广义分数阶微分方程的高效数值方法研究

分数阶微积分在数学物理和工程问题中获得了广泛的应用。广义分数阶微分方程的理论基础和高效数值方法是近几年分数阶微积分研究中的重要方向。本项目主要研究广义分数阶微分方程的高效数值方法，内容包括：运用不动点定理研究广义分数阶微分方程的可解性；基于有限差分法和插值、外推技术设计高精度有限差分格式；利用Ritz-Galerkin方法构造有限元方法求解具有对称结构的广义分数阶微分方程；运用线性代数的相关工具，研究离散化后的差分方程中系数矩阵的特征值分布和逆矩阵范数的上界；基于数值分析和逼近理论分析算法的稳定性、收敛性和误差；推广分数阶导数定义，研究非矩形区域上的广义分数阶微分方程数值解及特征。本项目旨在促进广义分数阶微分方程的研究，为高效精确地求解方程设计更好的算法和理论。本项目的开展将会极大地丰富已有的研究手段和研究方法，并对分数阶微积分理论、数值代数、逼近论等领域发展提供丰富的结果和研究课题。

Fractional calculus has been extensively and successfully applied to many fields of mathematical physics and engineering problems. The theory and high effective numerical approaches of generalized fractional differential equations become very important research directions in fractional calculus in the recent years. The current project is mainly devoted to investigating the high effective numerical methods of generalized fractional differential equations, including studying the solvability of generalized fractional differential equations by using various fixed point theorems; and based on the conventional Ritz-Galerkin technique, constructing finite element method associating with fractional energy functional for solving generalized fractional differential equations with symmetric structure; and applying related tools from linear algebra to study the distribution of eigenvalue and norm bound of the inverse matrix of the coefficient matrix in the corresponding difference equation; and employing useful conclusions of numerical analysis and approximation theory to analyze the stability, convergence and error of the designed numerical methods in this proposal; and redefining fractional derivatives on non-rectangular domain, then studying the numerical solutions and their properties of generalized fractional differential equations on the general convex domain. The project aims to promote the study of generalized fractional differential equations, and design more effective and precise algorithms and theory for solving generalized fractional differential equations. This project will enrich the existing research methods and research tools, and provide abundant conclusions and research topics in the related fields of fractional calculus theory, numerical algebra and approximation theory.