A general mesh independence principle for Newton's method applied to second order boundary value problems

Computing ◽  
1979 ◽  
Vol 23 (3) ◽  
pp. 233-246 ◽  
Author(s):  
E. L. Allgower ◽  
F. St. McCormick ◽  
D. V. Pryor
Keyword(s):  
Boundary Value Problems ◽  
Newton’S Method ◽  
Newton's Method ◽  
Boundary Value ◽  
Second Order ◽  
Independence Principle ◽  
Mesh Independence ◽  
Mesh Independence Principle

scholarly journals APPLICATION OF PROJECTION ALGORITHMS TO DIFFERENTIAL EQUATIONS: BOUNDARY VALUE PROBLEMS

2019 ◽  
Vol 61 (1) ◽  
pp. 23-46 ◽  
Author(s):  
BISHNU P. LAMICHHANE ◽  
SCOTT B. LINDSTROM ◽  
BRAILEY SIMS
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Newton’S Method ◽  
Newton's Method ◽  
Boundary Value ◽  
Second Order ◽  
Feasibility Problem ◽  
Projection Algorithms ◽  
The Stability

The Douglas–Rachford method has been employed successfully to solve many kinds of nonconvex feasibility problems. In particular, recent research has shown surprising stability for the method when it is applied to finding the intersections of hypersurfaces. Motivated by these discoveries, we reformulate a second order boundary value problem (BVP) as a feasibility problem where the sets are hypersurfaces. We show that such a problem may always be reformulated as a feasibility problem on no more than three sets and is well suited to parallelization. We explore the stability of the method by applying it to several BVPs, including cases where the traditional Newton’s method fails.

Newton’s Method Applied to Problems in Nonlinear Mechanics

1965 ◽  
Vol 32 (2) ◽  
pp. 383-388 ◽  
Author(s):  
G. A. Thurston
Keyword(s):  
Boundary Value Problems ◽  
Newton’S Method ◽  
Newton's Method ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Practical Method ◽  
Algebraic Equations ◽  
Nonlinear Mechanics ◽  
Nonlinear Boundary Value Problems ◽  
Nonlinear Boundary Value

Many problems in mechanics are formulated as nonlinear boundary-value problems. A practical method of solving such problems is to extend Newton’s method for calculating roots of algebraic equations. Three problems are treated in this paper to illustrate the use of this method and compare it with other methods.

scholarly journals A New Approach for Solving Nonlinear Singular Boundary Value Problems

2018 ◽  
Vol 23 (1) ◽  
pp. 33-43
Author(s):  
Hui Zhu ◽  
Jing Niu ◽  
Ruimin Zhang ◽  
Yingzhen Lin
Keyword(s):  
Boundary Value Problems ◽  
Newton’S Method ◽  
Newton's Method ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Boundary Value ◽  
Singular Boundary ◽  
Singular Boundary Value Problems ◽  
Reproducing Kernel Method ◽  
New Approach

In this paper, an efficient method based on Quasi-Newton's method and the simpliffied reproducing kernel method is proposed for solving nonlinear singular boundary value problems. For the Quasi-Newton's method the convergence order is studied. The uniform convergence of the numerical solution as well as its derivatives are also proved. Numerical examples are given to demonstrate the efficiency and stability of the proposed method. The numerical results are compared with exact solutions and the outcomes of other existing numerical methods.

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