Search

In this article, two different second-order numerical differential formulas have been derived for the Caputo derivatives C D 0 , t α f t , (0 < α < 1) and C D 0 , t β f t , (1 < β < 2) at point t k + 1 ∕ 2 , respectively. By applying these numerical differential formulas to the time fractional mixed sub-diffusion and diffusion–wave equations and combining with the fourth-order compact formula of the spatial derivative, then we get a difference scheme with convergence order O τ 2 + h 4 , which improves the accuracy of the known algorithms in the literature for the same type of equation. Based on the discrete energy method, we prove that the developed difference scheme is unconditionally stable and convergent for α ∈ (0 , 1) and β ∈ 1 , − 1 + 17 2 . Finally, some numerical results are provided to testify the accuracy and efficiency of the difference scheme. [ABSTRACT FROM AUTHOR]

In this paper, a high-order numerical algorithm is derived for a class of time fractional Fokker–Planck equations. To avoid discrete approximation of convection term, we first transform the original equation into an equivalent form, then the spatial derivative and time Riemann–Liouville derivative are approximated by a fourth-order compact formula and a second-order midpoint formula, respectively. The stability and convergence of the developed difference scheme is studied by the energy method. Finally, numerical experiments are provided to demonstrate the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

The main goal of this article is to establish a new 4th-order numerical differential formula to approximate Riesz derivatives which is obtained by means of a newly established generating function. Then the derived formula is used to solve the Riesz space fractional advection-dispersion equation. Meanwhile, by the energy method, it is proved that the difference scheme is unconditionally stable and convergent with order 𝓞(τ2 + h4). Finally, several numerical examples are given to show that the numerical convergence orders of the numerical differential formulas and the finite difference scheme are in line with the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, high‐order numerical methods for time‐Caputo and space‐Riesz fractional Bloch‐Torrey equations in one‐ and two‐dimensional space are constructed, where the second‐order backward fractional difference operator and the sixth‐order fractional‐compact difference operator are applied to approximate the time and space fractional derivatives, respectively. The stability and convergence of the methods are analyzed and it is shown that the convergence orders are higher than the earlier work. Finally, some numerical experiments are presented to demonstrate the effectiveness of the methods and confirm our theoretical results. [ABSTRACT FROM AUTHOR]

The main goal of this article is to establish a new 4th-order numerical differential formula to approximate Riesz derivatives which is obtained by means of a newly established generating function. Then the derived formula is used to solve the Riesz space fractional advection-dispersion equation. Meanwhile, by the energy method, it is proved that the difference scheme is unconditionally stable and convergent with order 𝓞(τ2 + h4). Finally, several numerical examples are given to show that the numerical convergence orders of the numerical differential formulas and the finite difference scheme are in line with the theoretical analysis. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we focus our attention on the development of the high-order numerical algorithm for the time–space tempered fractional diffusion-wave equation in two spatial dimensions. Based on the fourth-order fractional–compact difference operator, a new difference scheme with convergence order O (τ 2 + h 1 4 + h 2 4) is derived, where τ is the temporal stepsize, h 1 and h 2 are the spatial stepsizes, respectively. The stability and convergence of the algorithm are investigated by the energy method and numerical experiment is carried out to verify the feasibility of the numerical algorithm. [ABSTRACT FROM AUTHOR]

By constructing new numerical approximation formulas for the left and right Riemann–Liouville tempered fractional derivatives of order α ∈ ( 0 , 1 ) , a numerical algorithm is proposed to solve the space tempered fractional convection equation. The algorithm is shown to be unconditionally stable and convergent with order O τ 2 + h 2 . Finally, numerical example is provided to verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

Firstly, we construct a fourth-order numerical differential formula for approximating Riesz derivative. Substituting it into the damped nonlinear space fractional Schrödinger equation, the original equation becomes a matrix form of nonlinear ordinary differential equation system. Then, by means of implicit integrating factor method and Padé approximation, we obtain a new numerical scheme whose convergence order is higher than the existing algorithms. Finally, a numerical example is given to verify the effectiveness of the numerical algorithm and the correctness of the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, applying a novel second-order numerical approximation formula for the Riesz derivative and Crank–Nicolson technique for the temporal derivative, a numerical algorithm is constructed for the two-dimensional spatial fractional diffusion equation with convergence order, whereτ,andare the temporal and spatial step sizes, respectively. It is proved that the proposed algorithm is unconditionally stable and convergent by using the energy method. Meanwhile, by adding the high-order perturbation items for the above numerical scheme, an alternating direction implicit difference scheme is also constructed. Finally, some numerical results are presented to demonstrate the validity of theoretical analysis and show the accuracy and effectiveness of the method described herein. [ABSTRACT FROM AUTHOR]

In this article, based on the idea of combing symmetrical fractional centred difference operator with compact technique, a series of even-order numerical differential formulas (named the fractional-compact formulas) are established for the Riesz derivatives with order [ABSTRACT FROM AUTHOR]

