非线性分数阶微分方程的高精度数值方法的研究

fractional differential equations have been proved to be a valuable tool in the modeling of many phenomena，especially in physics, mechanics, biology, engineering, finance, hydrology, and fractional-order controllers．Due to its profound physical background, the subject of the fractional differential equations is gaining much importance and attention. Generally, most fractional differential equations can not be resolved analytically．Therefore，the research of numerical method of the fractional differential equation is of important theory significance and practical value． Spectral methods are a widely used tool for solving several types of differential and integral equations. It provides exceedingly accurate numerical results for smooth problems. This project is to study the application of spectral collocation methods to a few class of nonlinear fractional integro-differential equation (including nonlinear fractional integro-differential equation and singular initial value problems of the Lane-Emden type in the fractional order ordinary differential equations). Later, the rigorous error analysis are given. This study is great of both theoretical and practical significance.

分数阶微分方程可用于模拟物理、力学、生物学、工程、金融、水文学、分数阶控制器等领域中的许多现象。由于其深厚的物理背景，分数微分方程已经变成一个非常重要的热门课题。一般情况下，大多数分数阶微分方程的解析解是难以获得的，所以，有必要开展其数值方法的研究。而谱方法作为求解微分方程的一种重要数值方法，它的主要优点是高精度，已被应用于科学和工程计算的众多领域。本项目旨在研究谱配置方法求解几类非线性分数阶积分微分方程初值问题（包括非线性分数阶积分微分方程和Lane-Emden型非线性分数阶微分方程奇异初值问题）,并给出严格的误差分析。本课题的研究是一项具有重要理论意义和实际应用价值的工作，将对现有的理论和算法有所发展。