偏微分方程與數(shù)值方法

內(nèi)容概要

《偏微分方程與數(shù)值方法》的作者Stig Larsson現(xiàn)任瑞典Chalmers大學(xué)數(shù)學(xué)系教授、瑞典科學(xué)院院士。本書(shū)將微分方程的數(shù)學(xué)分析及有限差分理論和有限元方法結(jié)合起來(lái)，講述線性偏微分方程的基本理論及其常用的數(shù)值解法。分別用三章闡述橢圓型、拋物型及雙曲型偏微分方程，一章關(guān)于其數(shù)學(xué)理論，一章關(guān)于其有限差分方法，一章關(guān)于其有限元方法。在論述橢圓型方程之前，講述常微分方程的兩點(diǎn)邊值問(wèn)題；類(lèi)似地，在論述拋物型和雙曲型發(fā)展問(wèn)題之前，講述常微分方程的初值問(wèn)題。另有一章研究橢圓型特征值問(wèn)題和特征函數(shù)的展開(kāi)。附錄提供了閱讀本書(shū)所要求的線性泛函分析及索伯列夫空間的背景知識(shí)。閱讀本書(shū)不需要高深的數(shù)學(xué)分析和泛函分析知識(shí)。本書(shū)適用于應(yīng)用數(shù)學(xué)專(zhuān)業(yè)和工程專(zhuān)業(yè)的高年級(jí)本科生和低年級(jí)研究生。

書(shū)籍目錄

1 Introduction  1.1 Background  1.2 Notation and Mathematical Preliminaries  1.3 Physical Derivation of the Heat Equation  1.4 Problems2 A Two-Point Boundary Value Problem  2.1 The Maximum Principle  2.2 Green's Function  2.3 Variational Formulation  2.4 Problems3 Elliptic Equations  3.1 Preliminaries  3.2 A Maximum Principle  3.3 Dirichlet's Problem for a Disc. Poisson's Integral  3.4 Fundamental Solutions. Green's Function  3.5 Variational Formulation of the Dirichlet Problem  3.6 A Neumann Problem  3.7 Regularity  3.8 Problems4 Finite Difference Methods for Elliptic Equations  4.1 A Two-Point Boundary Value Problem  4.2 Poisson's Equation  4.3 Problems5 Finite Element Methods for Elliptic Equations  5.1 A Two-Point boundary Value Problem  5.2 A Model Problem in the Plane  5.3 Some Facts from Approximation Theory  5.4 Error Estimates  5.5 An A Posteriori Error Estimate  5.6 Numerical Integration  5.7 A Mixed Finite Element Method  5.8 Problems6 The Elliptic Eigenvalue Problem  6.1 Eigenfunction Expansions  6.2 Numerical Solution of the Eigenvalue Problem  6.3 Problems7 Initial-Value Problems for ODEs  7.1 The Initial Value Problem for a Linear System  7.2 Numerical Solution of ODEs  7.3 Problems8 Parabolic Equations  8.1 The Pure Initial Value Problem  8.2 Solution by Eigenfunction Expansion  8.3 Variational Formulation. Energy Estimates  8.4 A Maximum Principle  8.5 Problems9 Finite Difference Methods for Parabolic Problems  9.1 The Pure Initial Value Problem  9.2 The Mixed Initial-Boundary Value Problem  9.3 Problems10 The Finite Element Method for a Parabolic Problem  10.1 The Semidiscrete Galerkin Finite Element Method  10.2 Some Completely Discrete Schemes  10.3 Problems11 Hyperbolic Equations  11.1 Characteristic Directions and Surfaces  11.2 The Wave Equation  11.3 First Order Scalar Equation  11.4 Symmetric Hyperbolic Systems  11.5 Problems12 Finite Difference Methods for Hyperbolic Equations  12.1 First Order Scalar Equations  12.2 Symmetric Hyperbolic Systems  12.3 The Wendroff Box Scheme  12.4 Problems13 The Finite Element Method for Hyperbolic Equations  13.1 The Wave Equation  13.2 First Order Hyperbolic Equations  13.3 Problems14 Some Other Classes of Numerical Methods  14.1 Collocation Methods  14.2 Spectral Methods  14.3 Finite Volume Methods  14.4 Boundary Element Methods  14.5 ProblemsA Some Tools from Mathematical Analysis  A.1 Abstract Linear Spaces  A.2 Function Spaces  A.3 The Fourier Transform  A.4 ProblemsB Orientation on Numerical Linear Algebra  B.1 Direct Methods  B.2 Iterative Methods. Relaxation, Overrelaxation, and Acceleration  B.3 Preconditioned Conjugate Gradient Methods  B.4 Preconditioned Conjugate Gradient Methods  B.5 Multigrid and Domain Decomposition MethodsBibliographyIndex

