Well-conditioned fractional collocation methods using fractional Birkhoff interpolation basis
Well-conditioned fractional collocation methods using fractional Birkhoff interpolation basis
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DOI：
10.1016/j.jcp.2015.10.029
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发表时间：
2015-03
期刊：
影响因子：
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通讯作者：
Yu-jian Jiao;Lilian Wang;Can Huang
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文献类型：
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作者：
Yu-jian Jiao;Lilian Wang;Can Huang
本文的目的有两个方面。首先，我们提供了明确的和紧凑的公式计算Caputo和（修改后的）Riemann-Liouville（RL）的分数伪谱微分矩阵（F-PSDM）的任何阶一般雅可比-高斯-洛巴托（JGL）点。证明了在Caputo情形下，只要计算μ∈（0，1）阶的F-PSDM就可以计算任意k+ μ阶的F-PSDM，其中k≥ 0;而在修正RL情形下，只需计算μ∈（0，1）阶的分数次积分矩阵.其次，我们引入了合适的分数阶JGL Birkhoff插值问题，得到了新的插值多项式基函数，这些基函数具有如下性质：（i）由新基生成的矩阵在JGL“内部”点处是F-PSDM的精确逆，（ii）在新基下的配点格式中最高分数阶导数的矩阵是对角矩阵，（iii）在新基下的配点格式中最高分数阶导数的矩阵是对角矩阵，（iv）在新基下的配点格式中最高分数阶导数的矩阵是对角矩阵。（iii）在Caputo情形下得到的线性系统是良态的，而在修正RL情形下，系数矩阵的特征值是高度集中的.在这两种情况下，使用新基的配置格式的线性系统可以在几次迭代内由迭代求解器求解。值得注意的是，逆可以以非常稳定的方式计算，因此这为分数阶微分方程（FDE）的常用分数配置方法提供了最佳预条件。同样值得注意的是，选择某些特殊的JGL点，其参数与方程的阶数相关，可以简化实现。我们强调，使用贝特曼的分数阶积分公式和不同参数的雅可比多项式之间的快速变换，是必不可少的，我们的算法开发。
The purpose of this paper is twofold. Firstly, we provide explicit and compact formulas for computing both Caputo and (modified) Riemann–Liouville (RL) fractional pseudospectral differentiation matrices (F-PSDMs) of any order at general Jacobi–Gauss–Lobatto (JGL) points. We show that in the Caputo case, it suffices to compute F-PSDM of order μ∈(0, 1) to compute that of any order k+ μ with integer k≥ 0, while in the modified RL case, it is only necessary to evaluate a fractional integral matrix of order μ∈(0, 1). Secondly, we introduce suitable fractional JGL Birkhoff interpolation problems leading to new interpolation polynomial basis functions with remarkable properties:(i) the matrix generated from the new basis yields the exact inverse of F-PSDM at “interior” JGL points;(ii) the matrix of the highest fractional derivative in a collocation scheme under the new basis is diagonal; and (iii) the resulted linear system is well-conditioned in the Caputo case, while in the modified RL case, the eigenvalues of the coefficient matrix are highly concentrated. In both cases, the linear systems of the collocation schemes using the new basis can be solved by an iterative solver within a few iterations. Notably, the inverse can be computed in a very stable manner, so this offers optimal preconditioners for usual fractional collocation methods for fractional differential equations (FDEs). It is also noteworthy that the choice of certain special JGL points with parameters related to the order of the equations can ease the implementation. We highlight that the use of the Bateman's fractional integral formulas and fast transforms between Jacobi polynomials with different parameters, is essential for our algorithm development.