Overview

Multivariate integration has been a fundamental subject in mathematics, with broad connections to a number of areas: numerical analysis, approximation theory, partial differential equations, integral equations, harmonic analysis, etc. In this work the exposition focuses primarily on a powerful tool that has become especially important in our computerized age, namely, dimensionality reducing expansion (DRE). The method of DRE is a technique for changing a higher dimensional integration to a lower dimensional one with or without remainder. To date, there is no comprehensive treatment of this subject in monograph or textbook form. Key features of this self-contained monograph include: * fine exposition covering the history of the subject * up-to-date new results, related to many fields of current research such as boundary element methods for solving PDEs and wavelet analysis * presentation of DRE techniques using a broad array of examples * good balance between theory and application * coverage of such related topics as boundary type quadratures and asymptotic expansions of oscillatory integrals * excellent and comprehensive bibliography and index This work will appeal to a broad audience of students and researchers in pure and applied mathematics, statistics, and physics, and can be used in a graduate/advanced undergraduate course or as a standard reference text.

Full Product Details

Author:   Tian-Xiao He
Publisher:   Springer-Verlag New York Inc.
Imprint:   Springer-Verlag New York Inc.
Edition:   Softcover reprint of the original 1st ed. 2001
Dimensions:   Width: 15.50cm , Height: 1.20cm , Length: 23.50cm
Weight:   0.372kg
ISBN:  

9781461274148

ISBN 10:   1461274141
Pages:   227
Publication Date:   30 September 2011
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Paperback
Publisher's Status:   Active
Availability:   Manufactured on demand  
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Table of Contents

1 Dimensionality Reducing Expansion of Multivariate Integration.- 1.1 Darboux formulas and their special forms.- 1.2 Generalized integration by parts rule.- 1.3 DREs with algebraic precision.- 1.4 Minimum estimation of remainders in DREs with algebraic precision.- 2 Boundary Type Quadrature Formulas with Algebraic Precision.- 2.1 Construction of BTQFs using DREs.- 2.2 BTQFs with homogeneous precision.- 2.3 Numerical integration associated with wavelet functions.- 2.4 Some applications of DREs and BTQFs.- 2.5 BTQFs over axially symmetric regions.- 3 The Integration and DREs of Rapidly Oscillating Functions.- 3.1 DREs for approximating a double integral.- 3.2 Basic lemma.- 3.3 DREs with large parameters.- 3.4 Basic expansion theorem for integrals with large parameters.- 3.5 Asymptotic expansion formulas for oscillatory integrals with singular factors.- 4 Numerical Quadrature Formulas Associated with the Integration of Rapidly Oscillating Functions.- 4.1 Numerical quadrature formulas of double integrals.- 4.2 Numerical integration of oscillatory integrals.- 4.3 Numerical quadrature of strongly oscillatory integrals with compound precision.- 4.4 Fast numerical computations of oscillatory integrals.- 4.5 DRE construction and numerical integration using measure theory.- 4.6 Error analysis of numerical integration.- 5 DREs Over Complex Domains.- 5.1 DREs of the double integrals of analytic functions.- 5.2 Construction of quadrature formulas using DREs.- 5.3 Integral regions suitable for DREs.- 5.4 Additional topics.- 6 Exact DREs Associated With Differential Equations.- 6.1 DREs and ordinary differential equations.- 6.2 DREs and partial differential equations.- 6.3 Applications of DREs in the construction of BTQFs.- 6.4 Applications of DREs in the boundary element method.

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