編輯推薦

　　當(dāng)今科學(xué)家收集曲線樣本及其他函數(shù)觀測(cè)值，這本專(zhuān)著論述這類(lèi)數(shù)據(jù)分析的思想和技巧，主要內(nèi)容包括經(jīng)典的線性回歸方法、主成分分析、線性建模、典型相關(guān)分析及特殊的泛函技巧，如曲線注冊(cè)和主微分分析?！　”緯?shū)始終利用來(lái)源于實(shí)際應(yīng)用的數(shù)據(jù)，介紹方法的動(dòng)機(jī)并舉例論證，特別通過(guò)討論數(shù)據(jù)生成過(guò)程的光滑性，說(shuō)明如何通過(guò)泛函方法來(lái)發(fā)現(xiàn)數(shù)據(jù)的新特點(diǎn)；這些數(shù)據(jù)主要來(lái)源于增長(zhǎng)分析、氣象學(xué)、生物力學(xué)、馬類(lèi)科學(xué)、經(jīng)濟(jì)學(xué)及醫(yī)學(xué)等領(lǐng)域的應(yīng)用。本書(shū)論述新穎的統(tǒng)計(jì)技術(shù)，同時(shí)使其中的數(shù)學(xué)論證能被大多數(shù)人所理解?！　”緯?shū)許多內(nèi)容都基于作者自己的工作，某些內(nèi)容是首次出版。本書(shū)適合學(xué)生、應(yīng)用數(shù)據(jù)分析學(xué)者及科研人員閱讀，對(duì)統(tǒng)計(jì)學(xué)及其他廣闊領(lǐng)域的研究也頗有價(jià)值。　　本書(shū)第一作者Jim Ramsay是McGill大學(xué)的心理學(xué)教授，加拿大統(tǒng)計(jì)學(xué)會(huì)主席，多元分析等諸多領(lǐng)域的國(guó)際權(quán)威。他和言語(yǔ)清晰度、電動(dòng)控制、氣象學(xué)、心理學(xué)及人體生理學(xué)等多個(gè)領(lǐng)域的研究人員合作，在統(tǒng)計(jì)學(xué)及其應(yīng)用方面的許多雜志發(fā)表論文，對(duì)函數(shù)型數(shù)據(jù)分析研究作出了技術(shù)性貢獻(xiàn)?！　”緯?shū)第二作者Bernard Silverman是Bristol大學(xué)的統(tǒng)計(jì)學(xué)教授，著名的《統(tǒng)計(jì)學(xué)中的密度估計(jì)及數(shù)據(jù)分析》一書(shū)的作者，　《非參數(shù)回歸分析與廣義線性模型》的合著者。他因在光滑方法和應(yīng)用統(tǒng)計(jì)學(xué)、計(jì)算統(tǒng)計(jì)學(xué)及理論統(tǒng)計(jì)學(xué)等諸多方面的工作而獲得統(tǒng)計(jì)學(xué)會(huì)會(huì)長(zhǎng)聯(lián)合委員會(huì)頒發(fā)的會(huì)長(zhǎng)獎(jiǎng)及兩枚皇家統(tǒng)計(jì)學(xué)會(huì)Guy獎(jiǎng)?wù)隆?/pre>





圖書(shū)封面



圖書(shū)標(biāo)簽Tags
無(wú)




評(píng)論、評(píng)分、閱讀與下載



還沒(méi)讀過(guò)(29)
勉強(qiáng)可看(213)
一般般(363)
內(nèi)容豐富(1507)
強(qiáng)力推薦(123)


 


    偏微分方程與數(shù)值方法 PDF格式下載