Numerical solutions of the space-time fractional diffusion equation via a gradient-descent iterative procedure

K. Tansri, A. Kittisopaporn, P. Chansangiam
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Abstract

A one-dimensional space-time fractional diffusion equation describes anomalous diffusion on fractals in one dimension. In this paper, this equation is discretized by finite difference schemes based on the Gr¨unwald-Letnikov approximation for Riemann-Liouville and Caputo’s fractional derivatives. It turns out that the discretized equations can be put into a compact form, i.e., a linear system with a block lower-triangular coefficient matrix. To solve the linear system, we formulate a matrix iterative algorithm based on gradient-descent technique. In particular, we work out for the space fractional diffusion equation. Theoretically, the proposed solver is always applicable with satisfactory convergence rate and error estimates. Simulations are presented numerically and graphically to illustrate the accuracy, the efficiency, and the performance of the algorithm, compared to other iterative procedures for linear systems.
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时空分数阶扩散方程的梯度下降迭代解法
一维时空分数扩散方程描述一维分形上的异常扩散。在本文中,基于Riemann-Liouville和Caputo分数导数的Gr¨unwald-Letnikov近似,用有限差分格式离散了该方程。结果表明,离散化的方程可以被简化为紧致形式,即具有块下三角系数矩阵的线性系统。为了求解线性系统,我们提出了一种基于梯度下降技术的矩阵迭代算法。特别地,我们计算了空间分数阶扩散方程。理论上,所提出的求解器总是适用的,具有令人满意的收敛速度和误差估计。与线性系统的其他迭代程序相比,通过数值和图形模拟来说明该算法的准确性、效率和性能。
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CiteScore
3.10
自引率
4.00%
发文量
77
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