Report title: On Solving Semilinear Singularly Perturbed Neumann Problems for Multiple Solutions

Reporter: Yuan Yongjun, Associate Professor Hunan Normal University

Reporting time: 13:00-14:00, November 26, 2020

Report location: Tencent Conference ID: 221 827 035 Conference password: 123456

Conference link: https://meeting.tencent.com/s/KZ0ex0eyxe34

School contact: Xu Zhiguo xuzg2014@jlu.edu.cn

Report summary:

Based on the analysis of bifurcation points and Morse indices of trivial solutions for a general semilinear singularly perturbed Neumann boundary value problem at any perturbation value, the exact critical perturbation value $\varepsilon_c$ which determines the existence or non-existence of nontrivial positive solutions is obtained. As a result, the generating process of nontrivial positive solutions is studied and further used to guide algorithm design and numerical computation. An improved local minimax method is then proposed accordingly to compute both M-type and W-type saddle points by using an adaptive local refinement strategy and a Newton method to overcome singularity difficulty and accelerate local convergence. Extensive numerical results are reported to justify the critical perturbation value $\varepsilon_c$ and investigate some interesting solution properties of different types of problems.

Brief introduction of the speaker:

Yuan Yongjun, male, Ph.D., associate professor, master's tutor, was awarded the "Young Talents" World Heritage Talent Program of Hunan Normal University. He graduated from Hunan Normal University with a bachelor’s degree in mathematics and applied mathematics in 2007. He graduated from Hunan Normal University with a Ph.D. in computational mathematics in 2012. His tutor is Professor Xie Ziqing. From June 2012 to October 2015, he worked in Beijing Computing Science Research Center and Singapore. National University is engaged in post-doctoral research, and his supervisor is Professor Bao Weizhu; he has been working in Hunan Normal University since November 2015. Mainly engaged in the research of the numerical solution of differential equations, and obtained certain research results in the calculation of multiple solutions of nonlinear partial differential equations and the numerical simulation of Bose-Einstein condensation. The related results are published in Physical Review Letters, SIAM Journal of Scientific Computing, Mathematics of Computation and Physical Review A and other magazines. Presided over 1 National Natural Science Foundation Youth Fund Project and 1 General Project.

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