[PDF] Matrices Moments And Quadrature Applications To Time Dependent Partial Differential Equations eBook

Author : James V. Lambers
Publisher :
Page : pages
File Size : 32,58 MB
Release : 2016
Category : Mathematics
ISBN :

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The numerical solution of a time-dependent PDE generally involves the solution of a stiff system of ODEs arising from spatial discretization of the PDE. There are many methods in the literature for solving such systems, such as exponential propagation iterative (EPI) methods, that rely on Krylov projection to compute matrix function-vector products. Unfortunately, as spatial resolution increases, these products require an increasing number of Krylov projection steps, thus drastically increasing computational expense.

Author : Gene H. Golub
Publisher : Princeton University Press
Page : 376 pages
File Size : 44,30 MB
Release : 2009-12-07
Category : Mathematics
ISBN : 1400833884

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This computationally oriented book describes and explains the mathematical relationships among matrices, moments, orthogonal polynomials, quadrature rules, and the Lanczos and conjugate gradient algorithms. The book bridges different mathematical areas to obtain algorithms to estimate bilinear forms involving two vectors and a function of the matrix. The first part of the book provides the necessary mathematical background and explains the theory. The second part describes the applications and gives numerical examples of the algorithms and techniques developed in the first part. Applications addressed in the book include computing elements of functions of matrices; obtaining estimates of the error norm in iterative methods for solving linear systems and computing parameters in least squares and total least squares; and solving ill-posed problems using Tikhonov regularization. This book will interest researchers in numerical linear algebra and matrix computations, as well as scientists and engineers working on problems involving computation of bilinear forms.

Author : Vasilios Katsikis
Publisher : BoD – Books on Demand
Page : 175 pages
File Size : 50,87 MB
Release : 2016-07-06
Category : Mathematics
ISBN : 9535124196

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The present text book contains a collection of six high-quality articles. In particular, this book is devoted to Linear Mathematics by presenting problems in Applied Linear Algebra of general or special interest.

Author : Ulrich Langer
Publisher : Walter de Gruyter GmbH & Co KG
Page : 261 pages
File Size : 50,62 MB
Release : 2019-09-23
Category : Mathematics
ISBN : 3110548488

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This volume provides an introduction to modern space-time discretization methods such as finite and boundary elements and isogeometric analysis for time-dependent initial-boundary value problems of parabolic and hyperbolic type. Particular focus is given on stable formulations, error estimates, adaptivity in space and time, efficient solution algorithms, parallelization of the solution pipeline, and applications in science and engineering.

Author : Marco L. Bittencourt
Publisher : Springer
Page : 681 pages
File Size : 49,81 MB
Release : 2017-11-07
Category : Mathematics
ISBN : 3319658700

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This book features a selection of high-quality papers chosen from the best presentations at the International Conference on Spectral and High-Order Methods (2016), offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions.

Author : Chang Shu
Publisher : Springer Science & Business Media
Page : 356 pages
File Size : 48,34 MB
Release : 2012-12-06
Category : Technology & Engineering
ISBN : 1447104072

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In the past few years, the differential quadrature method has been applied extensively in engineering. This book, aimed primarily at practising engineers, scientists and graduate students, gives a systematic description of the mathematical fundamentals of differential quadrature and its detailed implementation in solving Helmholtz problems and problems of flow, structure and vibration. Differential quadrature provides a global approach to numerical discretization, which approximates the derivatives by a linear weighted sum of all the functional values in the whole domain. Following the analysis of function approximation and the analysis of a linear vector space, it is shown in the book that the weighting coefficients of the polynomial-based, Fourier expansion-based, and exponential-based differential quadrature methods can be computed explicitly. It is also demonstrated that the polynomial-based differential quadrature method is equivalent to the highest-order finite difference scheme. Furthermore, the relationship between differential quadrature and conventional spectral collocation is analysed. The book contains material on: - Linear Vector Space Analysis and the Approximation of a Function; - Polynomial-, Fourier Expansion- and Exponential-based Differential Quadrature; - Differential Quadrature Weighting Coefficient Matrices; - Solution of Differential Quadrature-resultant Equations; - The Solution of Incompressible Navier-Stokes and Helmholtz Equations; - Structural and Vibrational Analysis Applications; - Generalized Integral Quadrature and its Application in the Solution of Boundary Layer Equations. Three FORTRAN programs for simulation of driven cavity flow, vibration analysis of plate and Helmholtz eigenvalue problems respectively, are appended. These sample programs should give the reader a better understanding of differential quadrature and can easily be modified to solve the readers own engineering problems.

Author : James V Lambers
Publisher : World Scientific
Page : 675 pages
File Size : 29,86 MB
Release : 2018-09-17
Category : Mathematics
ISBN : 9813220031

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This textbook introduces advanced undergraduate and early-career graduate students to the field of numerical analysis. This field pertains to the design, analysis, and implementation of algorithms for the approximate solution of mathematical problems that arise in applications spanning science and engineering, and are not practical to solve using analytical techniques such as those taught in courses in calculus, linear algebra or differential equations. Topics covered include error analysis, computer arithmetic, solution of systems of linear equations, least squares problems, eigenvalue problems, polynomial interpolation and approximation, numerical differentiation and integration, nonlinear equations, optimization, ordinary differential equations, and partial differential equations. For each problem considered, the presentation includes the derivation of solution techniques, analysis of their efficiency, accuracy and robustness, and details of their implementation, illustrated through the MATLAB programming language. This text is suitable for a year-long sequence in numerical analysis, and can also be used for a one-semester course in numerical linear algebra.

Author : Narayan Kovvali
Publisher : Morgan & Claypool Publishers
Page : 67 pages
File Size : 10,79 MB
Release : 2011-10-10
Category : Technology & Engineering
ISBN : 1608457540

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Gaussian quadrature is a powerful technique for numerical integration that falls under the broad category of spectral methods. The purpose of this work is to provide an introduction to the theory and practice of Gaussian quadrature. We study the approximation theory of trigonometric and orthogonal polynomials and related functions and examine the analytical framework of Gaussian quadrature. We discuss Gaussian quadrature for bandlimited functions, a topic inspired by some recent developments in the analysis of prolate spheroidal wave functions. Algorithms for the computation of the quadrature nodes and weights are described. Several applications of Gaussian quadrature are given, ranging from the evaluation of special functions to pseudospectral methods for solving differential equations. Software realization of select algorithms is provided. Table of Contents: Introduction / Approximating with Polynomials and Related Functions / Gaussian Quadrature / Applications / Links to Mathematical Software

Author : Oleg P. Iliev
Publisher : Springer
Page : 327 pages
File Size : 10,6 MB
Release : 2013-06-04
Category : Mathematics
ISBN : 9781461471738

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One of the current main challenges in the area of scientific computing​ is the design and implementation of accurate numerical models for complex physical systems which are described by time dependent coupled systems of nonlinear PDEs. This volume integrates the works of experts in computational mathematics and its applications, with a focus on modern algorithms which are at the heart of accurate modeling: adaptive finite element methods, conservative finite difference methods and finite volume methods, and multilevel solution techniques. Fundamental theoretical results are revisited in survey articles and new techniques in numerical analysis are introduced. Applications showcasing the efficiency, reliability and robustness of the algorithms in porous media, structural mechanics and electromagnetism are presented. Researchers and graduate students in numerical analysis and numerical solutions of PDEs and their scientific computing applications will find this book useful.