分数阶微分方程的谱配置法研究

Fractional differential equations (FDEs) have been widely applied to numerical simulations of various problems arising in science and engineering. Spectral methods is natural candidate to simulate the FDEs for its high-order accuracy. However, the classical spectral methods are still facing many difficulties and challenges, for instance, the Jacobi spectral methods are usually restricted by parameter conditions, which lead to the lack of flexibility or the inconvenience of theoretical analysis; in addition, the spectral collocation methods for weakly singular problems have not been systematically studied. In this proposal, we investigate the efficient spectral collocation methods for fractional differential equations and do some work as follows: creating multi-step spectral collocation methods for fractional differential and integro-differential equations, modifying or constructing more efficient spectral collocation methods for fractional ordinary and partial differential equations based on Jacobi polynomials, trying to defining more suitable Generalized Jacobi Functions(GJFs), creating high order spectral collocation methods based on GJFs, establishing error analysis on weakly singular solutions for these new spectral methods, etc. These work will further expand the basic theories of spectral methods and enrich the numerical methods of fractional ordinary/partial differential equations.

分数阶微分方程在物理、生物、材料等领域应用越来越广泛，而谱方法因其高精度特性而成为该类方程数值求解的较好选择。本项目针对谱方法求解分数阶问题遇到的困难，如经典Jacobi谱方法的灵活性不够，或精度不足、或理论分析不便，且谱配置法研究尚不够系统等，展开如下工作：研究分数阶微分、积分方程与Volterra积分方程的关系，构造基于Jacobi多项式的多步谱配置法；进一步研究Jacobi多项式的分数次微积分并试图建立更好的表达式及更优的谱近似，丰富分数次函数空间的内容，对分数阶常、偏微分方程建立更为简便易行的谱配置法；探索基于广义Jacobi函数的谱方法理论，力求改进或定义新的广义Jacobi函数，促进谱配置法在分数阶常、偏微分方程的应用；揭示新算法对弱奇异解的适应性，并建立新算法对光滑解和奇异解的误差理论。这些问题的解决将在一定程度上拓展谱方法的基础理论和应用。

分数阶微分方程的数值解法是当前计算数学研究的热点之一，而谱方法因其高精度特性成为较好的选择。本项目针对谱方法求解分数阶微分方程相关问题遇到的困难，如经典谱方法的灵活性不够，或对奇异解精度不足、或理论分析不便，且谱配置法研究尚不够系统等，进行了如下几方面的研究：对分数阶微分、积分微分方程的初边值问题，构造基于移位正交多项式的多步谱配置法；进一步研究Jacobi多项式的分数次微积分的谱近似，以此为基础对分数阶扩散方程等建立更为简便易行的谱配置法；探索基于广义Jacobi函数的谱方法理论，促进谱配置法在分数阶常、偏微分方程的应用。项目基本按计划进行执行，完成了部分研究任务，即对分数阶积分微分方程的初值问题和分数阶微分方程的两点边值问题分别建立多步谱配置法及误差估计理论，并用典型数值算例验证了算法的有效性和误差理论的合理性。这些工作将在一定程度上拓展谱方法在工程技术领域的应用，丰富谱方法的基础理论。