Introduction to Partial Differential Equations with Applications

E. C. Zachmanoglou, Dale W. Thoe

• Publisher: Dover Publications
• Year: 2012
• ISBN:
  9780486132174
• Topic: Mathematics
• Subtopic: Differential Equations
• Index: Mathematics

This book has been widely acclaimed for its clear, cogent presentation of the theory of partial differential equations, and the incisive application of its principal topics to commonly encountered problems in the physical sciences and engineering. It was developed and tested at Purdue University over a period of five years in classes for advanced undergraduate and beginning graduate students in mathematics, engineering and the physical sciences.
The book begins with a short review of calculus and ordinary differential equations, then moves on to explore integral curves and surfaces of vector fields, quasi-linear and linear equations of first order, series solutions and the Cauchy Kovalevsky theorem. It then delves into linear partial differential equations, examines the Laplace, wave and heat equations, and concludes with a brief treatment of hyperbolic systems of equations.
Among the most important features of the text are the challenging problems at the end of each section which require a wide variety of responses from students, from providing details of the derivation of an item presented to solving specific problems associated with partial differential equations. Requiring only a modest mathematical background, the text will be indispensable to those who need to use partial differential equations in solving physical problems. It will provide as well the mathematical fundamentals for those who intend to pursue the study of more advanced topics, including modern theory.

CHAPTER 1

---

Some concepts from calculus and ordinary differential equations

---

In this chapter we review some basic definitions and theorems from Calculus and Ordinary Differential Equations. At the same time we introduce some of the notation which will be used in this book. Most topics may be quite familiar to the student and in this case the chapter may be covered quickly.

In Section 1 we review some concepts associated with sets, functions, limits, continuity and differentiability. In Section 2 we discuss surfaces and their normals and recall one of the most useful and important theorems of mathematics, the Implicit Function Theorem. In Section 3 we discuss two ways of representing curves in three or higher dimensional space and give the formulas for finding the tangent vectors for each of these representations. In Section 4 we review the basic existence and uniqueness theorem for the initial value problem for ordinary differential equations and systems.

1. Sets and Functions

We will denote by Rⁿ the n-dimensional Euclidean space. A point in Rⁿ has n coordinates x₁, x₂, . . . , x_n and its position vector will be denoted by x. Thus x = (x₁, . . . , x_n). For n = 2 or n = 3 we may also use different letters for the coordinates. For example, we may use (x, y) for the coordinates of a point in R² and (x, y, z) or (x, y, u) for the coordinates of a point in R³.

The distance between two points x = (x₁, . . . , x_n) and y = (y₁, . . . , y_n) in Rⁿ is given by

An open ball with center x⁰

Rⁿ and radius ρ > 0 is the set of points in Rⁿ which are at a distance less than ρ from x⁰,

(1.1) B(x⁰, ρ) = {x : x

Rⁿ, d(x, x⁰ < ρ}.

The corresponding closed ball is

(1.2)

and the surface of this ball, called a sphere, is

(1.3) S(x⁰, ρ) = {x : x

Rⁿ, d(x, x⁰ = ρ}.

A set A of points in Rⁿ is called open if for every x

A there is a ball with center at x which is contained in A. A set in Rⁿ is called closed if its complement is an open set. A point x is called a boundary point of a set A if every ball with center at x contains points of A and points of the complement of A. The set o...