Riesz Space Fractional Derivative Research Articles

A New Fractional Calculus Model for the Two-dimensional Anomalous Diffusion and its Analysis

In this paper, a special model for the two-dimensional anomalous diffusion is first deduced from the basic continuous time random walk equations in terms of a time- and space-fractional partial differential equation with the Caputo time-fractional derivative of order α / 2 and the Riesz space-fractional derivative of order α . For α -fractional diffusion equation describes the so called Levy flights that correspond to the continuous time random walk model, where both the mean waiting time and the jump length variance of the diffusing particles are divergent. The fundamental solution to the α -fractional diffusion equation is shown to be a two-dimensional probability density function that can be expressed in explicit form in terms of the Mittag-Leffler function depending on the auxiliary variable |x |/(2√t ) as in the case of the fundamental solution to the classical isotropic diffusion equation. Moreover, we show that the entropy production rate associated with the anomalous diffusion process described by the α -fractional diffusion equation is exactly the same as in the case of the classical isotropic diffusion equation. Thus the α -fractional diffusion equation can be considered to be a natural generalization of the classical isotropic diffusion equation that exhibits some characteristics of both anomalous and classical diffusion.

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.

Space-Time Fractional Schrödinger Equation With Composite Time Fractional Derivative

Abstract The fractional Schrödinger equation has recently received substantial attention. We generalize the fractional Schrödinger equation to the Hilfer time derivative and the Caputo space derivative, and solve this equation for an infinite potential by using the Adomian decomposition method. The infinite domain solution of the space-time fractional Schrödinger equation in the case of Riesz space fractional derivative is obtained in terms of the Fox H-functions. We interpret our results for the fractional Schrödinger equation by introducing a complex effective potential in the standard Schrödinger equation, which can be used to describe quantum transport in quantum dots.

Soliton Solutions of Nonlinear and Nonlocal Sine-Gordon Equation Involving Riesz Space Fractional Derivative

AbstractIn this article, a novel approach comprising modified homotopy analysis method with Fourier transform has been implemented for the approximate solution of fractional sine-Gordon equation (SGE) ${u_{tt}}\; - \;{}^RD_x^\alpha u\; + \;{\rm{sin}}u\; = \;0,$ where $^RD_x^\alpha $ is the Riesz space fractional derivative, 1 ≤ α ≤ 2. For α=2, it becomes classical SGE utt−uxx + sinu=0, and corresponding to α=1, it becomes nonlocal SGE utt−Hu + sinu=0, which arises in the Josephson junction theory, where H is the Hilbert transform. The fractional SGE is considered as an interpolation between the classical SGE (corresponding to α=2) and nonlocal SGE (corresponding to α=1). Here, the approximate solution of fractional SGE is derived by using modified homotopy analysis method with the Fourier transform. Then, we analyse the results by numerical simulations, which demonstrate the simplicity and effectiveness of the present method.

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.

A new analytical modelling for nonlocal generalized Riesz fractional sine-Gordon equation

In this paper, a novel approach comprising the modified decomposition method with Fourier transform has been implemented for the approximate solution of fractional sine-Gordon equation utt-RDxαu+sinu=0 where RDxα is the Riesz space fractional derivative, 1≤α≤2. For α=2, it becomes classical sine-Gordon equation utt−uxx+sin u=0 and corresponding to α=1, it becomes nonlocal sine-Gordon equation utt−Hu+sin u=0 which arises in Josephson junction theory, where H is the Hilbert transform. The fractional sine-Gordon equation is considered as an interpolation between the classical sine-Gordon equation (corresponding to α=2) and nonlocal sine-Gordon equation (corresponding to α=1). Here the analytic solution of fractional sine-Gordon equation is derived by using the modified decomposition method with Fourier transform. Then, we analyze the results by numerical simulations, which demonstrate the simplicity and effectiveness of the present method.

