空间分数阶偏微分方程高精度快速算法的研究

项目名称： 空间分数阶偏微分方程高精度快速算法的研究

项目编号： No.11271068

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 孙志忠

作者单位： 东南大学

项目金额： 60万元

中文摘要： 分数阶偏微分方程具有明确的应用背景.发展数值方法求解分数阶偏微分方程是近年来国际上学术界的一个热点问题. 目前, 对时间分数阶偏微分方程已发展了很多稳定高效的数值算法, 对于空间分数阶偏微分方程仍有许多亟待解决的问题. 然而求解空间分数阶偏微分方程时不能简单照搬求解时间分数阶偏微分方程的数值方法, 因为二者是有本质区别的. 本课题旨在构造空间分数阶导数一致逼近的高阶数值微分公式, 进而对空间分数阶偏微分方程建立高精度的差分格式, 证明其可解性、稳定性和收敛性.由于分数阶导数的非局部性质, 对空间和时空分数阶微分方程所建立的数值算法还要考虑减少其存储量和降低计算的复杂度. 最后, 筛选出稳定高效的数值算法, 从而建立起一套新的求解空间分数阶偏微分方程理论框架体系.

中文关键词： 分数阶导数；分数阶微分方程；差分方法；收敛性；稳定性

英文摘要： The fractional partial differential equations have great applications in science and technology. The development of new numerical algorithms for fractional partial differential equations (FPDEs) has received great attention recently in scientific computing. Currently, there are a lot of stable and efficient numerical algorithms for solving time-FPDEs, meanwhile there are still some issues that need to be resolved in solving space-FPDEs. Moreover, the methods for solving time-FPDEs cannot be used directly for solving space-FPDEs as they are fundamentally different. To this end, this research plan devotes to develop high order numerical differential formulas, which are uniformly convergent, for space fractional derivatives; then to construct some high accuracy methods and fast algorithms for space-FPDEs and prove their solvability, stability and convergence; next, to reduce the storage amount and decrease computational complexity for all numerical methods since the nonlocal behavior of the fractional derivative; at last, to select computational stable and effective methods among all numerical algorithms via the numerical experiments and establish a new system of theoretical framework for solving fractional space-FPDEs.

英文关键词： factional derivative；fractional differtial equation；difference scheme；convergence；stability