riesz space fractional derivative (RSFD). Here two types of FPDE–RSFD are considered, the first one is a two‐dimensional nonlinear Riesz space‐fractional reaction–diffusion equation (RSFRDE) and the second one is a two‐dimensional nonlinear Riesz space‐fractional reaction‐advection‐diffusion equation (RSFRADE). SFRDE is obtained by simply replacing second‐order derivative term of the standard nonlinear diffusion equation by the Riesz fractional derivative of order whereas the SFRADE is obtained by replacing the first‐order and second‐order space derivatives from the standard order advection–dispersion equation with the Riesz fractional derivatives of order . A numerical method is provided to deal with the RSFD with the weighted and shifted Grünwald–Letnikov (WSGD) approximations, for the spatial discretization. The SFRDE and SFRADE are transformed into a system of ordinary differential equations (ODEs), which have been solved using a fast compact implicit integration factor (FcIIF) with nonuniform time meshes. Finally, the demonstration of the validation and effectiveness of the numerical method is given by considering some existing models.

Analysis and numerical simulation of cross reaction–diffusion systems with the Caputo–Fabrizio and Riesz operators

AbstractThe evolutionary dynamics of cross‐reaction–diffusion equations of predator–prey type are investigated in the sense of fractional operator. In the models, we replace the classical time and spatial derivatives with the Caputo–Fabrizio and Riesz fractional derivatives, respectively. The nature of the resulting problem (is nonlinear, nonlocal, and nonsingular) do not either admit a closed form solution, while in most cases the analytical solution is too involved to be useful. As a result, there is need to provide a reliable numerical scheme that can approximate these derivatives in time and space. Hence, we formulate an approximation scheme with second‐order convergence rate for the time‐Caputo–Fabrizio fractional operator of order 0 < α ≤ 1 and L1 formula for the Riesz fractional derivative of order 1 < β ≤ 2 in space. As a case study, we consider two examples of strongly coupled cross fractional reaction–diffusion systems describing the interaction between two individual species that prey on the other one. We examine the system for stability analysis and establish the condition for the occurrence of Turing instability. The complexity of the dynamics of time–space cross fractional reaction–diffusion systems is theoretically studied and numerically in one and two dimensions for some instances of fractional orders.

High-order solvers for space-fractional differential equations with Riesz derivative

This paper proposes the computational approach for fractional-in-space reaction-diffusion equation, which is obtained by replacing the space second-order derivative in classical reaction-diffusion equation with the Riesz fractional derivative of order \begin{document}$ α $\end{document} in \begin{document}$ (0, 2] $\end{document} . The proposed numerical scheme for space fractional reaction-diffusion equations is based on the finite difference and Fourier spectral approximation methods. The paper utilizes a range of higher-order time stepping solvers which exhibit third-order accuracy in the time domain and spectral accuracy in the spatial domain to solve some fractional-in-space reaction-diffusion equations. The numerical experiment shows that the third-order ETD3RK scheme outshines its third-order counterparts, taking into account the computational time and accuracy. Applicability of the proposed methods is further tested with a higher dimensional system. Numerical simulation results show that pattern formation process in the classical sense is the same as in fractional scenarios.

A novel finite volume method for the Riesz space distributed-order advection–diffusion equation

In this paper, we investigate the finite volume method (FVM) for a distributed-order space-fractional advection–diffusion (AD) equation. The mid-point quadrature rule is used to approximate the distributed-order equation by a multi-term fractional model. Next, the transformed multi-term fractional equation is solved by discretizing in space by the finite volume method and in time using the Crank–Nicolson scheme. We use a novel technique to deal with the convection term, by which the Riesz fractional derivative of order 0 < γ < 1 is transformed into a fractional integral form. An important contribution of our work is the use of nodal basis function to derive the discrete form of our model. The unique solvability of the scheme is also discussed and we prove that the Crank–Nicolson scheme is unconditionally stable and convergent with second-order accuracy. Finally, we give some examples to show the effectiveness of the numerical method.

WITHDRAWN: High-order numerical methods for the Riesz space fractional advection–dispersion equations

Abstract In this paper, we propose high-order numerical methods for the Riesz space fractional advection–dispersion equations (RSFADE) on a finite domain. The RSFADE is obtained from the standard advection–dispersion equation by replacing the first-order and second-order space derivative with the Riesz fractional derivatives of order α ∈ ( 0 , 1 ) and β ∈ ( 1 , 2 ] , respectively. Firstly, we utilize the weighted and shifted Grunwald difference operators to approximate the Riesz fractional derivative and present the finite difference method for the RSFADE. Specifically, we discuss the Crank–Nicolson scheme and solve it in matrix form. Secondly, we prove that the scheme is unconditionally stable and convergent with the accuracy of O ( τ 2 + h 2 ) . Thirdly, we use the Richardson extrapolation method (REM) to improve the convergence order which can be O ( τ 4 + h 4 ) . Finally, some numerical examples are given to show the effectiveness of the numerical method, and the results are in excellent with theoretical analysis.

