A Novel Method for Nonlinear Boundary Value Problems Based on Multiscale Orthogonal Base

2021 ◽  
pp. 2150036
Author(s):  
Yingchao Zhang ◽  
Liangcai Mei ◽  
Yingzhen Lin
Keyword(s):  
Differential Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
Nonlinear Boundary ◽  
Nonlinear Differential Equations ◽  
Nonlinear Boundary Value Problems ◽  
Numerical Examples ◽  
Second Order Differential Equations ◽  
Orthogonal Base ◽  
Novel Method

In this paper, a new algorithm is presented to solve the nonlinear second-order differential equations. The approach combines the Quasi-Newton’s method and the multiscale orthogonal bases. It is worth mentioning that the convergence of Newton’s method is verified for solving the nonlinear differential equations. The uniform convergence of the numerical solution as well as its derivatives are also proved. Numerical examples are given to demonstrate the efficiency and feasibility of the proposed method.

Marine Pipeline Analysis Based on Newton’s Method With an Arctic Application

1970 ◽  
Vol 92 (4) ◽  
pp. 827-833 ◽  
Author(s):  
D. W. Dareing ◽  
R. F. Neathery
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Newton’S Method ◽  
Finite Differences ◽  
Newton's Method ◽  
Nonlinear Differential Equations ◽  
Bending Moments ◽  
Iterative Solutions ◽  
Linear Differential ◽  
Pipe Deflection

Newton’s method is used to solve the nonlinear differential equations of bending for marine pipelines suspended between a lay-barge and the ocean floor. Newton’s method leads to linear differential equations, which are expressed in terms of finite differences and solved numerically. The success of Newton’s method depends on initial trial solutions, which in this paper are catenaries. Iterative solutions converge rapidly toward the exact solution (pipe deflection) even though large bending moments exist in the pipe. Example calculations are given for a 48-in. pipeline suspended in 300 ft of water.

A New Method for Computing Axisymmetric Buckling of Spherical Caps

1971 ◽  
Vol 38 (1) ◽  
pp. 179-184 ◽  
Author(s):  
G. A. Thurston
Keyword(s):  
Quadratic Form ◽  
Differential Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
Nonlinear Differential Equations ◽  
Uniform Pressure ◽  
Axisymmetric Buckling ◽  
Spherical Caps ◽  
Limit Points ◽  
Check Experiment

A modification of Newton’s method is applied to the solution of the nonlinear differential equations for clamped, shallow spherical caps under uniform pressure. The linear form of Newton’s method or quasi-linearization breaks down at limit points of the differential equations. A simplified “quadratic form” is derived in the paper and shown to be satisfactory for continuing the solution past the limit point and into the postbuckling region. Results for the buckling pressures defined by the limit points agree with published results for perfect caps. New results are presented for imperfect caps that check experiment.

scholarly journals Solving nonlinear boundary value problems by the Galerkin method with sinc functions

Open Physics ◽  
2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Sertan Alkan ◽  
Aydin Secer
Keyword(s):  
Differential Equations ◽  
Galerkin Method ◽  
Nonlinear Boundary ◽  
Nonlinear Differential Equations ◽  
Nonlinear Problems ◽  
Approximate Solutions ◽  
Nonlinear Boundary Value Problems ◽  
Sinc Approximation ◽  
Sinc Functions ◽  
The Galerkin Method

AbstractIn this paper, the sinc-Galerkin method is used for numerically solving a class of nonlinear differential equations with boundary conditions. The importance of this study is that sinc approximation of the nonlinear term is stated as a new theorem. The method introduced here is tested on some nonlinear problems and is shown to be a very efficient and powerful tool for obtaining approximate solutions of nonlinear ordinary differential equations.

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.

Haar wavelet based numerical solution of nonlinear differential equations arising in fluid dynamics

2016 ◽  
Vol 05 (02) ◽  
pp. 1650010 ◽  
Author(s):  
S. C. Shiralashetti ◽  
M. H. Kantli ◽  
A. B. Deshi
Keyword(s):  
Fluid Dynamics ◽  
Differential Equations ◽  
Error Analysis ◽  
Numerical Solution ◽  
Boundary Value Problems ◽  
Nonlinear Boundary ◽  
Nonlinear Differential Equations ◽  
Elastohydrodynamic Lubrication ◽  
Haar Wavelet ◽  
Nonlinear Boundary Value Problems

In this paper, we obtained the Haar wavelet-based numerical solution of the nonlinear differential equations arising in fluid dynamics, i.e., electrohydrodynamic flow, elastohydrodynamic lubrication and nonlinear boundary value problems. Error analysis is observed, it shows that the Haar wavelet-based results give better accuracy than the existing methods, which is justified through illustrative examples.

Continuation of Newton’s Method Through Bifurcation Points

1969 ◽  
Vol 36 (3) ◽  
pp. 425-430 ◽  
Author(s):  
G. A. Thurston
Keyword(s):  
Differential Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
Nonlinear Differential Equations ◽  
Elastic Stability ◽  
Variational Equations ◽  
Bifurcation Points ◽  
Limit Points

A modification of Newton’s method is suggested that provides a practical means of continuing solutions of nonlinear differential equations through limit points or bifurcation points. The method is applicable when the linear “variational” equations for the problem are self-adjoint. The procedure is illustrated by examples from the field of elastic stability.

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