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Zbl 1112.65069
Zou, Yongkui; Hu, Qingwan; Zhang, Ran
On numerical studies of multi-point boundary value problem and its fold bifurcation.
(English)
[J] Appl. Math. Comput. 185, No. 1, 527-537 (2007). ISSN 0096-3003

The nonlinear second order multi-point boundary value problem $u''+f(t,u)=0$, $t\in (0,1)$ with the boundary conditions $u(0)=\sum_{i=1}^{n-2}a_iu(\xi_i)+\lambda_1,\;u'(1)=\sum_{i=1}^{n-2} b_iu'(\xi_i)+\lambda_2$ is investigated. Here the function $f$ is Lipschitz continuous with respect to $(t,u)$ and $f'_u$ is continuous in $[0,1]\times\mathbb{R}$, and $0<\xi_1<\xi_2<\dots<\xi_{n-2}<1$, $n\in \mathbb{Z}$. The authors suggest a numerical approximating method based upon the shooting method for finding regular solutions and simple fold bifurcation solutions and their structure depending on a parameter. The other types of boundary conditions $u(0),u(1);u'(0),u(1)$ and $u'(0),u'(1)$ can be investigated analogously. The results of many numerical experiments are given, which show the genericity of the fold bifurcation phenomenon in these problems.
MSC 2000:
*65L10 Boundary value problems for ODE (numerical methods)
34B10 Multipoint boundary value problems
34B15 Nonlinear boundary value problems of ODE
34C23 Bifurcation (periodic solutions)

Keywords: numerical computation; shooting method; numerical experiments

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