分数阶微分-代数方程的高精度数值算法

Due to the non-local property of fractional differential/integral operator, the fractional calculus can be more precisely used to model various materials and processes having the memory and hereditary properties in scientific and engineering fields. Recently, fractional differential-algebraic equations has gained a wide applications in the research of constrained dynamical system, it makes more urgent to study the numerical solutions of this kind of equations. This program will adopt spectral deferred correction techniques and analytical structural mechanics methods to construct a high accuracy method for fractional differential-algebraic equations, and carry out the corresponding theoretical analysis. To improve computation efficiency, the resulting preconditioned nonlinear system is solved by using Newton-Krylov schemes. Finally, numerical experiments will be given to verify the advantages of the proposed methods.

由于分数阶微积分算子具有非局部性，因此分数阶微积分能用于更精确地建模在科学和工程中具有记忆性和遗传性的材料和过程。最近，分数阶微分-代数方程已成功地运用到约束动力系统的研究中，使得对该类方程的数值求解更加迫切。本申请项目将采用谱延迟校正技巧和分析结构力学方法来构造分数阶微分-代数方程的高精度格式，给出相应的理论分析，并且用Newton-Krylov法来提高计算效率。最后，通过数值实验来验证格式的优越性。

由于分数阶微积分算子具有非局部性，因此分数阶微积分能用于更精确地建模在科学和工程中具有记忆性和遗传性的材料和过程。最近，分数阶微分方程已成功地运用到约束动力系统的研究中，使得对该类方程的求解更加迫切。本项目研究了分数阶扩散波动方程的一个Runge-Kutta法，目前还没有公开发表的关于时间分数阶扩散方程的Runge-Kutta法。此外，设计了一个对分数阶次扩散方程和超扩散方程都有效的统一格式。