Part I. "Asymptotic Boundary Conditions for Ordinary Differential Equations"ududThe numerical solution of two point boundary value problems on semi-infinite intervals is often obtained by truncating the interval at some finite point. In this thesis we determine a hierarchy of increasingly accurate boundary conditions for the truncated interval problem. Both linear and nonlinear problems are considered. Numerical techniques for error estimation and the determination of an appropriate truncation point are discussed.ududA Fredholm theory for boundary value problems on semi-infinite intervals is developed, and used to prove the stability of our numerical methods.ududPart II. "Numerical Hopf Bifurcation"ududSeveral numerical methods for locating a Hopf bifurcation point of a system of o.d.e.'s or p.d.e.'s are discussed. A new technique for computing the Hopf bifurcation parameters is also presented. Finally, well-known numerical techniques for simple bifurcation problems are adapted for Hopf bifurcation problems. This provides numerical techniques for computing the bifurcating branch of periodic solutions, possibly including turning points and simple bifurcation points. The stability of the periodic solutions is also discussed.ud
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机译:第I部分“常微分方程的渐近边界条件” ud ud半无限区间上的两点边值问题的数值解通常是通过在某个有限点处截断区间来获得的。在本文中,我们为截断区间问题确定了越来越精确的边界条件的层次结构。同时考虑了线性和非线性问题。 ud ud建立了半无限区间边值问题的Fredholm理论,用于证明我们数值方法的稳定性。 ud ud第二部分。讨论了“数值霍普夫分叉” ud ud用于定位一个或多个d.e.或p.d.e.系统的霍普夫分叉点的几种数值方法。还提出了一种计算Hopf分叉参数的新技术。最后,用于简单分叉问题的众所周知的数值技术适用于Hopf分叉问题。这提供了用于计算周期解的分支分支的数值技术,可能包括转折点和简单分支点。还讨论了周期解的稳定性。 ud
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