Maximum norm error analysis of difference schemes for fractional diffusion equations

In this study, we present several numerical approximation methods for solving the fractional diffusion equations with the Riesz space fractional derivative. Based on the fractional centered difference approximation to the Riesz fractional derivative, an explicit and an implicit finite difference schemes for solving the space fractional diffusion equations are obtained. The stability and global convergence of these two schemes in the maximum norm are analyzed rigorously. The convergence order is O(τ+h2), where τ is the temporal grid size and h is spatial grid size, respectively. Furthermore, a numerical scheme for the one-dimensional multi-term time–space fractional diffusion equations is obtained. The unconditionally stability and convergence of the scheme in maximum norm are established. The convergence orders are 2-α in the temporal direction and two in the spatial direction, where α is the maximum time fractional derivative orders in the equation. In addition, the extension to the two-dimensional case is also discussed. Finally, several numerical examples are provided to show the effectiveness and accuracy of our methods.

NUMERICAL SOLUTION OF A MODEL FOR TURBULENT DIFFUSION

A model is considered for turbulent diffusion which consists of a Riesz space fractional derivative to describe the turbulent phenomenon and also includes advection and classical diffusion. We present a first order explicit numerical method and a second order implicit numerical method to solve our problem and prove convergence results for both methods, including the derivation of stability constraints needed for the explicit numerical method to converge. In the end, to give some insights into the phenomenon of turbulent diffusion described by the Riesz fractional derivative, we show the behavior of the solution when we consider a Gaussian initial condition.

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.

Crank–Nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative

In this paper, the Crank–Nicolson (CN) difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative is studied. The existence of this difference solution is proved by the Brouwer fixed point theorem. The stability and convergence of the CN scheme are discussed in the L2 norm. When the fractional order is two, all those results are in accord with the difference scheme developed for the classical non-fractional coupled nonlinear Schrödinger equations. Some numerical examples are also presented.

Novel numerical methods for time-space fractional reaction diffusion equations in two dimensions

We consider time-space fractional reaction diffusion equations in two dimensions. This equation is obtained from the standard reaction diffusion equation by replacing the first order time derivative with the Caputo fractional derivative, and the second order space derivatives with the fractional Laplacian. Using the matrix transfer technique proposed by Ilic, Liu, Turner and Anh [Fract. Calc. Appl. Anal., 9:333--349, 2006] and the numerical solution strategy used by Yang, Turner, Liu, and Ilic [SIAM J. Scientific Computing, 33:1159--1180, 2011], the solution of the time-space fractional reaction diffusion equations in two dimensions can be written in terms of a matrix function vector product $f(A)b$ at each time step, where $A$ is an approximate matrix representation of the standard Laplacian. We use the finite volume method over unstructured triangular meshes to generate the matrix $A$, which is therefore non-symmetric. However, the standard Lanczos method for approximating $f(A)b$ requires that $A$ is symmetric. We propose a simple and novel transformation in which the standard Lanczos method is still applicable to find $f(A)b$, despite the loss of symmetry. Numerical results are presented to verify the accuracy and efficiency of our newly proposed numerical solution strategy. References D. Baleanu, Z. B. Guvenc, J. A. T. Machado. (Eds.) New Trends in Nanotechnology and Fractional Calculus Applications. Springer, 2010. R. E. Ewing, T. Lin and Y. Lin. On the accuracy of the finite volume element method based on piecewise linear polynomials. SIAM J. Numerical Analysis 39(6):1865--1888, 2002. doi:10.1137/S0036142900368873 M. Ilic, F. Liu, I. Turner, and V. Anh. Numerical approximation of a fractional-in-space diffusion equation (II)---with nonhomogeneous boundary conditions. Fract. Calc. Appl. Anal., 9:333--349, 2006. http://hdl.handle.net/10525/1287 R. Klages, G. Radons, I. M. Sokolov. (Eds.) Anomalous Transport: Foundations and Applications. Wiley, 2008. Y. Lin and C. Xu. Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comp. Phys., 225:1533--1552, 2007. doi:10.1016/j.jcp.2007.02.001 I. Podlubny. Fractional Differential Equations. Academic Press, New York, 1999. Y. Saad. Analysis of some Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal., 29:209--228, 1992. doi:10.1137/0729014 J. Sabatier, O. P. Agrawal, J. A. T. Machado. (Eds.) Advances in fractional calculus: Theoretical developments and applications in physics and engineering. Springer, 2007. V. Simoncini and D. B. Szyld. Recent computational developments in Krylov subspace methods for linear systems. Numer. Linear Algebra Appl., 14:1--59, 2007. doi:10.1002/nla.499 H. A. van der Vorst. An iterative solution method for solving $f(A)x=b$ using Krylov subspace information obtained for the symmetric positive definite matrix $A$. J. Comput. Appl. Math., 18:249--263, 1987. doi:10.1016/0377-0427(87)90020-3 Q. Yang, F. Liu, and I. Turner. Stability and convergence of an effective numerical method for the time-space fractional Fokker--Planck equation with a nonlinear source term. International Journal of Differential Equations, Article ID 464321, 22 pages, 2010. doi:10.1155/2010/464321 Q. Yang, F. Liu, and I. Turner. Numerical methods for fractional partial differential equations with Riesz space fractional derivatives. Applied Mathematical Modelling, 34:200--218, 2010. doi:10.1016/j.apm.2009.04.006 Q. Yang, I. Turner, F. Liu, and M. Ilic. Novel numerical methods for solving the time-space fractional diffusion equation in 2D. SIAM J. Scientific Computing, 33:1159--1180, 2011. doi:10.1137/100800634

