再生核空间框架下弱奇异积分方程组的数值解法研究

The system of weakly singular integral equations can be applied in many science and engineering fields. Their numerical solution is solved because it is very difficult to obtain the analytic solution of the system of weakly singular integral equations. Due to the weakly singular property of integral operators, a lot of classic numerical methods rarely play their advantages. Therefore, it is theoretical and practical significant that the new and efficient numerical method for solving the system of weakly singular integral equations can be given. In the frame of the reproducing kernel space, considering the system of weakly singular integral equations, such as Abel kernel, logarithmic kernel and another kernel, this project provides a new numerical method, which is efficient to deal with the singularity, and the solution of this system is obtained by the provided method with small computation and high accuracy. In addition, combining with the collocation method and the reproducing kernel space method, this project gives the new numerical method for solving the system of integral equations. The special feature of this project is that the reproducing kernel space method is used and exact and approximate solutions are given for the system of weakly singular integral equations. We hope that this method can apply to other similar problems.

科学和工程的许多问题都可归结为计算具有弱奇异核的积分分方程组，而这些问题的解析解很难得到，只能靠数值求解。特别由于积分算子核函数的弱奇异性质，很多经典的数值算法难以发挥其优势。因此能够对此类方程组的求解给出新的、高效的数值算法具有一定的理论意义与实际应用价值。在再生核空间框架下，对具有Abel核、对数核形式的弱奇异核积分方程组，及另一形式的弱奇异核积分方程组，本项目提供了计算此类问题的数值算法，一种新算法，是克服弱奇异性的有效方法，同时方程组的求解计算量小，精度高，有收敛性证明；将配置法与再生核空间法有机结合起来，本项目将形成一种全新的方程组的数值求解算法；利用再生核空间框架和对研究的所有问题都给出精确解与近似解是本项目的特色，我们的方法和成果可望为解决这类问题提供可行的解决途径。

科学和工程的许多问题都可归结为计算具有弱奇异核的积分方程（组），而这些问题的解析解很难得到，只能靠数值求解。特别由于积分算子核函数的弱奇异性质，很多经典的数值算法难以发挥其优势。因此能够对此类方程（组）的求解给出新的、高效的数值算法具有一定的理论意义与实际应用价值。在再生核空间框架下，设计了基于再生核方法求解弱奇异核的方程组的数值算法，数值仿真结果表明该算法收敛速度快，精确度高；针对此类问题提出再生核配置法，避免了正交化过程，提高了求解精度，并例证再生核配置法求解奇异积分方程的有效性。针对具有Abel核的弱奇异积分方程等价转化为分数阶微分方程，并给出了在再生核空间中求解分数阶微分方程的数值方法；构造再生核空间的稠密子空间，结合最小二乘方法给出具有Chauchy核的奇异积分方程的近似解。构造再生核空间中的多尺度正交基，证明算法的收敛性并给出误差估计。本项目在再生核空间框架下形成了一系列全新的数值求解算法，为项目所提出的问题提供了有效的解决途径。