基于再生核理论求解奇异摄动边值问题

Singularly perturbed problems arise in appications to engineering and they are difficult to solve in differential equations. Therefore, singularly perturbed problems have attracted much attention. For many nonlinear singularly perturbed problems, it is almost impossible to obtain their exact solutions, and therfore it is very important to obtain their effective approximate solutions. However, the solution of singularly perturbed problem has a multiscale character. So it is also difficult to obtain their good numerical solutions. Reproducing kernel space is a perfect space in studying numerical analysis. The project aims at finding methods for solving singularly perturbed problems with two-point boundary conditions and nonlocal boundary conditions in the reproducing kernel space. The project will present effective numerical methods for solving local and nonlocal singularly perturbed boundary value problems by using the good properties of the reproducing kernel space and its reproducing kernel. By the methods, the global approximation can be established on the whole interval and the convergence of the approximate solution and its derivatives is uniform. Hence, it permits the study of the behavior of derivatives of the approximate solution. The expected results will be of great significance for singular perturbation theory and applications and the solution of many engineering problems.

奇异摄动问题源于各种工程应用，是微分方程中比较难处理的一类问题。因此，奇异摄动问题得到了广泛的关注。对很多奇异摄动问题，特别是非线性问题，无法求出其精确解，获得其有效的近似解就显得尤为重要。然而，奇异摄动问题的解具有多尺度特性，这就决定了获得其理想的数值解也是比较困难的。再生核空间是研究数值分析比较理想的空间框架，本项目致力于在再生核空间框架下研究奇异摄动两点边值问题和具有非局部边界条件的奇异摄动问题的求解方法。本项目将充分利用再生核空间及其再生核的良好性质，提出获得局部、非局部奇异摄动边值问题的理想近似解的有效方法，通过该方法可以得到整个求解区间上的整体近似解，且该近似解及其导数分别一致收敛于精确解及其导数，这就为进一步研究解的导数的特性提供了方便。预期结果对奇异摄动理论与应用，以及很多实际工程问题的解决都具有十分重要的意义。

奇异摄动问题在地球流体力学、海洋与大气环流、化学反应、最优控制等很多领域具有重要的应用，因此，这类问题的数值求解方法的研究引起了很多研究人员的兴趣。然而，这类问题解的边界层特性使得构造有效的数值方法具有很大的挑战性。本项目致力于在再生核空间框架下，充分利用再生核理论在数值近似方面的优越性，提出求解奇异摄动边值问题的简单有效的数值方法。基于再生核理论，借助一个合适的变量变换，提出了求解具有单个边界层的奇异摄动边值问题的有效数值方法。结合渐近近似技术、变量伸缩技术和再生核理论，分别提出了求解具有内部层和两个边界层的奇异摄动转向点问题的数值方法。另外，基于再生核理论，也提出了求解几类奇异摄动延迟初值问题和边值问题的数值方法。通过本项目提出的数值方法得到的近似解具有自身的优点，都是光滑的全局近似解。