Partial Differential Equations: An Introduction to Analytical and Numerical Methods

Arendt, Wolfgang, Urban, Karsten, Kennedy, James B.

  • 出版商: Springer
  • 出版日期: 2024-01-03
  • 售價: $2,340
  • 貴賓價: 9.5$2,223
  • 語言: 英文
  • 頁數: 452
  • 裝訂: Quality Paper - also called trade paper
  • ISBN: 3031133811
  • ISBN-13: 9783031133817
  • 海外代購書籍(需單獨結帳)

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商品描述

This textbook introduces the study of partial differential equations using both analytical and numerical methods. By intertwining the two complementary approaches, the authors create an ideal foundation for further study. Motivating examples from the physical sciences, engineering, and economics complete this integrated approach.

A showcase of models begins the book, demonstrating how PDEs arise in practical problems that involve heat, vibration, fluid flow, and financial markets. Several important characterizing properties are used to classify mathematical similarities, then elementary methods are used to solve examples of hyperbolic, elliptic, and parabolic equations. From here, an accessible introduction to Hilbert spaces and the spectral theorem lay the foundation for advanced methods. Sobolev spaces are presented first in dimension one, before being extended to arbitrary dimension for the study of elliptic equations. An extensive chapter on numerical methods focuses onfinite difference and finite element methods. Computer-aided calculation with Maple(TM) completes the book. Throughout, three fundamental examples are studied with different tools: Poisson's equation, the heat equation, and the wave equation on Euclidean domains. The Black-Scholes equation from mathematical finance is one of several opportunities for extension.

Partial Differential Equations offers an innovative introduction for students new to the area. Analytical and numerical tools combine with modeling to form a versatile toolbox for further study in pure or applied mathematics. Illuminating illustrations and engaging exercises accompany the text throughout. Courses in real analysis and linear algebra at the upper-undergraduate level are assumed.

商品描述(中文翻譯)

這本教科書介紹了使用分析和數值方法來研究偏微分方程。通過交織這兩種互補的方法,作者為進一步研究打下了理想的基礎。物理科學、工程學和經濟學中的實例完善了這種整合方法。

書中展示了一系列模型,展示了偏微分方程在涉及熱、振動、流體流動和金融市場的實際問題中的應用。使用幾個重要的特徵性質來分類數學相似性,然後使用基本方法來解析双曲型、椭圆型和抛物型方程的例子。從這裡,對希爾伯特空間和譜定理的易於理解的介紹為高級方法打下了基礎。首先在一維空間中介紹 Sobolev 空間,然後擴展到任意維度以研究椭圆型方程。一個廣泛的數值方法章節聚焦於有限差分和有限元方法。使用 Maple(TM) 進行計算完成了本書。在整個過程中,使用不同工具研究了三個基本例子:泊松方程、熱方程和歐幾里得域上的波動方程。數學金融中的 Black-Scholes 方程是擴展的其中之一。

《偏微分方程》為初學者提供了一種創新的介紹。分析和數值工具結合建模形成了進一步研究純數學或應用數學的多功能工具箱。豐富的插圖和引人入勝的練習貫穿全書。假設讀者已具備高年級本科水平的實分析和線性代數知識。

作者簡介

Wolfgang Arendt is Senior Professor of Analysis at Ulm University. His research areas are functional analysis and partial differential equations.

Karsten Urban is Professor of Numerical Mathematics at Ulm University. His research interests include numerical methods for partial differential equations, especially with concrete applications in science and technology.

作者簡介(中文翻譯)

Wolfgang Arendt是Ulm大學的高級分析學教授。他的研究領域包括函數分析和偏微分方程。

Karsten Urban是Ulm大學的數值數學教授。他的研究興趣包括偏微分方程的數值方法,尤其是在科學和技術中的具體應用。