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 &amp; Sons, Ltd.

Galerkin finite element approximation of symmetric space-fractional partial differential equations

In this paper, symmetric space-fractional partial differential equations (SSFPDE) with the Riesz fractional operator are considered. The SSFPDE 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 2 β ∈ (0, 1) and 2 α ∈ (1, 2], respectively. We prove that the variational solution of the SSFPDE exists and is unique. Using the Galerkin finite element method and a backward difference technique, a fully discrete approximating system is obtained, which has a unique solution according to the Lax–Milgram theorem. The stability and convergence of the fully discrete schemes are derived. Finally, some numerical experiments are given to confirm our theoretical analysis.

Solutions of a Particle with Fractional δ-Potential in a Fractional Dimensional Space

A Fourier transformation in a fractional dimensional space of order λ (0<λ≤1) is defined to solve the Schrödinger equation with Riesz fractional derivatives of order α. This new method is applied for a particle in a fractional δ-potential well defined by V(x)=−γ δ λ(x), where γ>0 and δ λ(x) is the fractional Dirac delta function. A complete solutions for the energy values and the wave functions are obtained in terms of the Fox H-functions. It is demonstrated that the eigen solutions are exist if 0<λ<α. The results for λ=1 and α=2 are in exact agreement with those presented in the standard quantum mechanics.

Riesz Fractional Derivatives and Fractional Dimensional Space

The Fourier transform method is used to solve fractional Poisson’s equation with Riesz fractional derivative of order α. It is shown that the solution is given in terms of the fractional dimensional space D. Gauss law for the electrostatic problem is given and the total electric flux is obtained in terms of α and D.

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.

The fundamental solution and numerical solution of the Riesz fractional advection-dispersion equation

In this paper, we consider a Riesz fractional advection-dispersion equation (RFADE), which is derived from the kinetics of chaotic dynamics. The RFADE is obtained from the standard advection-dispersion equation by replacing the first-order and second-order space derivatives by the Riesz fractional derivatives of order a e (0, 1) and fi e (1, 2], respectively. We derive the fundamental solution for the Riesz fractional advection-dispersion equation with an initial condition (RFADE-IC). We investigate a discrete random walk model based on an explicit finite-difference approximation for the RFADE-IC and prove that the random walk model belongs to the domain of attraction of the corresponding stable distribution. We also present explicit and implicit difference approximations for the Riesz fractional advection-dispersion equation with initial and boundary conditions (RFADE-IBC) in a finite domain. Stability and convergence of these numerical methods for the RFADE-IBC are discussed. Some numerical examples are given to show that the numerical results are in good agreement with our theoretical analysis.

Coupled oscillators with power-law interaction and their fractional dynamics analogues

A one-dimensional chain of coupled oscillators with the long-range power-law interactions is considered. Equations of motion in the infrared limit are mapped onto the continuum equation with the Riesz fractional derivative of order α, when 0 < α < 2. The evolution of soliton-like and breather-like structures is obtained numerically and compared for two types of simulations: using the chain of oscillators and using the continuous medium equation with the fractional derivative.

Fractional dynamics of coupled oscillators with long-range interaction

We consider a one-dimensional chain of coupled linear and nonlinear oscillators with long-range powerwise interaction. The corresponding term in dynamical equations is proportional to 1//n-m/alpha+1. It is shown that the equation of motion in the infrared limit can be transformed into the medium equation with the Riesz fractional derivative of order alpha, when 0<alpha<2. We consider a few models of coupled oscillators and show how their synchronization can appear as a result of bifurcation, and how the corresponding solutions depend on alpha. The presence of a fractional derivative also leads to the occurrence of localized structures. Particular solutions for fractional time-dependent complex Ginzburg-Landau (or nonlinear Schrodinger) equation are derived. These solutions are interpreted as synchronized states and localized structures of the oscillatory medium.

Fractional dynamics of systems with long-range interaction

We consider one-dimensional chain of coupled linear and nonlinear oscillators with long-range powerwise interaction defined by a term proportional to 1/∣ n − m∣ α+1 . Continuous medium equation for this system can be obtained in the so-called infrared limit when the wave number tends to zero. We construct a transform operator that maps the system of large number of ordinary differential equations of motion of the particles into a partial differential equation with the Riesz fractional derivative of order α, when 0 < α < 2. Few models of coupled oscillators are considered and their synchronized states and localized structures are discussed in details. Particularly, we discuss some solutions of time-dependent fractional Ginzburg–Landau (or nonlinear Schrodinger) equation.