Agarwal Donal O’Regan Ordinary and Partial Differential Equations With Special Functions, Fourier Series, and Boundary Value Problems £ Springer Universitext For other books in this series: http://www.com/series/223 Ravi P. Agarwal Donal O’Regan Ordinary and Partial Differential Equations With Special Functions, Fourier Series, and Boundary Value Problems VÀ Springer Ravi P. Agarwal Florida Institute of Technology Department of Mathematical Sciences 150 West University Blvd. Melbourne, FL 32901 agarwal @fit.edu ISBN: 978-0-387-79 145-6 DOI 10.1007/978-0-387-79 146-3 Library of Congress Control Number: 2008938952 Donal O’Regan National University of Ireland, Galway Mathematics Department University Road Galway, Ireland donal.ie e-ISBN: 978-0-387-79 146-3 Mathematics Subject Classification (2000): 00-01, 34-XX, 35-XX © Springer Science+Business Media, LLC 2009 All rights reserved.
This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper springer.com Dedicated to our Sons Hans Agarwal and Daniel Joseph O’Regan Preface This book comprises 50 class-tested lectures which both the authors have given to engineering and mathematics major students under the titles Boundary Value Problems and Methods of Mathematical Physics at various institutions all over the globe over a period of almost 35 years.
The main topics covered in these lectures are power series solutions, special func- tions, boundary value problems for ordinary differential equations, Sturm— Liouville problems, regular and singular perturbation techniques, Fourier series expansion, partial differential equations, Fourier series solutions to initial-boundary value problems, and Fourier and Laplace transform tech- niques. The prerequisite for this book is calculus, so it can be used for a senior undergraduate course. It should also be suitable for a beginning graduate course because, in undergraduate courses, students do not have any exposure to various intricate concepts, perhaps due to an inadequate level of mathematical sophistication. The content in a particular lecture, together with the problems therein, provides fairly adequate coverage of the topic under study.
These lectures have been delivered in one year courses and provide flexibility in the choice of material for a particular one-semester course. Throughout this book, the mathematical concepts have been ex- plained very carefully in the simplest possible terms, and illustrated by a number of complete workout examples. Like any other mathematical book, it does contain some theorems and their proofs. A detailed description of the topics covered in this book is as follows: In Lecture 1 we find explicit solutions of the first-order linear differential equations with variable coefficients, second-order homogeneous differential equations with constant coefficients, and second-order Cauchy—Euler differ- ential equations.
In Lecture 2 we show that if one solution of the homoge- neous second-order differential equation with variable coefficients is known, then its second solution can be obtained rather easily. Here we also demon- strate the method of variation of parameters to construct the solutions of nonhomogeneous second-order differential equations. In Lecture 3 we provide some basic concepts which are required to con- struct power series solutions to differential equations with variable coeffi- cients. Here through various examples we also explain ordinary, regular singular, and irregular singular points of a given differential equation.
In Lecture 4 first we prove a theorem which provides sufficient conditions so that the solutions of second-order linear differential equations can be ex- pressed as power series at an ordinary point, and then construct power se- ries solutions of Airy, Hermite, and Chebyshev differential equations. These equations occupy a central position in mathematical physics, engineering, and approximation theory. In Lectures 5 and 6 we demonstrate the method viii Preface of Frobenius to construct the power series solutions of second-order linear differential equations at a regular singular point. Here we prove a gen- eral result which provides three possible different forms of the power series solution.
We illustrate this result through several examples, including La- guerre’s equation, which arises in quantum mechanics. In Lecture 7 we study Legendre’s differential equation, which arises in problems such as the flow of an ideal fluid past a sphere, the determination of the electric field due to a charged sphere, and the determination of the temperature distribution in a sphere given its surface temperature. Here we also develop the polyno- mial solution of the Legendre differential equation. In Lecture 8 we study polynomial solutions of the Chebyshev, Hermite, and Laguerre differential equations.
In Lecture 9 we construct series solutions of Bessel’s differential equation, which first appeared in the works of Euler and Bernoulli. Since many problems of mathematical physics reduce to the Bessel equation, we investigate it in somewhat more detail. In Lecture 10 we develop series so- lutions of the hypergeometric differential equation, which finds applications in several problems of mathematical physics, quantum mechanics, and fluid dynamics. Mathematical problems describing real world situations often have so- lutions which are not even continuous.
Thus, to analyze such problems we need to work in a set which is bigger than the set of continuous func- tions. In Lecture 11 we introduce the sets of piecewise continuous and piecewise smooth functions, which are quite adequate to deal with a wide variety of applied problems. Here we also define periodic functions, and introduce even and odd extensions. In Lectures 12 and 13 we introduce orthogonality of functions and show that the Legendre, Chebyshev, Her- mite, and Laguerre polynomials and Bessel functions are orthogonal.
Here we also prove some fundamental properties about the zeros of orthogonal polynomials. In Lecture 14 we introduce boundary value problems for second-order ordinary differential equations and provide a necessary and sufficient con- dition for the existence and uniqueness of their solutions. In Lecture 15 we formulate some boundary value problems with engineering applications, and show that often solutions of these problems can be written in terms of Bessel functions. In Lecture 16 we introduce Green’s functions of ho- mogeneous boundary value problems and show that the solution of a given nonhomogeneous boundary value problem can be explicitly expressed in terms of Green’s function of the corresponding homogeneous equation.
