A Spectral Method (of Exponential Convergence) for Singular Solutions of the Diffusion Equation with General Two-Sided Fractional Derivative
A Spectral Method (of Exponential Convergence) for Singular Solutions of the Diffusion Equation with General Two-Sided Fractional Derivative
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DOI：
10.1137/16m1103622
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发表时间：
2018-01
期刊：
影响因子：
--
通讯作者：
Zhiping Mao;G. Karniadakis
中科院分区：
文献类型：
--
作者：
Zhiping Mao;G. Karniadakis
分数阶偏微分方程数值解的开放性问题是如何获得高阶精度的奇异解；即使对于光滑的右手边，fpga的解也是奇异的。这里，我们考虑具有一般双侧分数阶导数的一维扩散方程，其特征参数为p\in [0,1]；对于$p=1/2$，我们恢复了Riesz分数阶导数，而对于$p=1 $, 0，我们得到了单侧分数阶导数。我们在适当加权的Sobolev空间中使用（双边）Jacobi多项式作为基和测试函数的Petrov—Galerkin投影。特别地，我们导出了这些双面Jacobi多分形作为Sturm—Liouville问题的特征函数，该问题的权重唯一地由参数p决定。我们提供了严格的分析，并获得了依赖于强迫项的正则性的最优误差估计，即，对于光滑数据（对应于奇异解），我们获得了指数收敛，其中…
An open problem in the numerical solution of fractional partial differential equations (FPDEs) is how to obtain high-order accuracy for singular solutions; even for smooth right-hand sides solutions of FPDEs are singular. Here, we consider the one-dimensional diffusion equation with general two-sided fractional derivative characterized by a parameter $p\in [0,1]$; for $p=1/2$ we recover the Riesz fractional derivative, while for $p = 1$, 0 we obtain the one-sided fractional derivative. We employ a Petrov--Galerkin projection in a properly weighted Sobolev space with (two-sided) Jacobi polyfracnomials as basis and test functions. In particular, we derive these two-sided Jacobi polyfractonomials as eigenfunctions of a Sturm--Liouville problem with weights uniquely determined by the parameter $p$. We provide a rigorous analysis and obtain optimal error estimates that depend on the regularity of the forcing term, i.e., for smooth data (corresponding to singular solutions) we obtain exponential convergence, wh...