The Weak Newton Method and Boundary Value Problems

1969 ◽  
Vol 6 (4) ◽  
pp. 539-550 ◽  
Author(s):  
Richard A. Tapia
Keyword(s):  
Boundary Value Problems ◽  
Newton Method ◽  
Boundary Value

Bifurcation Analysis Of Nonlinear Parameterized Two-Point Bvps With Liapunov — Schmidt Reduced Functions

2008 ◽  
Vol 8 (4) ◽  
pp. 350-359
Author(s):  
M. HERMANN ◽  
T.H. MILDE
Keyword(s):  
Boundary Value Problems ◽  
Newton Method ◽  
Bifurcation Analysis ◽  
Control Parameter ◽  
Boundary Value ◽  
External Control ◽  
Point Boundary ◽  
Bifurcation Points ◽  
Graphical Technique ◽  
Branch Switching

Abstract In this paper, we study nonlinear two-point boundary value problems (BVPs) which depend on an external control parameter. In order to determine numeri-cally the singular points (turning or bifurcation points) of such a problem with so-called extended systems and to realize branch switching, some information on the type of the singularity is required. In this paper, we propose a strategy to gain numerically this information. It is based on strongly equivalent approximations of the corresponding Liapunov — Schmidt reduced function which are generated by a simplified Newton method. The graph of the reduced function makes it possible to determine the type of singularity. The efficiency of our numerical-graphical technique is demonstrated for two BVPs.

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