Introduction to Partial Differential Equations

This textbook is designed for a one year course covering the fundamentals of partial differential equations, geared towards advanced undergraduates and beginning graduate students in mathematics, science, engineering, and elsewhere. The exposition carefully balances solution techniques, mathematical rigor, and significant applications, all illustrated by numerous examples. Extensive exercise sets appear at the end of almost every subsection, and include straightforward computational problems to develop and reinforce new techniques and results, details on theoretical developments and proofs, challenging projects both computational and conceptual, and supplementary material that motivates the student to delve further into the subject.

No previous experience with the subject of partial differential equations or Fourier theory is assumed, the main prerequisites being undergraduate calculus, both one- and multi-variable, ordinary differential equations, and basic linear algebra. While the classical topics of separation of variables, Fourier analysis, boundary value problems, Green's functions, and special functions continue to form the core of an introductory course, the inclusion of nonlinear equations, shock wave dynamics, symmetry and similarity, the Maximum Principle, financial models, dispersion and solutions, Huygens' Principle, quantum mechanical systems, and more make this text well attuned to recent developments and trends in this active field of contemporary research. Numerical approximation schemes are an important component of any introductory course, and the text covers the two most basic approaches: finite differences and finite elements.

這本教科書旨在為一年制的課程提供基礎的偏微分方程知識，適用於數學、科學、工程等領域的高年級本科生和初級研究生。該教材精心平衡了解題技巧、數學嚴謹性和重要應用，並通過大量實例進行了說明。幾乎每個小節末尾都有大量的練習題，包括簡單的計算問題以發展和鞏固新技巧和結果，理論發展和證明的細節，具有計算和概念性難度的挑戰性項目，以及激發學生進一步深入研究該主題的補充材料。

本書不需要對偏微分方程或傅立葉理論有任何先備知識，主要先備條件是本科微積分（單變量和多變量）、常微分方程和基礎線性代數。儘管傳統的分離變量、傅立葉分析、邊界值問題、格林函數和特殊函數仍然是入門課程的核心內容，但本書還包括非線性方程、衝擊波動力學、對稱性和相似性、最大值原理、金融模型、色散和解、惠更斯原理、量子力學系統等，使其與當代研究領域的最新發展和趨勢相契合。數值逼近方法是任何入門課程的重要組成部分，本書涵蓋了兩種最基本的方法：有限差分和有限元。

Peter J. Olver is professor of mathematics at the University of Minnesota. His wide-ranging research interests are centered on the development of symmetry-based methods for differential equations and their manifold applications. He is the author of over 130 papers published in major scientific research journals as well as 4 other books, including the definitive Springer graduate text, Applications of Lie Groups to Differential Equations, and another undergraduate text, Applied Linear Algebra.

Peter J. Olver是明尼蘇達大學的數學教授。他廣泛的研究興趣集中在基於對稱性的微分方程方法的發展及其多樣的應用。他是超過130篇論文的作者，這些論文發表在主要的科學研究期刊上，並且還出版了其他4本書，包括權威的Springer研究生教材《Lie群在微分方程中的應用》，以及另一本本科教材《應用線性代數》。