In recent years, fractional differential equations have attracted much attention due to their wide application. In this paper, we present a novel numerical method for the space–time Riesz–Caputo fractional diffusion equation, which discrete the Riesz derivative by a fourth-order fractional-compact difference scheme, then the above space–time Riesz–Caputo fractional diffusion equation change into a fractional ordinary differential equation (FODE) system. Again using Laplace and inverse Laplace transforms, one can get the analytical solution of the FODE system, furthermore, we approximate the Mittag-Leffler function by the global Padé approximation and obtain the numerical method for the space–time Riesz–Caputo fractional diffusion equation. Finally, two numerical examples are presented to show that the numerical results are in good line with our theoretical analysis. [ABSTRACT FROM AUTHOR]

In this article, two kinds of high-order compact finite difference schemes for second-order derivative are developed. Then a second-order numerical scheme for a Riemann-Liouvile derivative is established based on a fractional centered difference operator. We apply these methods to a fractional anomalous subdiffusion equation to construct two kinds of novel numerical schemes. The solvability, stability, and convergence analysis of these difference schemes are studied by using Fourier method. The convergence orders of these numerical schemes are O(t² +h6 and O(t² + h8), respectively. Finally, numerical experiments are displayed which are in line with the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we firstly develop two high-order approximate formulas for the Riesz fractional derivative. Secondly, we propose a temporal second order numerical method for a fractional reaction-dispersion equation, where we discretize the Riesz fractional derivative by using two numerical schemes. We prove that the numerical methods for a spatial Riesz fractional reaction dispersion equation are both unconditionally stable and convergent, and the orders of convergence are O ( τ 2 + h 6 ) and O ( τ 2 + h 8 ) , in which τ and h are spatial and temporal step sizes, respectively. Finally, we test our numerical schemes and observe that the numerical results are in good agreement with the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, our aim is to study the high order finite difference method for the reaction and anomalous-diffusion equation. According to the equivalence of the Riemann-Liouville and Grünwald-Letnikov derivatives under the suitable smooth condition, a second-order difference approximation for the Riemann-Liouville fractional derivative is derived. A fourth-order compact difference approximation for second-order derivative in spatial is used. We analyze the solvability, conditional stability and convergence of the proposed scheme by using the Fourier method. Then we obtain that the convergence order is O (τ 2 + h 4), where τ is the temporal step length and h is the spatial step length. Finally, numerical experiments are presented to show that the numerical results are in good agreement with the theoretical analysis. [ABSTRACT FROM AUTHOR]

Abstract: The Riesz space fractional reaction dispersion equation is a generalization of the classical reaction dispersion equation. In this paper, we develop a class of different form difference scheme for solving the Riesz space fractional reaction dispersion equation based on the parameter spline function and improved matrix transform method. Stability of the difference schemes are discussed by the matrix method. Finally, some numerical examples are given to show that the numerical results are in good agreement with our theoretical analysis. [Copyright &y& Elsevier]

Abstract: In this paper, we develop two classes of finite difference schemes for the reaction–subdiffusion equations by using a mixed spline function in space direction, forward and backward difference in time direction, respectively. It has been shown that some of the previous known difference schemes can be derived from our schemes if we suitably choose the spline parameters. By Fourier method, we prove that one class of difference scheme is unconditionally stable and convergent, the other is conditionally stable and convergent. Finally, some numerical results are provided to demonstrate the effectiveness of the proposed difference schemes. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a three level compact difference scheme of for the difference solution of a two-dimensional second-order non-homogeneous linear hyperbolic equation , where . Stability analysis of the method has been carried out. Finally, numerical examples are used to illustrate the efficiency of the new difference scheme. [Copyright &y& Elsevier]

Abstract: In this paper, we use a semi-discrete and a padé approximation method to propose a new difference scheme for solving convection–diffusion problems. The truncation error of the difference scheme is . It is shown through analysis that the scheme is unconditionally stable. Numerical experiments are conducted to test its high accuracy and to compare it with Crank–Nicolson method. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a three level compact difference scheme of for the difference solution of one-dimensional second-order non-homogeneous linear hyperbolic equation , , where . Stability analysis of the method has been carried out. Finally, numerical examples are used to illustrate the efficiency of the new difference scheme. [Copyright &y& Elsevier]

Abstract: In this paper, by using a parameters cubic spline in space and compact finite difference in time direction, we get a class of finite difference schemes for solving second-order hyperbolic equations with mixed boundary conditions. Stability analysis of the methods have been carried out. It has been shown that by suitable choosing the cubic spline parameters most of the previous known methods for homogeneous and non-homogeneous cases can be derived from our methods. We also obtain new high accuracy schemes of and . Numerical comparison with Rashidinia’s method [J. Rashidinia et al., Spline methods for the solutions of hyperbolic equations, Appl. Math. Comput. 190 (2007) 882–886] shows the superiority of our presented schemes. [Copyright &y& Elsevier]

In this paper, we develop a new numerical algorithm for solving the Riesz tempered space fractional diffusion equation. The stability and convergence of the numerical scheme are discussed via the technique of matrix analysis. Finally, numerical experiments are performed to confirm the effectiveness of our numerical algorithm. [ABSTRACT FROM AUTHOR]