题目:"A New Augmented Singular Transform and its Partial Newton-Correction Method for Finding More Solutions"

报告人：Jianxin Zhou, Texas A&M University, College Station, TX, USA

时间: 2015年11月12日, 周四 上午10:30-11:30  

地点：管研楼1218

摘要：Using the information provided by previously found solutions, an augmented singular transform is
introduced in [XYZ,2015] to change the local basin structure of the original problem for finding new solutions.
The formulation in [XYZ,2015] involves the kernel of an unknown solution to be found or the kernels of all previously
found solutions, and then left several theoretical issues unsolved. In this paper, we derive a new augmented singular
transform which changes only the local basin/barrier structure around $u=0$ for finding more solutions. Thus the new
formulation is much easier to apply and resolves all unsolved theoretical issues left in [XYZ,2015]. Then a corresponding
partial Newton-correction method is designed to solve the augmented problem on the solution set. Mathematical justification
of the new formulation, method and its local convergence are established. The new method is first tested on two very different
variational problems. It is then applied to solve a non-variational nonlinear convection-diffusion equation for multiple solutions, which are, for the first time, numerically computed and visualized with their profile and contour plots. Several interesting phenomenons are observed for the first time and open for mathematical verification. Since the new formulation is general and simple, it can be modified to treat other problems, e.g., quasilinear differential PDEs, a large system of PDEs with equality constraints, for multiple solutions.

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