The asymptotic expansions of singularly perturbed boundary value problems

1989 ◽  
Vol 10 (6) ◽  
pp. 577-581 ◽  
Author(s):  
Zhou Qin-de ◽  
Li Yong
Keyword(s):  
Boundary Value Problems ◽  
Asymptotic Expansions ◽  
Boundary Value ◽  
Singularly Perturbed

scholarly journals MULTIPLE SOLUTIONS OF A GENERALIZED SINGULAR PERTURBED BRATU PROBLEM

2012 ◽  
Vol 22 (04) ◽  
pp. 1250095 ◽  
Author(s):  
TAOUFIK BAKRI ◽  
YURI A. KUZNETSOV ◽  
FERDINAND VERHULST ◽  
EUSEBIUS DOEDEL
Keyword(s):  
Boundary Value Problems ◽  
Asymptotic Expansions ◽  
Multiple Solutions ◽  
Boundary Value ◽  
Slow Manifold ◽  
Singularly Perturbed ◽  
Point Boundary ◽  
Numerical Bifurcation ◽  
Manifold Theory ◽  
Bratu Problem

Nonlinear two-point boundary value problems (BVPs) may have none or more than one solution. For the singularly perturbed two-point BVP εu″ + 2u′ + f(u) = 0, 0 < x < 1, u(0) = 0, u(1) = 0, a condition is given to have one and only one solution; also cases of more solutions have been analyzed. After attention to the form and validity of the corresponding asymptotic expansions, partially based on slow manifold theory, we reconsider the BVP within the framework of small and large values of the parameter. In the case of a special nonlinearity, numerical bifurcation patterns are studied that improve our understanding of the multivaluedness of the solutions.

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