Abstract In this paper, we construct two efficient numerical schemes by combining the finite difference method and operational matrix method (OMM) to solve Riesz-space fractional diffusion equation (RFDE) and Riesz-space fractional advection-dispersion equation (RFADE) with initial and Dirichlet boundary conditions. We applied matrix transform method (MTM) for discretization of Riesz-space fractional derivative and OMM based on shifted Legendre polynomials (SLP) and shifted Chebyshev polynomial (SCP) of second kind for approximating the time derivatives. The proposed schemes transform the RFDE and RFADE into the system of linear algebraic equations. For a better understanding of the methods, numerical algorithms are also provided for the considered problems. Furthermore, optimal error bound for the numerical solution is derived, and theoretical unconditional stability has been proved with respect to L 2 -norm. The stability of the schemes is also verified numerically. The schemes are observed to be of second-order accurate in space. The effectiveness and accuracy of the schemes are tested by taking two numerical examples of RFDE and RFADE and found to be in good agreement with the exact solutions. It is observed that the numerical schemes are simple, easy to implement, yield high accurate results with both the basis functions. Moreover, the CPU time taken by the schemes with SLP basis is very less as compared to schemes with SCP basis.

中文翻译：

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基于有限差分/运算矩阵的 Riesz 空间分数式偏微分方程的高效数值算法

摘要 在本文中，我们结合有限差分法和运算矩阵法（OMM）构造了两种有效的数值方案来求解 Riesz 空间分数阶扩散方程（RFDE）和 Riesz 空间分数阶对流扩散方程（RFADE），其中初始Dirichlet 边界条件。我们应用矩阵变换方法 (MTM) 离散化 Riesz 空间分数阶导数和基于移动勒让德多项式 (SLP) 和第二类移动切比雪夫多项式 (SCP) 的 OMM 以逼近时间导数。所提出的方案将 RFDE 和 RFADE 转换为线性代数方程组。为了更好地理解这些方法，还为所考虑的问题提供了数值算法。此外，推导出数值解的最佳误差界限，并且已经证明了关于L 2 -范数的理论无条件稳定性。方案的稳定性也得到了数值验证。观察到这些方案在空间上是二阶精确的。通过RFDE和RFADE的两个数值例子对方案的有效性和准确性进行了测试，发现与精确解非常吻合。可以看出，数值方案简单，易于实现，两种基函数都能产生高精度的结果。此外，与基于 SCP 的方案相比，基于 SLP 的方案占用的 CPU 时间非常少。通过RFDE和RFADE的两个数值例子对方案的有效性和准确性进行了测试，发现与精确解非常吻合。可以看出，数值方案简单，易于实现，两种基函数都能产生高精度的结果。此外，与基于 SCP 的方案相比，基于 SLP 的方案占用的 CPU 时间非常少。通过RFDE和RFADE的两个数值例子对方案的有效性和准确性进行了测试，发现与精确解非常吻合。可以看出，数值方案简单，易于实现，两种基函数都能产生高精度的结果。此外，与基于 SCP 的方案相比，基于 SLP 的方案占用的 CPU 时间非常少。