Generalized Collocation Methods
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| Author | : Nicola Bellomo |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 206 |
| Release | : 2007-10-16 |
| Genre | : Mathematics |
| ISBN | : 081764525X |
Analysis of nonlinear models and problems is crucial in the application of mathematics to real-world problems. This book approaches this important topic by focusing on collocation methods for solving nonlinear evolution equations and applying them to a variety of mathematical problems. These include wave motion models, hydrodynamic models of vehicular traffic flow, convection-diffusion models, reaction-diffusion models, and population dynamics models. The book may be used as a textbook for graduate courses on collocation methods, nonlinear modeling, and nonlinear differential equations. Examples and exercises are included in every chapter.
| Author | : Nicola Bellomo |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 206 |
| Release | : 2007-09-26 |
| Genre | : Mathematics |
| ISBN | : 0817646108 |
Analysis of nonlinear models and problems is crucial in the application of mathematics to real-world problems. This book approaches this important topic by focusing on collocation methods for solving nonlinear evolution equations and applying them to a variety of mathematical problems. These include wave motion models, hydrodynamic models of vehicular traffic flow, convection-diffusion models, reaction-diffusion models, and population dynamics models. The book may be used as a textbook for graduate courses on collocation methods, nonlinear modeling, and nonlinear differential equations. Examples and exercises are included in every chapter.
| Author | : Paul Shuleshko |
| Publisher | : |
| Total Pages | : 76 |
| Release | : 1960 |
| Genre | : Differential equations |
| ISBN | : |
| Author | : Z-C. Li |
| Publisher | : WIT Press |
| Total Pages | : 433 |
| Release | : 2008-02-22 |
| Genre | : Technology & Engineering |
| ISBN | : 1845641531 |
This title was reviewed in the January 2009 issue of Mathematical Reviews.
| Author | : J.jr. Douglas |
| Publisher | : Springer |
| Total Pages | : 152 |
| Release | : 2006-08-01 |
| Genre | : Mathematics |
| ISBN | : 3540383379 |
| Author | : H. T. Huynh |
| Publisher | : BiblioGov |
| Total Pages | : 42 |
| Release | : 2013-06 |
| Genre | : |
| ISBN | : 9781289031084 |
We study the numerical solutions of ordinary differential equations by one-step methods where the solution at tn is known and that at t(sub n+1) is to be calculated. The approaches employed are collocation, continuous Galerkin (CG) and discontinuous Galerkin (DG). Relations among these three approaches are established. A quadrature formula using s evaluation points is employed for the Galerkin formulations. We show that with such a quadrature, the CG method is identical to the collocation method using quadrature points as collocation points. Furthermore, if the quadrature formula is the right Radau one (including t(sub n+1)), then the DG and CG methods also become identical, and they reduce to the Radau IIA collocation method. In addition, we present a generalization of DG that yields a method identical to CG and collocation with arbitrary collocation points. Thus, the collocation, CG, and generalized DG methods are equivalent, and the latter two methods can be formulated using the differential instead of integral equation. Finally, all schemes discussed can be cast as s-stage implicit Runge-Kutta methods.
| Author | : A. Solomonoff |
| Publisher | : |
| Total Pages | : 72 |
| Release | : 1986 |
| Genre | : |
| ISBN | : |
| Author | : National Aeronautics and Space Adm Nasa |
| Publisher | : Independently Published |
| Total Pages | : 40 |
| Release | : 2019-01-13 |
| Genre | : Science |
| ISBN | : 9781793962157 |
We study the numerical solutions of ordinary differential equations by one-step methods where the solution at tn is known and that at t(sub n+1) is to be calculated. The approaches employed are collocation, continuous Galerkin (CG) and discontinuous Galerkin (DG). Relations among these three approaches are established. A quadrature formula using s evaluation points is employed for the Galerkin formulations. We show that with such a quadrature, the CG method is identical to the collocation method using quadrature points as collocation points. Furthermore, if the quadrature formula is the right Radau one (including t(sub n+1)), then the DG and CG methods also become identical, and they reduce to the Radau IIA collocation method. In addition, we present a generalization of DG that yields a method identical to CG and collocation with arbitrary collocation points. Thus, the collocation, CG, and generalized DG methods are equivalent, and the latter two methods can be formulated using the differential instead of integral equation. Finally, all schemes discussed can be cast as s-stage implicit Runge-Kutta methods. Huynh, H. T. Glenn Research Center NASA/TM-2011-216340, E-17277
| Author | : National Aeronautics and Space Adm Nasa |
| Publisher | : Independently Published |
| Total Pages | : 40 |
| Release | : 2018-10-17 |
| Genre | : Science |
| ISBN | : 9781728890067 |
Collocation schemes are presented for solving linear fourth order differential equations in one and two dimensions. The variational formulation of the model fourth order problem is discretized by approximating the integrals by a Gaussian quadrature rule generalized to include the values of the derivative of the integrand at the boundary points. Collocation schemes are derived which are equivalent to this discrete variational problem. An efficient preconditioner based on a low-order finite difference approximation to the same differential operator is presented. The corresponding multidomain problem is also considered and interface conditions are derived. Pseudospectral approximations which are C1 continuous at the interfaces are used in each subdomain to approximate the solution. The approximations are also shown to be C3 continuous at the interfaces asymptotically. A complete analysis of the collocation scheme for the multidomain problem is provided. The extension of the method to the biharmonic equation in two dimensions is discussed and results are presented for a problem defined in a nonrectangular domain. Malek, Alaeddin and Phillips, Timothy N. Unspecified Center NAS1-19480; RTOP 505-90-52-01
| Author | : Wen Chen |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 98 |
| Release | : 2013-11-09 |
| Genre | : Technology & Engineering |
| ISBN | : 3642395724 |
This book surveys the latest advances in radial basis function (RBF) meshless collocation methods which emphasis on recent novel kernel RBFs and new numerical schemes for solving partial differential equations. The RBF collocation methods are inherently free of integration and mesh, and avoid tedious mesh generation involved in standard finite element and boundary element methods. This book focuses primarily on the numerical algorithms, engineering applications, and highlights a large class of novel boundary-type RBF meshless collocation methods. These methods have shown a clear edge over the traditional numerical techniques especially for problems involving infinite domain, moving boundary, thin-walled structures, and inverse problems. Due to the rapid development in RBF meshless collocation methods, there is a need to summarize all these new materials so that they are available to scientists, engineers, and graduate students who are interest to apply these newly developed methods for solving real world’s problems. This book is intended to meet this need. Prof. Wen Chen and Dr. Zhuo-Jia Fu work at Hohai University. Prof. C.S. Chen works at the University of Southern Mississippi.
