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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]

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]

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 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, 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]