In Lecture 17 we discuss the regular perturbation technique which re- lates the unknown solution of a given initial value problem to the known solutions of the infinite initial value problems. In many practical problems one often meets cases where the methods of regular perturbations cannot be applied. In the literature such problems are known as singular pertur- bation problems. In Lecture 18 we explain the methodology of singular perturbation technique with the help of some examples.
Preface ix If the coefficients of the homogeneous differential equation and/or of the boundary conditions depend on a parameter, then one of the pioneer problems of mathematical physics is to determine the values of the param- eter (eigenvalues) for which nontrivial solutions (eigenfunctions) exist. In Lecture 19 we explain some of the essential ideas involved in this vast field, which is continuously growing. In Lectures 20 and 21 we show that the sets of orthogonal polynomials and functions we have provided in earlier lectures can be used effectively as the basis in the expansions of general functions. This in particular leads to Fourier’s cosine, sine, trigonometric, Legendre, Chebyshev, Hermite and Bessel series.
In Lectures 22 and 23 we examine pointwise convergence, uniform convergence, and the convergence in the mean of the Fourier se- ries of a given function. Here the importance of Bessel’s inequality and Parseval’s equality are also discussed. In Lecture 24 we use Fourier series expansions to find periodic particular solutions of nonhomogeneous differ- ential equations, and solutions of nonhomogeneous self-adjoint differential equations satisfying homogeneous boundary conditions, which leads to the well-known Fredholm’s alternative. In Lecture 25 we introduce partial differential equations and explain sev- eral concepts through elementary examples.
Here we also provide the most fundamental classification of second-order linear equations in two indepen- dent variables. In Lecture 26 we study simultaneous differential equations, which play an important role in the theory of partial differential equations. Then we consider quasilinear partial differential equations of the Lagrange type and show that such equations can be solved rather easily, provided we can find solutions of related simultaneous differential equations. Finally, we explain a general method to find solutions of nonlinear first-order par- tial differential equations which is due to Charpit.
In Lecture 27 we show that like ordinary differential equations, partial differential equations with constant coefficients can be solved explicitly. We begin with homogeneous second-order differential equations involving only second-order terms, and then show how the operator method can be used to solve some particular nonhomogeneous differential equations. Then, we extend the method to general second and higher order partial differential equations. In Lecture 28 we show that coordinate transformations can be employed successfully to reduce second-order linear partial differential equations to some standard forms, which are known as canonical forms.
These transformed equations sometimes can be solved rather easily. Here the concept of characteristic of second-order partial differential equations plays an important role. The method of separation of variables involves a solution which breaks up into a product of functions each of which contains only one of the vari- ables. This widely used method for finding solutions of linear homoge- neous partial differential equations we explain through several simple ex- amples in Lecture 29.
In Lecture 30 we derive the one-dimensional heat equation and formulate initial-boundary value problems, which involve the x Preface heat equation, the initial condition, and homogeneous and nonhomogeneous boundary conditions. Then we use the method of separation of variables to find the Fourier series solutions to these problems. In Lecture 31 we construct the Fourier series solution of the heat equation with Robin’s boundary conditions. In Lecture 32 we provide two different derivations of the one-dimensional wave equation, formulate an initial-boundary value problem, and find its Fourier series solution.
In Lecture 33 we continue using the method of separation of variables to find Fourier series solutions to some other initial-boundary value problems related to one-dimensional wave equation. In Lecture 34 we give a derivation of the two-dimensional Laplace equation, formulate the Dirichlet problem on a rectangle, and find its Fourier series solution. In Lecture 35 we discuss the steady-state heat flow problem in a disk. For this, we consider the Laplace equation in po- lar coordinates and find its Fourier series solution.
In Lecture 36 we use the method of separation of variables to find the temperature distribution of rectangular and circular plates in the transient state. Again using the method of separation of variables, in Lecture 37 we find vertical displace- ments of thin membranes occupying rectangular and circular regions. The three-dimensional Laplace equation occurs in problems such as gravitation, steady-state temperature, electrostatic potential, magnetostatics, fluid flow, and so on. In Lecture 38 we find the Fourier series solution of the Laplace equation in a three-dimensional box and in a circular cylinder.
In Lecture 39 we use the method of separation of variables to find the Fourier series solutions of the Laplace equation in and outside a given sphere. Here, we also discuss briefly Poisson’s integral formulas. In Lecture 40 we demon- strate how the method of separation of variables can be employed to solve nonhomogeneous problems. The Fourier integral is a natural extension of Fourier trigonometric series in the sense that it represents a piecewise smooth function whose domain is semi-infinite or infinite.
In Lecture 41 we develop the Fourier integral with an intuitive approach and then discuss Fourier cosine and sine inte- grals which are extensions of Fourier cosine and sine series, respectively. This leads to Fourier cosine and sine transform pairs.