分数阶微分方程多点边值问题的数值算法研究

分数阶微分方程边值问题是经典微分方程边值问题自然的数学推广，具有深刻的物理背景和丰富的理论内涵，在物理、生物、化学等多个学科领域具有广泛的应用。本项目基于非线性泛函分析的理论基础，以分数阶积微分方程多点边值问题为研究对象，在所构造的满足多点边值条件的再生核空间中，利用此空间良好的性质，给出此类分数阶模型的 Fourier 级数逼近算法，并且通过理论分析证明了算法的收敛性和稳定性，同时给出了详尽的误差估计以及算法的时间复杂性分析。. 本项目的研究能够促进分数阶模型多点边值问题处理技术的发展，为解决力学、生物学和工程技术中的一些实际问题提供有力的支持。

Fractional differential equations(FDE) with the boundary value problem are the natural mathematical promotion of the classic problem. FDE has profound physical background and rich theoretical connotation, which have been widely applied to problems in physical, biological, chemical, and other disciplines. Based on nonlinear functional analysis theory, this project mainly investigates fractional integro-differential equations with multi-point boundary value problem. By constructing the reproducing kernel space with multi-point boundary value conditions, we give the Fourier series approximation algorithmsthe for a class of FDE by the good properties of the reproducing kernel spaces. The theoretical analysis shows that the scheme is stable. At the same time, we present the detailed error estimation and complexity analysis of the algorithms. . The research of this project, not only enriches and develops the processing technology of FDE, but also provides a powerful support for many practical problems in mechanics, biology and engineering.