Caputo型分数阶微分方程非局部边值问题的数值方法

Fractional differential equations have been applied successfully in several areas of Physics, Chemistry, Medicine, Control Engineering, and Signal Processing. Fractional differential equations have attracted much attention of scientists and engineers. In most case the solutions of fractional differential equations are not possible to be obtained in a closed form and that requires the application of different numerical methods. However, there are still some computational difficulties in the numerical treatment of these equations since fractional order differential and integral operators are non-local operators. The available literature on fractional calculus deals mainly with the numerical solutions of initial value problems for fractional order differential equations. In contrast to the case for initial value problems, not much attention has been paid to the nonlocal fractional boundary value problems. The main objective of this project is to present efficient and reliable numerical methods for Caputo fractional differential equations with nonlocal boundary conditions. By using advantages of reproducing kernel theory, this project will give continuous approximate solutions over the whole interval in the reproducing kernel Hilbert space. The expected results will be of great significance for fractional calculus and its applications in practical problems.

分数阶微分方程已经成功应用于物理、化学、医学、控制工程和信号处理等越来越多的领域。因此，分数阶微分方程引起了很多科学和工程技术人员的关注。对很多分数阶微分方程，无法获得其解析解，寻求分数阶微分方程的数值求解方法就成为一个迫切需要解决的问题。然而，分数阶微积分算子是非局部算子（具有历史依赖性和全局相关性），这就决定了构造分数阶微分方程的有效数值求解方法也是不容易的。目前已有的文献主要关注分数阶初值问题的数值处理方法，而关于分数阶微分方程非局部边值问题的数值求解方法却很少。本项目主要致力于提出Caputo型分数阶非局部边值问题的有效数值处理方法。在再生核希尔伯特空间框架下，利用再生核理论的优势，给出Caputo型分数阶非局部边值问题的连续整体近似解。预期结果对分数阶微积分理论及其在很多实际问题中的应用都具有十分重要的意义。

分数阶微分方程在物理和工程领域具有重要的应用，本项目的主要目的是构造Caputo型分数阶微分方程的有效数值方法。在再生核希尔伯特空间框架下，充分利用再生核理论的优点，提出求解Caputo型分数阶微分方程的全局性方法。结合再生核理论和拟线性化技术，提出了求解Caputo型分数阶Riccati微分方程的有效方法。利用打靶法的思想，在再生核空间中，提出了求解非局部Caputo型分数阶微分方程的有效近似方法，该方法的主要优点是：简化了非局部边界条件的处理，获得的近似解及其导数一致收敛于精确解及其导数。