A Jacobi spectral collocation method for solving multi-dimensional nonlinear fractional sub-diffusion equations
A Jacobi spectral collocation method for solving multi-dimensional nonlinear fractional sub-diffusion equations
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DOI：
10.1007/s11075-015-0087-2
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发表时间：
2016-09
影响因子：
2.1
通讯作者：
A. Bhrawy
中科院分区：
文献类型：
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作者：
A. Bhrawy
本文采用一维和二维非线性分数阶次扩散方程的配置法的运算矩阵形式。在所提出的配置法中，一维和二维情形的近似解均采用双平移和三平移雅可比多项式作为基函数。在下划线问题中给出的空间和时间分数导数用雅可比运算矩阵表示。本文研究了用于时间和空间离散的谱配置方案。因此，通过将FSDE及其初始条件和边界条件简化为更容易求解的非线性代数方程组来确定展开系数。此外，还从理论上估计了近似解的误差，并结合图解分析，证实了该方法在空间和时间离散上的指数收敛速度。为了说明我们算法的高精度，我们给出了一些数值例子的数值结果，并将我们的数值结果与文献报道的结果进行了比较。
This article adapts an operational matrix formulation of the collocation method for the one- and two-dimensional nonlinear fractional sub-diffusion equations (FSDEs). In the proposed collocation approach, the double and triple shifted Jacobi polynomials are used as base functions for approximate solutions of the one- and two-dimensional cases. The space and time fractional derivatives given in the underline problems are expressed by means of Jacobi operational matrices. This investigates spectral collocation schemes for both temporal and spatial discretizations. Thereby, the expansion coefficients are then determined by reducing the FSDEs, with their initial and boundary conditions, into systems of nonlinear algebraic equations which are far easier to be solved. Furthermore, the error of the approximate solution is estimated theoretically along with graphical analysis to confirm the exponential convergence rate of the proposed method in both spatial and temporal discretizations. In order to show the high accuracy of our algorithms, we report the numerical results of some numerical examples and compare our numerical results with those reported in the literature.