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Showing papers on "Homotopy analysis method published in 1980"


Journal ArticleDOI
TL;DR: The Chow-Yorke algorithm is a homotopy method that has been proved globally convergent for Brouwer fixed point problems, classes of zero finding, nonlinear programming, and two-point boundary value problems as discussed by the authors.
Abstract: The Chow–Yorke algorithm is a homotopy method that has been proved globally convergent for Brouwer fixed point problems, classes of zero finding, nonlinear programming, and two-point boundary value problems. The method is numerically stable, and has been successfully applied to several practical nonlinear optimization and fluid dynamics problems. Previous application of the homotopy method to two-point boundary value problems has been based on shooting, which is inappropriate for fluid dynamics problems with sharp boundary layers. Here the Chow–Yorke algorithm is proved globally convergent for a class of finite difference approximations to nonlinear two-point boundary value problems. The numerical implementation of the algorithm is briefly sketched, and computational results are given for two fairly difficult fluid dynamics boundary value problems.

35 citations


Journal ArticleDOI
Layne T. Watson1
TL;DR: The Chow—Yorke algorithm is a nonsimplicial homotopy type method for computing Brouwer fixed points that is globally convergent and effective for fixed point problems.
Abstract: The Chow—Yorke algorithm is a nonsimplicial homotopy type method for computing Brouwer fixed points that is globally convergent. It is efficient and accurate for fixed point problems. L.T. Watson, T.Y. Li, and C.Y. Wang have adapted the method for zero finding problems, the nonlinear complementarity problem, and nonlinear two-point boundary value problems. Here theoretical justification is given for applying the method to some mathematical programming problems, and computational results are presented.

24 citations


Journal ArticleDOI
TL;DR: A triangulation is introduced for homotopy methods to compute fixed points on the unit simplex or inRn, which allows for factors of incrementation of more than two and can be accelerated without using restart methods.
Abstract: In this paper a triangulation is introduced for homotopy methods to compute fixed points on the unit simplex or inR n . This triangulation allows for factors of incrementation of more than two. The factor may be of any size and even different at each level. Also the starting point on a new level may be any gridpoint of the last found completely labelled subsimplex on the last level. So, the decision which new factor of incrementation and which starting point is used, can be made on the ground of previous approximations. Doing so, the convergence rate can be accelerated without using restart methods.

23 citations


Journal ArticleDOI
TL;DR: In this paper, a new homotopy method was developed for solving the nonlinear system which is globally convergent with probability one, and an algorithm was developed based on this method to produce an optimal design solution for a given starting point.
Abstract: An optimal design problem is formulated as a system of nonlinear equations rather than the extremum of a functional. Based on a new homotopy method, an algorithm is developed for solving the nonlinear system which is globally convergent with probability one. Since no convexity is required, the nonlinear system may have more than one solution. The algorithm will produce an optimal design solution for a given starting point. For most engineering problems, the initial prototype design is already well conceived and close to the global optimal solution. Such a starting point usually leads to the optimal design by the homotopy method, even though Newton's method may diverge from that starting point. A simple example is given.

22 citations


Journal ArticleDOI
TL;DR: In this paper, sufficient conditions are given that guarantee the homotopy process to be feasible for a class of two-point boundary value problems and the numerical solution of two practical problems arising in physiology is described.
Abstract: The homotopy method is a frequently used technique in overcoming the local convergence nature of multiple shooting. In this paper sufficient conditions are given that guarantee the homotopy process to be feasible. The results are applicable to a class of two-point boundary value problems. Finally, the numerical solution of two practical problems arising in physiology is described.

10 citations


Journal ArticleDOI
TL;DR: In a lecture given at the M.R.C. Symposiucn on Recent Advances in Numerical Analysis in May, 1978, H. B. Keller discussed a global homotopy method for finding solutions of f (u) = 0.
Abstract: In a lecture given at the M.R.C. Symposiucn on Recent Advances in Numerical Analysis in May, 1978, H. B. Keller discussed a global homotopy method for finding solutions of f (u) = 0. At that time he was unable to show that one of the hypotheses of his main result is satisfied by almost all initial points for paths determined by his method. The purpose of this note is to fill that gap in Keller's result.

6 citations


01 Jan 1980
TL;DR: In this paper, the homotopy analysis method was used to solve delay-integro-dierential equations and the convergence of the method was investigated in numerical experiments to illustrate the computational eectiveness of the algorithm.
Abstract: In this paper, we implement the homotopy analysis method to solve delay-integro-dierential equations The convergence of the method is investigated In the end, numerical experiments are presented to illustrate the computational eectiveness of the method

1 citations