This paper presents an iterative reproducing kernel algorithm for obtaining the numerical solutions of Riesz fractional diffusion and advection-dispersion equations in porous media on a finite domain. The representation of the exact and the numerical solutions is given in the W (Ω) and H (Ω) inner product spaces. The computation of the required grid points relies on the R_(y,s) (x, t) and r_(y,s) (x,t) reproducing kernel functions. An efficient construction is given to obtain the numerical solution together with an existence proof of the exact solution based upon the reproducing kernel theory. Numerical solution of such Riesz fractional equations is acquired by interrupting the n-term of the exact solution. In this approach, numerical examples were analyzed to illustrate the design procedure and confirm the performance of the proposed algorithm in the form of tabulated data, numerical comparisons, and graphical results. Finally, the utilized results show the significant improvement of the algorithm while saving the convergence accuracy and time.

中文翻译：

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迭代介质核算法在多孔介质中Riesz分数阶扩散和扩散-扩散方程的数值解

本文提出了一种迭代再现核算法，用于获得有限域上多孔介质中Riesz分数阶扩散和对流扩散方程的数值解。精确解和数值解的表示形式在W（Ω）和H（Ω）内积空间中给出。所需网格点的计算取决于R _（y，s）（x，t）和r _（y，s）（x，t）复制内核函数。基于重现核理论，给出了一种有效的构造来获得数值解以及精确解的存在性证明。通过中断精确解的n项，可以获得此类Riesz分数方程的数值解。在这种方法中，通过分析数值示例来说明设计过程并以列表数据，数值比较和图形结果的形式确认所提出算法的性能。最后，所利用的结果表明该算法得到了显着改进，同时节省了收敛精度和时间。