A space-time Legendre spectral tau method for the two-sided space-time Caputo fractional diffusion-wave equation
A space-time Legendre spectral tau method for the two-sided space-time Caputo fractional diffusion-wave equation
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DOI：
10.1007/s11075-015-9990-9
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发表时间：
2015
影响因子：
2.1
通讯作者：
A. Bhrawy;M. Zaky;R. A. Gorder
中科院分区：
文献类型：
--
作者：
A. Bhrawy;M. Zaky;R. A. Gorder
时空分数扩散波动方程（FDWE）是对经典扩散和波动方程的推广，用于模拟流体流动、油层等实际扩散和波动现象。本文报道了一种求解具有各种非齐次边界条件的双面时空Caputo FDWE的精确谱方法。该方法基于移位勒让德（SLT）过程，结合Riemann-Liouville分数阶积分的移位勒让德运算矩阵、左侧和右侧分数阶导数。我们主要关注在时间和空间离散中实现该算法。此外，从理论上对Dirichlet边界条件进行了收敛性分析，并对使用其他条件的几种特殊情况进行了图解分析。这表明，如果给定FDWE中的数据是光滑的，则Legendre方法是指数收敛的。最后，通过数值算例验证了该方法的精度。
The space-time fractional diffusion-wave equation (FDWE) is a generalization of classical diffusion and wave equations which is used in modeling practical phenomena of diffusion and wave in fluid flow, oil strata and others. This paper reports an accurate spectral tau method for solving the two-sided space and time Caputo FDWE with various types of nonhomogeneous boundary conditions. The proposed method is based on shifted Legendre tau (SLT) procedure in conjunction with the shifted Legendre operational matrices of Riemann-Liouville fractional integral, left-sided and right-sided fractional derivatives. We focus primarily on implementing this algorithm in both temporal and spatial discretizations. In addition, convergence analysis is provided theoretically for the Dirichlet boundary conditions, along with graphical analysis for several special cases using other conditions. These suggest that the Legendre Tau method converges exponentially provided that the data in the given FDWE are smooth. Finally, several numerical examples are given to demonstrate the high accuracy of the proposed method.