Stability and Convergence of an Effective Numerical Method for the Time‐Space Fractional Fokker‐Planck Equation with a Nonlinear Source Term

Fractional Fokker‐Planck equations (FFPEs) have gained much interest recently for describing transport dynamics in complex systems that are governed by anomalous diffusion and nonexponential relaxation patterns. However, effective numerical methods and analytic techniques for the FFPE are still in their embryonic state. In this paper, we consider a class of time‐space fractional Fokker‐Planck equations with a nonlinear source term (TSFFPENST), which involve the Caputo time fractional derivative (CTFD) of order α∈ (0, 1) and the symmetric Riesz space fractional derivative (RSFD) of order μ∈ (1, 2]. Approximating the CTFD and RSFD using the L1‐algorithm and shifted Grünwald method, respectively, a computationally effective numerical method is presented to solve the TSFFPE‐NST. The stability and convergence of the proposed numerical method are investigated. Finally, numerical experiments are carried out to support the theoretical claims.

Computationally efficient numerical methods for time- and space-fractional Fokker–Planck equations

Fractional Fokker–Planck equations have been used to model several physical situations that present anomalous diffusion. In this paper, a class of time- and space-fractional Fokker–Planck equations (TSFFPE), which involve the Riemann–Liouville time-fractional derivative of order 1-α (α∊(0, 1)) and the Riesz space-fractional derivative (RSFD) of order μ∊(1, 2), are considered. The solution of TSFFPE is important for describing the competition between subdiffusion and Lévy flights. However, effective numerical methods for solving TSFFPE are still in their infancy.We present three computationally efficient numerical methods to deal with the RSFD, and approximate the Riemann–Liouville time-fractional derivative using the Grünwald method. The TSFFPE is then transformed into a system of ordinary differential equations (ODE), which is solved by the fractional implicit trapezoidal method (FITM). Finally, numerical results are given to demonstrate the effectiveness of these methods. These techniques can also be applied to solve other types of fractional partial differential equations.

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.

Evolution equations for the probabilistic generalization of the Voigt profile function

The spectrum profile that emerges in molecular spectroscopy and atmospheric radiative transfer as the combined effect of Doppler and pressure broadenings is known as the Voigt profile function. Because of its convolution integral representation, the Voigt profile can be interpreted as the probability density function of the sum of two independent random variables with Gaussian density (due to the Doppler effect) and Lorentzian density (due to the pressure effect). Since these densities belong to the class of symmetric Lévy stable distributions, a probabilistic generalization is proposed as the convolution of two arbitrary symmetric Lévy densities. We study the case when the widths of the distributions considered depend on a scale factor τ that is representative of spatial inhomogeneity or temporal non-stationarity. The evolution equations for this probabilistic generalization of the Voigt function are here introduced and interpreted as generalized diffusion equations containing two Riesz space-fractional derivatives, thus classified as space-fractional diffusion equations of double order.