论文标题
一种细节扩散方程的新光谱方法
A novel spectral method for the subdiffusion equation
论文作者
论文摘要
在本文中，我们设计和分析了一种新的频谱方法，以用于细胞扩散方程。众所周知，该方程的解通常在初始时间附近是单数。因此，传统高阶数值方法的直接应用效率低下。我们试图通过将可变转换技术与光谱方法相结合，从而克服这种困难。这个想法是首先使用合适的变量转换来重新缩放基础方程，然后为重新缩放方程式构造光谱方法。我们建立了一个基于$ψ$ - 分离Sobolev空间的新变分框架。这使我们能够证明相关的变分问题的适合性。所提出的光谱方法基于变异问题和广义的jacobi多项式，以近似重新缩放的分数微分方程。我们的理论和数值研究表明，即使确切的解决方案的规律性非常有限，该提出的方法对于一般右侧侧函数是指数收敛的。还提供了实施细节，以及一系列数值示例，以显示所提出的方法的效率。
In this paper, we design and analyze a novel spectral method for the subdiffusion equation. As it has been known, the solutions of this equation are usually singular near the initial time. Consequently, direct application of the traditional high-order numerical methods is inefficient. We try to overcome this difficulty in a novel approach by combining variable transformation techniques with spectral methods. The idea is to first use suitable variable transformation to re-scale the underlying equation, then construct spectral methods for the re-scaled equation. We establish a new variational framework based on the $ψ$-fractional Sobolev spaces. This allows us to prove the well-posedness of the associated variational problem. The proposed spectral method is based on the variational problem and generalized Jacobi polynomials to approximate the re-scaled fractional differential equation. Our theoretical and numerical investigation show that the proposed method is exponentially convergent for general right hand side functions, even though the exact solution has very limited regularity. Implementation details are also provided, along with a series of numerical examples to show the efficiency of the proposed method.
