【作者】 沈铨；

【导师】 丁睿；

【作者基本信息】 苏州大学 ， 计算数学， 2008， 硕士

【摘要】 半线性椭圆边值问题在物理、力学、化工、天文等众多领域中有广泛的应用.由于半线性椭圆边值问题常具有多解且解缺乏稳定性,因而数值上很难计算出它的多解.本文讨论了二维半线性椭圆边值问题的径向基函数无网格数值方法及其可视化.按照半线性椭圆边值问题解的三种不同情况,分别给出了基于径向基函数的无网格数值方法并分析了其多解性和解的可视化.第二章,利用径向基函数(Radial Basis Functions)无网格法和牛顿法(Newton’s method)求解了单解情形时半线性椭圆边值问题,并给出了数值算例,通过与常用算法进行比较,说明了该方法具有易于编程、计算精度高及不需要对区域进行网格划分等优点.第三章,提出了基于径向基函数的配点型(RBF Collocation Method)无网格法和尺度迭代算法(Scaling Iterative Algorithm)的耦合方法.利用该方法在各种不同的区域上求解了半线性椭圆边值问题的正解以及奇异摄动半线性椭圆问题.我们发现利用径向基函数无网格方法耦合尺度迭代算法(SIA)来求解半线性椭圆边值问题与奇异摄动半线性椭圆问题是可行的,且具有良好的精度、易于编程.该方法明显优于现有的边界元(BEM),有限元(FEM),有限差分(FDM)的耦合方法.第四章,对于半线性椭圆边值问题多解情形时的变号解,构造了基于径向基函数(RBF)配点型的搜索牛顿法(Search Newton Method)的耦合方法,大量数值算例表明基于径向基函数配点型的搜索牛顿法精度好,易于编程,无需对区域进行划分,对不规则区域同样有效,是一种求解半线性椭圆边值问题多解的有效方法.更多还原

【Abstract】 The semilinear elliptic boundary value problems(SEBVPs) appear widely in physics, mechanics, chemical engineering, astronomy and so on. Because the SEBVPs often have multiple solutions, which reveal a lack of stability, it is difficult to compute the multiple solutions numerically. This thesis describes the radial basis functions(RBF) meshless methods of the SEBVPs and the visualization of the solutions. According to three different cases of the solutions of the SEBVPs, this thesis presents RBF meshless methods respectively and analyzes the multiplicity and visualization of the solutions.In chapter two, it introduces the coupling method of RBF meshless method and Newton’s method to solve the SEBVP, which has a single solution. Then the numerical example is given. Compared with other algorithms, the results show that the method is easy to programme, has a high precision and do not need the mesh in the area.In chapter three, it introduces the coupling of meshless method with RBF collocation method and scaling iterative algorithm(SIA) to solve the positive solution of the SEBVPs and the singularly perturbed SEBVPs on different areas. We discover it is feasible to solve the positive solution of the SEBVPs and the singularly perturbed SEBVPs by the coupling method. The method is easy to implement, has a high precision and is superior to the coupling method of BEM, FEM and FDM. In chapter four, it introduces RBF collocation meshless method combined with search newton method to solve the sign-changing solution of the SEBVPs and visualizes the solution. The numerical examples show that our algorithm has the properties of easy programming, high precision and meshfree in the area. It is also suitable for the irregular area. So it is an efficient method for the SEBVP.更多还原