scholarly journals Modified HPMs Inspired by Homotopy Continuation Methods

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Héctor Vázquez-Leal ◽  
Uriel Filobello-Niño ◽  
Roberto Castañeda-Sheissa ◽  
Luis Hernández-Martínez ◽  
Arturo Sarmiento-Reyes
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Research Area ◽  
Ease Of Use ◽  
Continuation Methods ◽  
Homotopy Continuation ◽  
Homotopy Methods ◽  
Mechanism Model ◽  
Starting Point ◽  
Homotopy Continuation Methods

Nonlinear differential equations have applications in the modelling area for a broad variety of phenomena and physical processes; having applications for all areas in science and engineering. At the present time, the homotopy perturbation method (HPM) is amply used to solve in an approximate or exact manner such nonlinear differential equations. This method has found wide acceptance for its versatility and ease of use. The origin of the HPM is found in the coupling of homotopy methods with perturbation methods. Homotopy methods are a well established research area with applications, in particular, an applied branch of such methods are the homotopy continuation methods, which are employed on the numerical solution of nonlinear algebraic equation systems. Therefore, this paper presents two modified versions of standard HPM method inspired in homotopy continuation methods. Both modified HPMs deal with nonlinearities distribution of the nonlinear differential equation. Besides, we will use a calcium-induced calcium released mechanism model as study case to test the proposed techniques. Finally, results will be discussed and possible research lines will be proposed using this work as a starting point.

Patterned Bootstrap: A New Method Which Gives Efficiency for Precision Position Synthesis of Planar Linkages

2005 ◽  
Author(s):  
Zhenjun Luo ◽  
Jian S. Dai
Keyword(s):  
New Method ◽  
Continuation Methods ◽  
Interval Methods ◽  
Homotopy Continuation ◽  
Root Finding ◽  
Starting Point ◽  
Homotopy Continuation Methods ◽  
Upper Level ◽  
Planar Linkages ◽  
Position Synthesis

This paper presents a new method, termed as patterned bootstrap (PB), which is suitable for precision position synthesis of planar linkages. The method solves a determined system of equations using a new bootstrapping strategy. In principle, a randomly generated starting point is advanced to a final solution through solving a number of intermediate systems. The structure and the associated parameters of each intermediate system is defined as a pattern. In practice, a PB procedure generally consists of two levels: an upper level which controls the transition of patterns, and a lower level which solves intermediate systems using globally convergent root-finding algorithms. Besides introducing the new method, tunnelling functions have been added to several systems of polynomials derived by formal researchers in order to exclude degenerated solutions. Our numerical experiments demonstrate that many precision position synthesis problems can be solved efficiently without resorting to time-consuming polynomial homotopy continuation methods or interval methods. Finding over 95 percentages of the complete solutions of the 11 precision position function generation problem of a Stephenson-III linkage has been achieved for the first time.

Numerical solution of multivariate polynomial systems by homotopy continuation methods

Acta Numerica ◽  
1997 ◽  
Vol 6 ◽  
pp. 399-436 ◽  
Author(s):  
T. Y. Li
Keyword(s):  
Power Flow ◽  
Polynomial Systems ◽  
Continuation Methods ◽  
Homotopy Continuation ◽  
Multivariate Polynomial ◽  
Science And Engineering ◽  
Flow Problems ◽  
Homotopy Continuation Methods ◽  
Buchberger Algorithm ◽  
Image Position

Let P(x) = 0 be a system of n polynomial equations in n unknowns. Denoting P = (p1,…, pn), we want to find all isolated solutions offor x = (x1,…,xn). This problem is very common in many fields of science and engineering, such as formula construction, geometric intersection problems, inverse kinematics, power flow problems with PQ-specified bases, computation of equilibrium states, etc. Elimination theory-based methods, most notably the Buchberger algorithm (Buchberger 1985) for constructing Gröbner bases, are the classical approach to solving (1.1), but their reliance on symbolic manipulation makes those methods seem somewhat unsuitable for all but small problems.

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