Space-time Chebyshev spectral collocation method for nonlinear time-fractional Burgers equations based on efficient basis functions
Space-time Chebyshev spectral collocation method for nonlinear time-fractional Burgers equations based on efficient basis functions
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基于高效基函数的非线性时间分式Burgers方程的时空切比雪夫谱配置方法
DOI：
10.1002/mma.7015
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发表时间：
2021
影响因子：
2.9
通讯作者：
Tohidi Emran
中科院分区：
文献类型：
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作者：
Huang Yu;Mohammadi Zadeh Fatemeh;Hadi Noori Sk;ari Mohammad;Ahsani Tehrani Hojjat;Tohidi Emran
本文提出了一种平衡时空谱配置方法，用于求解给定初始边界条件下的非线性时间分数伯格方程。大多数现有的求解偏微分方程的近似方法都是不平衡的，因为它们使用低阶方案（例如有限差分法）来积分时间变量，并使用高阶数值框架（例如谱伽辽金（或无网格）方法）来离散空间变量。因此，在本文中，我们建议的方案在时间和空间变量上都是平衡的。由于时间分数Burgers方程解的非光滑性，我们应用有效的基函数作为分数拉格朗日函数来插值时间变量。通过将主方程和初始边界条件与空间和分数时间变量的相应运算矩阵的实现结合起来，假设的模型被转换为相关的非线性代数方程组，可以通过高效的迭代求解器（例如Levenberg-Marquardt方法）来求解。同时，我们还充分分析了方法的收敛性。此外，我们考虑了几个测试问题来检查建议的方案，确认其相对于文献中最新数值方法的高精度和低计算成本。
This article contributes to a balanced space–time spectral collocation method for solving nonlinear time‐fractional Burgers equations with given initial‐boundary conditions. Most of existing approximate methods for solving partial differential equations are unbalanced, since they have used a low order scheme such as finite difference methods for integrating the temporal variable and a high order numerical framework such as spectral Galerkin (or meshless) method for discretization of space variables. So in the current paper, our suggested scheme is balanced in both time and space variables. Due to the non‐smoothness of solutions of time‐fractional Burgers equations, we apply efficient basis functions as the fractional Lagrange functions for interpolating time variable. By collocating the main equation and the initial‐boundary conditions together with the implementation of the corresponding operational matrices of spatial and fractional temporal variables, the assumed model is transformed into the associated system of nonlinear algebraic equations, which can be solved via efficient iterative solvers such as the Levenberg–Marquardt method. Also, we fully analyze the convergence of method. Moreover, we consider several test problems for examining the suggested scheme that confirms its high accuracy and low computational cost with respect to recent numerical methods in the literature.