A First Course in Partial Differential Equations

J Robert Buchanan, Zhoude Shao

• Editore: WSPC
• Anno: 2017
• ISBN:
  9789813226456
• Argomento: Mathematics
• Categoria: Mathematical Analysis

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This textbook gives an introduction to Partial Differential Equations (PDEs), for any reader wishing to learn and understand the basic concepts, theory, and solution techniques of elementary PDEs. The only prerequisite is an undergraduate course in Ordinary Differential Equations. This work contains a comprehensive treatment of the standard second-order linear PDEs, the heat equation, wave equation, and Laplace's equation. First-order and some common nonlinear PDEs arising in the physical and life sciences, with their solutions, are also covered.

This textbook includes an introduction to Fourier series and their properties, an introduction to regular Sturm–Liouville boundary value problems, special functions of mathematical physics, a treatment of nonhomogeneous equations and boundary conditions using methods such as Duhamel's principle, and an introduction to the finite difference technique for the numerical approximation of solutions. All results have been rigorously justified or precise references to justifications in more advanced sources have been cited. Appendices providing a background in complex analysis and linear algebra are also included for readers with limited prior exposure to those subjects.

The textbook includes material from which instructors could create a one- or two-semester course in PDEs. Students may also study this material in preparation for a graduate school (masters or doctoral) course in PDEs.

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--> Contents:

• Introduction
• First-Order Partial Differential Equations
• Fourier Series
• The Heat Equation
• The Wave Equation
• The Laplace Equation
• Sturm–Liouville Theory
• Special Functions
• Applications of PDEs in the Physical Sciences
• Nonhomogeneous Initial Boundary Value Problems
• Nonlinear Partial Differential Equations
• Numerical Solutions to PDEs Using Finite Differences
• Appendices:
  • Complex Arithmetic and Calculus
  • Linear Algebra Primer

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--> Readership: Mathematics, physical and life sciences, and engineering undergraduate students interested in partial differential equations. -->
Keywords:Heat Equation;Wave Equation;Laplace's Equation;Poisson's Equation;Applications of PDEs;Fourier Series;Sturm–Liouville Theory;Special Functions;Finite Difference Techniques;Nonlinear PDEs;Duhamel's Principle;Maximum/Minimum PrincipleReview: Key Features:

• It is written for undergraduate science and mathematics students as a self-contained introduction to the field of PDEs assuming only a prerequisite background in ordinary differential equations. This textbook is appropriate for readers in mathematics, the physical and life sciences, and engineering. The style of writing is clear and easily accessible to the student. After successfully completing a study of PDEs in the proposed textbook, students will be prepared for more advanced study in PDEs
• The treatment of topics is precise and rigorous, to the extent that it is accessible to the typical undergraduate science and mathematics student. For the topics where rigorous mathematical treatment is beyond the scope of the book, references to more advanced and precise treatments are provided
• The first six chapters of the textbook (five chapters if an instructor elects to omit the chapter on first-order PDEs) provide a standard undergraduate introduction to PDEs and their solution techniques and can be delivered in a typical one-semester course. The material from the remaining six chapters of the textbook can form a second-semester course in PDEs or material for independent study

Chapter 1

Introduction

When modeling physical phenomena, often it is necessary to consider functions that depend on more than one variable. Therefore, if rates of changes are involved, partial derivatives of functions must be used and this leads to mathematical models that are partial differential equations. Loosely speaking, a partial differential equation is an equation that involves an unknown function of two or more variables and its partial derivatives. The following are some examples of partial differential equations:

1.1Preliminaries: Notation, Definitions, and the Principle of Superposition

In each of the equations above, u represents the unknown function and

represent the partial derivatives of u with respect to the indicated variables. Given any partial differential equation, one of the basic problems is to find the function (or functions) that satisfies (or satisfy) the equation. For any partial differential equation, a function is called a solution to the equation if the function and its partial derivatives involved in the equation are all defined on a certain domain and satisfy the partial differential equation on that domain. For example, a solution to the third equation above is defined on a domain in the three-dimensional Euclidean space R³ and solutions of all the other examples are defined on domains in the two-dimensional Euclidean space R². In general, the n-dimensional Euclidean space will be denoted by Rⁿ.

Recall that ordinary differential equations are classified based on the concepts of order and linearity. Similar concepts arise in partial differential equations. The order of a partial differential equation is defined to be the highest order of all the partial derivatives of the unknown function that appear in the partial differential equation. In the examples above, the first equation is a first-order partial differential equation, the last one is a third-order equation, and the rest are all second-order equations. A partial differential equation is said to be linear if all the terms in the equation are linear in the unknown function and its partial derivatives, that is, each term in the equation contains at most one instance of the unknown function or one of its partial derivatives raised to the first power. If an equation is not linear, it is said to be nonlinear. For example, the second and third equations in the examples above are linear, while the rest are nonlinear. Later, nonlinear partial differential equations will be further classified as semi-linear, quasilinear, and (truly) nonlinear. Systems of partial differential equations can also be considered. This is necessary in many situations. For example, the real part u(x, y) and the imaginary part v(x, y) of any complex analytic function¹ must satisfy the so-called Cauchy²–Riemann³ equations:

which is a simple system of (first-order, linear) partial differential equations.

A very important class of partial differential equations in physics and other fields of science and engineering consists of linear partial differential equations of the second order. The general form of second-order, linear partial differential equations with two independent variables, t and x (generally interpreted as time and position respectively), is the following equation

where A, B, C, . . . , G are functions of t and x only (one or more of them can be constants or even 0). Equation (1.1) is said to be homogeneous if G(x, t) ≡ 0. In this case, the equation becomes

The theory of second-order, linear partial differential equations arises frequently in applications and involves a minimal amount of technicality without sacrificing the richness of the theory of partial differential equations. In fact, the discussion of second-order, linear partial differential equations of various types constitutes a large part of this text.

Recall that in the theory of linear ordinary differential equations, the general solution of an equation can be written as a linear combination of a set of specific solutions. This is possible due to the Principle of Supe...