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Introduction to Partial Differential Equations (Hardcover) (David Borthwick)
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Designed for students without an analysis background, this textbook provides a balanced presentation of partial differential equations through modern methods that do not require knowledge beyond vector calculus and linear algebra. By introducing definitions from analysis only when they are needed in the text, Borthwick reaches more advanced methods and emphasizes the role that tools from analysis play in PDE applications. The material included in this textbook allows students to study the most important classical PDE within a single semester.
Borthwick focuses on a particular equation in each section, including conservative equations and their characteristics, the wave equation, the heat equation, function spaces, and Fourier series. He creates a narrative within each section that answers the five questions:(1) What is the scientific problem we are trying to understand?
(2) How do we model that with PDE?
(3) What techniques can we use to analyze the PDE?
(4) How do those techniques apply to this equation?
(5) What information or insight did we obtain by developing and analyzing the PDE?
The text stresses the interplay between modeling and mathematical analysis, providing a thorough source of problems and an inspiration for the development of methods.