分数阶对流-扩散方程的时空广义Jacobi谱与谱元法

Fractional differential equations more precisely describe the property of some materials with memory and heredity. Although one can derive the analytical solutions of some linear fractional differential equations, but it is difficult to numerically expand these analytical solutions composed by special functions. Moreover, one can not find analytical solutions of some nonlinear fractional differential equations. Therefore, it is of great importance to find efficient and reliable numerical methods to solve these fractional differential equations. As all we know, the long history and globality dependence of fractional differential often cause local numerical schemes, such as, finite difference methods suffer from heavy cost of computing for long-time numerical simulation. Therefore, as a kind of global high accuracy numerical methods, spectral methods are more appropriate tools for solving fractional differential equations. This work will be dedicated to space-time generalized Jacobi spectral and spectral element methods for solving initial-boundary value problems of fractional advection-diffusion equations. We first establish the approximation theory on any fractional derivative of generalized Jacobi orthogonal, quasi-orthogonal projections and related interpolations. We then propose space-time generalized Jacobi spectral and spectral element methods for linear and nonlinear fractional advection-diffusion equations and design their efficient numerical algorithms. At last, we analyze the stability and convergence of corresponding numerical schemes by using the above approximation results. The study is bound to expand the application of spectral methods in numerically solving fractional differential equations.

分数阶微分方程非常适合描述具有记忆和遗传特性的问题。尽管一些分数阶微分方程可以精确求解，但它们大多由难以数值表示的特殊函数构成，还有些非线性分数阶微分方程无法求出解析解。因此，寻找分数阶微分方程的可靠、有效数值解法就显得非常重要。我们知道，分数阶微分的历史依赖和全局相关性常导致诸如有限差分等非全局性数值格式难以长时间数值模拟。而谱方法作为一种全局性高精度数值方法，自然成为求解分数阶微分方程的理想工具。本项目拟采用时空广义Jacobi谱与谱元法求解线性与非线性分数阶对流-扩散方程初边值问题。我们首先建立广义Jacobi正交、拟正交投影和相关插值及其任意分数阶导数逼近理论；然后发展线性与非线性分数阶对流-扩散方程的时空广义Jacobi谱与谱元方法及其高效算法；最后利用上述逼近结果分析这些数值格式的稳定性和收敛性。该研究成果必将拓展谱方法在数值求解分数阶微分方程中的应用。

谱方法作为求解偏微分方程的一种重要数值方法，已被广泛应用于科学及工程计算的众多领域。分数阶微分的历史依赖和全局相关性常导致诸如有限差分等非全局性数值格式难以长时间数值模拟。而谱方法作为一种全局性高精度数值方法，自然成为求解分数阶微分方程的理想工具。在本项目的资助下，我们依照研究计划，针对分数阶微分方程初边值问题的时空高精度谱与谱配置法开展了系统、深入的研究工作，具体包括：.(1)针对时间分数阶对流-扩散方程初边值问题构造了混合广义Jacobi和Chebyshev谱配置格式，并设计了相应的高精度算法，利用一些数值算例验证了数值格式的有效性和高精度。.(2)针对时间分数阶KdV方程初边值问题构造了差分Petrov-Galerkin谱格式。然后分析了该数值格式的稳定性和收敛性，同时设计了相应的高精度算法，利用一些数值算例验证了数值格式的有效性和高精度。.(3)针对变系数多项时间分数阶扩散和扩散-波动方程构造了一致差分Galerkin谱格式。然后分析了该数值格式的稳定性和收敛性，同时设计了相应的高精度算法，最后利用一些数值算例验证了数值算法的有效性和高精度。. 总体而言，我们在International Journal of Computer Mathematics、Journal of Computational and Applied Mathematics和 Mathematical Modelling and Analysis等国际重要影响学术刊物发表SCI论文5篇，基本完成项目拟定计划和目标。