Deceleration of a porous rotating disk in a viscous fluid
TL;DR: In this paper, the Navier-Stokes equations are transformed to non-linear ordinary differential equations using similarity transformations, and the resulting equations are solved numerically using a globally convergent homotopy method.
About: This article is published in International Journal of Engineering Science.The article was published on 1985-01-01 and is currently open access. It has received 16 citations till now. The article focuses on the topics: Flow (mathematics) & Viscous liquid.
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TL;DR: Probability one homotopy algorithms as mentioned in this paper are a class of methods for solving nonlinear systems of equations that are globally convergent with probability one, and if constructed and implemented properly, are robust, numerically stable, accurate, and practical.
Abstract: Probability one homotopy algorithms are a class of methods for solving nonlinear systems of equations that are globally convergent with probability one These methods are theoretically powerful, and if constructed and implemented properly, are robust, numerically stable, accurate, and practical The concomitant numerical linear algebra problems deal with rectangular matrices, and good algorithms require a delicate balance (not always achieved) of accuracy, robustness, and efficiency in both space and time The author's experience with globally convergent homotopy algorithms is surveyed here, and some of the linear algebra difficulties for dense and sparse problems are discussed
159Â citations
TL;DR: Homotopy algorithms for solving nonlinear systems of (algebraic) equations, which are convergent for almost all choices of starting point, are globally convergent with probability one and exhibit a large amount of coarse grain parallelism.
123Â citations
TL;DR: The theory of globally convergent homotopy algorithms was introduced in this article, which is a class of algorithms for solving nonlinear systems of equations that are accurate, robust, and converge from an arbitrary starting point almost surely.
Abstract: Probability-one homotopy methods are a class of algorithms for solving nonlinear systems of equations that are accurate, robust, and converge from an arbitrary starting point almost surely. These new globally convergent homotopy techniques have been successfully applied to solve Brouwer fixed point problems, polynomial systems of equations, constrained and unconstrained optimization problems, discretizations of nonlinear two-point boundary value problems based on shooting, finite differences, collocation, and finite elements, and finite difference, collocation, and Galerkin approximations to nonlinear partial differential equations. This paper introduces, in a tutorial fashion, the theory of globally convergent homotopy algorithms, deseribes some computer algorithms and mathematical software, and presents several nontrivial engineering applications.
82Â citations
TL;DR: The basic theory for probability one globally convergent homotopy algorithm was developed in 1976, and since then the theory, algorithms, and applications have considerably expanded as discussed by the authors, which is applicable to Brouwer fixed point problems, certain classes of zero-finding problems, unconstrained optimization, linearly constrained optimization, nonlinear complementarity, and the dlscretizations of nonlinear two-point boundary value problems based on shooting, finite differences, collocation and finite elements.
77Â citations
TL;DR: In this article, the laminar unsteady flow over a stretchable rotating disk with deceleration is investigated, and the three dimensional Navier-Stokes (NS) equations are reduced into a similarity ordinary differential equation group, which is solved numerically using a shooting method.
49Â citations
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16Â citations