An Elementary Course on Partial Differential Equations

Partial differential equations are a vital part of any course in pure or applied mathematics. This book will be invaluable to anyone looking for a lucid but comprehensive introduction to PDEs. Designed to strike a balance between theory and practical problems, it covers all major methods as well as their historical backgrounds, theoretical rigour, and geometric significance. The book is divided into three parts. It starts with basic topics like ordinary differential equations, multivariable calculus, and geometry. This is followed by important techniques to solve certain types of partial differential equations. The last part is devoted to first, second, and higher-order PDEs. The chapters have been arranged to help students develop their knowledge gradually and systematically. Each method is discussed through theoretical descriptions in the form of theorems followed by illustrative problems to help the readers. Finally, numerous solved examples and practice problems helps the student learn to apply this knowledge.

偏微分方程是純數學或應用數學課程中不可或缺的一部分。這本書對於尋求清晰而全面的偏微分方程介紹的人來說是無價的。它旨在在理論和實際問題之間取得平衡，涵蓋了所有主要方法以及它們的歷史背景、理論嚴謹性和幾何意義。該書分為三個部分。首先介紹基本主題，如常微分方程、多變量微積分和幾何。接下來介紹解決特定類型的偏微分方程的重要技巧。最後一部分專門介紹一階、二階和高階偏微分方程。章節的安排有助於學生逐漸而系統地發展他們的知識。每種方法都通過定理形式的理論描述進行討論，然後是解題示例，以幫助讀者理解。最後，大量的解題示例和練習問題幫助學生學習如何應用這些知識。