How to Make Use of the Solution Set to Solve Boundary Value Problems

2000 ◽  
pp. 15-25
Author(s):  
Giuseppe Anichini ◽  
Giuseppe Conti
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Solution Set

Multidimensional Manifold Continuation for Adaptive Boundary-Value Problems

2020 ◽  
Vol 15 (5) ◽  
Author(s):  
Harry Dankowicz ◽  
Yuqing Wang ◽  
Frank Schilder ◽  
Michael E. Henderson
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Approximate Solutions ◽  
General Purpose ◽  
Solution Set ◽  
Benchmark Problems ◽  
Finite Dimensional ◽  
Predictor Corrector ◽  
Parameter Continuation ◽  
Autonomous Dynamical Systems

Abstract Parameter continuation of finitely parameterized, approximate solutions to integro-differential boundary-value problems typically involves regular adaptive updates to the number and meaning of the unknowns and/or the associated constraints. Different continuation steps produce solutions with different discretizations or to formally different sets of equations. Existing general-purpose, multidimensional continuation algorithms fail to account for such differences without significant additional coding and are therefore prone to redundant coverage of the set of solutions. We describe a new algorithm, implemented in the software package coco, which overcomes this problem by characterizing the solution set in an invariant, finite dimensional, projected geometry rather than in the space of unknowns corresponding to any particular discretization. It is in this geometry that distances between solutions and angles between tangent spaces are quantified and used to construct possible directions of outward expansion. A pointwise lift identifies such directions in the projected geometry with directions of continuation in the full set of unknowns, used by a nonlinear predictor-corrector algorithm to expand into uncharted parts of the solution set. Several benchmark problems from the analysis of periodic orbits in autonomous dynamical systems are used to illustrate the theory.

scholarly journals Global Bifurcation in2m-Order Generic Systems of Nonlinear Boundary Value Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-8
Author(s):  
Xiaoling Han ◽  
Jia Xu ◽  
Guowei Dai
Keyword(s):  
Boundary Value Problems ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Global Bifurcation ◽  
Solution Set ◽  
Nonlinear Boundary Value Problems ◽  
Countable Collection ◽  
Nonlinear Boundary Value

We consider the systems of(-1)mu(2m)=λu+λv+uf(t,u,v),  t∈(0,1),  u(2i)(0)=u(2i)(1)=0, and0≤i≤m-1,  (-1)mv(2m)=μu+μv+vg(t,u,v),  t∈(0,1),  v(2i)(0)=v(2i)(1)=0,  0≤i≤m-1, whereλ,μ∈Rare real parameters.f,g:[0,1]×R2→RareCk,k≥3functions andf(t,0,0)=g(t,0,0)=0,t∈[0,1]. It will be shown that if the functions,fandgare “generic” then the solution set of the systems consists of a countable collection of 2-dimensional,Ckmanifolds.

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