The Application of Newton’s Method to the Problem of Elastic Stability

1972 ◽  
Vol 39 (4) ◽  
pp. 1060-1065 ◽  
Author(s):  
E. Riks
Keyword(s):  
Numerical Solution ◽  
Newton’S Method ◽  
Iteration Method ◽  
Newton's Method ◽  
Elastic Stability ◽  
Auxiliary Equation

The numerical solution of problems of elastic stability through the use of the iteration method of Newton is examined. It is found that if the equations of equilibrium are completed by a simple auxiliary equation, problems governed by a snapping condition can, in principle, always be calculated as long as the problem at hand is properly formulated. The effectiveness of the proposed procedure is demonstrated by means of an elementary example.

JULIA SETS OF GENERALIZED NEWTON'S METHOD

Fractals ◽  
2007 ◽  
Vol 15 (04) ◽  
pp. 323-336 ◽  
Author(s):  
XINGYUAN WANG ◽  
TINGTING WANG
Keyword(s):  
Newton’S Method ◽  
Iteration Method ◽  
Newton's Method ◽  
Julia Sets ◽  
Basins Of Attraction ◽  
Principal Value ◽  
Extraneous Fixed Points ◽  
Sets Theory ◽  
Steffensen Method ◽  
Generalized Newton’S Method

The Julia sets theory of generalized Newton's method is analyzed and the Julia sets of generalized Newton's method are constructed using the iteration method. From the research we find that: (1) the basins of attraction of the Julia sets of generalized Newton's method depend on the roots of the equation and their orders and also the existence of the extraneous fixed points; (2) the Steffensen method is an exception to the law given in (1); and (3) if the order of the root is decimal, then the different choice of the range of the principal value of the phase angle will cause a different evolvement of the Julia sets.

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.

Algorithms for Solving the Catch Equation Forward and Backward in Time

1982 ◽  
Vol 39 (1) ◽  
pp. 197-202 ◽  
Author(s):  
S. E. Sims
Keyword(s):  
Mortality Rate ◽  
Numerical Solution ◽  
Exponential Function ◽  
Newton’S Method ◽  
Newton's Method ◽  
Padé Approximation ◽  
Approximate Solutions ◽  
Fishing Mortality ◽  
Convergence Criteria ◽  
Pade Approximation

Approximate solutions to the catch equation for the fishing mortality rate both forward and backward in time are obtained with an application of the diagonal Padé approximation of degree four to the exponential function. In either case the resulting approximation as well as Pope's estimate are shown to serve quite well as starting values for Newton's Method which is used to obtain a numerical solution of the catch equation. Convergence criteria for Newton's Method are discussed in each setting.Key words: catch equation, Newton's method, Padé approximation, Pope's estimate

scholarly journals Numerical solution of system of two Nonlinear Volterra Integral equations

2014 ◽  
Vol 12 (10) ◽  
pp. 3967-3975
Author(s):  
Dalal Adnan Maturi
Keyword(s):  
Numerical Solution ◽  
Integral Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
Trapezoidal Rule ◽  
Volterra Integral Equations ◽  
Implicit Trapezoidal Rule

In this paper, using the implicit trapezoidal rule in conjunction with Newton's method to solve nonlinear system.We have used a Maple 17 program to solve the System of two nonlinear Volterra integral equations. Finally, several illustrative examples are presented to show the effectiveness and accuracy of this method.

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