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.

Application of New Homotopy Perturbation Method for Solving Partial Differential Equations

2018 ◽  
Vol 15 (2) ◽  
pp. 500-508 ◽  
Author(s):  
Musa R. Gad-Allah ◽  
Tarig M. Elzaki
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Initial Conditions ◽  
Nonlinear Differential Equations ◽  
Novel Technique ◽  
Homotopy Perturbation ◽  
Linear And Nonlinear ◽  
Linear Differential

In this paper, a novel technique, that is to read, the New Homotopy Perturbation Method (NHPM) is utilized for solving a linear and non-linear differential equations and integral equations. The two most important steps in the application of the new homotopy perturbation method are to invent a suitable homotopy equation and to choose a suitable initial conditions. Comparing between the effects of the method (NHPM), is given exact solution, and the method (HPM), is given approximate solution, in this paper, we make some instances are provided to prove the ability of the method (NHPM). Show that the method (NHPM) is valid and effective, easy and accurate in solving linear and nonlinear differential equations, compared with the Homotopy Perturbation Method (HPM).

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.

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.

scholarly journals On a class of non-linear differential equations with exact solution

2012 ◽  
Vol 34 (1) ◽  
pp. 7-17
Author(s):  
Dao Huy Bich ◽  
Nguyen Dang Bich
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Exact Solutions ◽  
Solid Mechanics ◽  
Linear Equations ◽  
Linear Differential Equations ◽  
Second Order Differential Equations ◽  
Non Linear ◽  
Linear Differential ◽  
Non Linear Equations

The present paper deals with a class of non-linear ordinary second-order differential equations with exact solutions. A procedure for finding the general exact solution based on a known particular one is derived. For illustration solutions of some non-linear equations occurred in many problems of solid mechanics are considered.

Entire solutions of certain type of non-linear differential equations

2020 ◽  
Vol 70 (1) ◽  
pp. 87-94
Author(s):  
Bo Xue
Keyword(s):  
Differential Equations ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
Nonlinear Differential Equations ◽  
Differential Polynomial ◽  
Value Distribution ◽  
Linear Differential Equations ◽  
Entire Solutions ◽  
Value Distribution Theory ◽  
Linear Differential

AbstractUtilizing Nevanlinna’s value distribution theory of meromorphic functions, we study transcendental entire solutions of the following type nonlinear differential equations in the complex plane$$\begin{array}{} \displaystyle f^{n}(z)+P(z,f,f',\ldots,f^{(t)})=P_{1}\text{e}^{\alpha_{1}z}+P_{2}\text{e}^{\alpha_{2}z}+P_{3}\text{e}^{\alpha_{3}z}, \end{array}$$where Pj and αi are nonzero constants for j = 1, 2, 3, such that |α1| > |α2| > |α3| and P(z, f, f′, …, f(t) is an algebraic differential polynomial in f(z) of degree no greater than n – 1.

Sign in / Sign up

Export Citation Format

Share Document