| Summary: | 博士 === 國立臺灣大學 === 數學研究所 === 107 === In this thesis, we first revisit multi-peak solutions to the Lane-Endem problem as in [7]. In order to find numerical solutions, the Scaling Iterative Algorithm (SIA) is implemented in our solvers. By adding different symmetrical restrictions on the numerical solutions, more multi-peak solutions to the Lane-Endem problem are found. After the Lane-Endem problem, we investigate into nontrivial solutions of the Kirchhoff-type problems. The SIA is again implemented in the solver for positive solutions which standard iteration methods failed to compute. Implicit SIA (ISIA), an improvement of the SIA, is developed due to the nonlinear and nonlocal nature of the Kirchhoff -type problem. Three implementations of the ISIA solvers are also discussed.
The second part of this thesis is on numerical approaches to Poisson-Boltzmann-type equations. We develop solvers for Poisson-Boltzmann-type equations and give numerical supports to a few theoretical results. We check the fact that the PB_ns model (4.19) can be reduced to Li Bo''s model (4.20) (also see [12][13]). Such solvers are also used to study multiple solutions of steady-state Poisson-Nernst-Planck equations with steric effects (PNP-steric model), whose existence was proven in [3]. Two distinct solutions of the PNP-steric model are numerically found and plotted after we clarity where they might appear. The efficiency and complexity of the solvers are also discussed.
Most problems mentioned above are modeled and solved in OpenFOAM, which is a free, open source software for computational fluid dynamics problems. Others are solved in MATLAB. Profiles of all the solutions are delicately plotted.
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