Rational Spectral Methods for PDEs Involving Fractional Laplacian in Unbounded Domains
Rational Spectral Methods for PDEs Involving Fractional Laplacian in Unbounded Domains
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DOI：
10.1137/19m1244299
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发表时间：
2019-05
期刊：
影响因子：
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通讯作者：
T. Tang;Lilian Wang;Huifang Yuan;Tao Zhou
中科院分区：
文献类型：
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作者：
T. Tang;Lilian Wang;Huifang Yuan;Tao Zhou
许多包含分数阶拉普拉斯算子的偏微分方程自然地设置在无界域中，其基础解衰减非常缓慢，服从一定的幂律。它们的数值解尚未得到充分研究。本文的目的是利用有理基（或改进的映射Gegenbauer函数）在无界域中对这类模型开发精确的谱方法。谱算法的主要组成部分是对有理基的傅里叶变换和分数阶拉普拉斯变换的显式表示，这些表示来自于与修正贝塞尔函数相关的一些有用的积分恒等式。有了这些，我们可以通过预先计算相关的分数阶微分矩阵来构造有理谱伽辽金和直接搭配方案。得到了分数阶Sobolev空间中有理谱逼近的最优误差估计，并分析了所提Galerkin格式的最优收敛性。我们还提供了大量的数值结果来证明理性方法优于Hermite函数方法。
Many PDEs involving fractional Laplacian are naturally set in unbounded domains with underlying solutions decay very slowly, subject to certain power laws. Their numerical solutions are under-explored. This paper aims at developing accurate spectral methods using rational basis (or modified mapped Gegenbauer functions) for such models in unbounded domains. The main building block of the spectral algorithms is the explicit representations for the Fourier transform and fractional Laplacian of the rational basis, derived from some useful integral identites related to modified Bessel functions. With these at our disposal, we can construct rational spectral-Galerkin and direct collocation schemes by pre-computing the associated fractional differentiation matrices. We obtain optimal error estimates of rational spectral approximation in the fractional Sobolev spaces, and analyze the optimal convergence of the proposed Galerkin scheme. We also provide ample numerical results to show that the rational method outperforms the Hermite function approach.