An inexact Newton method for nonlinear two-point boundary-value problems

1992 ◽  
Vol 75 (3) ◽  
pp. 471-486 ◽  
Author(s):  
E. J. Dean
Keyword(s):  
Boundary Value Problems ◽  
Newton Method ◽  
Boundary Value ◽  
Inexact Newton Method ◽  
Point Boundary

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.

scholarly journals Existence of Multiple Positive Solutions for Even Order Multi-Point Boundary Value Problems

2007 ◽  
Vol 14 (4) ◽  
pp. 775-792
Author(s):  
Youyu Wang ◽  
Weigao Ge
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Multiple Positive Solutions ◽  
Point Boundary ◽  
Even Order

Abstract In this paper, we consider the existence of multiple positive solutions for the 2𝑛th order 𝑚-point boundary value problem: where (0,1), 0 < ξ 1 < ξ 2 < ⋯ < ξ 𝑚–2 < 1. Using the Leggett–Williams fixed point theorem, we provide sufficient conditions for the existence of at least three positive solutions to the above boundary value problem. The associated Green's function for the above problem is also given.

Boundary-Value Problems for Third-Order Lipschitz Ordinary Differential Equations

2014 ◽  
Vol 58 (1) ◽  
pp. 183-197 ◽  
Author(s):  
John R. Graef ◽  
Johnny Henderson ◽  
Rodrica Luca ◽  
Yu Tian
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Point Boundary ◽  
Third Order ◽  
Optimal Length ◽  
The Third

AbstractFor the third-order differential equationy′″ = ƒ(t, y, y′, y″), where, questions involving ‘uniqueness implies uniqueness’, ‘uniqueness implies existence’ and ‘optimal length subintervals of (a, b) on which solutions are unique’ are studied for a class of two-point boundary-value problems.

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