Jacobi-Galerkin spectral method for eigenvalue problems of Riesz fractional differential equations
Jacobi-Galerkin spectral method for eigenvalue problems of Riesz fractional differential equations
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发表时间：
2018-03
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通讯作者：
Lizhen Chen;Zhiping Mao;Hui-yuan Li
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作者：
Lizhen Chen;Zhiping Mao;Hui-yuan Li
本文提出了一种计算具有齐次Dirichlet边值的Riesz分数阶偏微分方程特征值的Jacobi-Galerkin谱方法。为了保持离散线性系统的对称性和正定性，引入适当定义的Sobolev空间，用标准Galerkin弱公式代替文献中的Petrov-Galerkin公式来近似特征值问题.证明了所提出的Galerkin公式的Poincar\'{e}不等式和逆不等式，这最终帮助我们建立了代数系统的条件数的精确估计。利用Babu\v{s}ka和奥斯本关于自伴和正定特征值问题的近似理论，得到了特征值和特征向量的严格误差估计.数值结果证明了算法的准确性和有效性，并验证了算法的渐近指数收敛性。此外，还得到了2阶Riesz分数阶微分算子的第n个特征值的Weyl型渐近律$ \lambda_n=\mathcal{O}（n^{2\alpha}）$，以及其代数系统关于多项式次数$N$的条件数$N^{4\alpha}$.
An efficient Jacobi-Galerkin spectral method for calculating eigenvalues of Riesz fractional partial differential equations with homogeneous Dirichlet boundary values is proposed in this paper. In order to retain the symmetry and positive definiteness of the discrete linear system, we introduce some properly defined Sobolev spaces and approximate the eigenvalue problem in a standard Galerkin weak formulation instead of the Petrov-Galerkin one as in literature. Poincar\'{e} and inverse inequalities are proved for the proposed Galerkin formulation which finally help us establishing a sharp estimate on the algebraic system's condition number. Rigorous error estimates of the eigenvalues and eigenvectors are then readily obtained by using Babu\v{s}ka and Osborn's approximation theory on self-adjoint and positive-definite eigenvalue problems. Numerical results are presented to demonstrate the accuracy and efficiency, and to validate the asymptotically exponential oder of convergence. Moreover, the Weyl-type asymptotic law $ \lambda_n=\mathcal{O}(n^{2\alpha})$ for the $n$-th eigenvalue $\lambda_n$ of the Riesz fractional differential operator of order $2\alpha$, and the condition number $N^{4\alpha}$ of its algebraic system with respect to the polynomial degree $N$ are observed.