Solutions Manual to Accompany Beginning Partial Differential Equations

Third Edition

Peter V. O'Neil

University of Alabama at Birmingham

Copyright © 2015 by John Wiley & Sons, Inc. All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New Jersey

Published simultaneously in Canada

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permissions.

Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.

For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002.

Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com.

Library of Congress Cataloging in Publication Data is available.

ISBN 978-1-118-63009-9

Preface

This manual contains solutions for many of the problems in Beginning Partial Differential Equations, third edition.

Because solutions for many odd-numbered problems are included in Chapter Nine of the book, most of the problems included here are even-numbered. However, particularly in the case of problems exploring ideas beyond the text discussion, some odd-numbered solutions are also included.

Chapter 1
First Ideas

1.1 Two Partial Differential Equations

2. Verifying that the function is a solution of the heat equation is a straightforward exercise in differentiation. One way to show that c01-math-0001 is unbounded is to observe that if c01-math-0002 and c01-math-0003, then

and this can be made as large as we like by choosing c01-math-0005 sufficiently close to zero.

4. By the chain rule,

It is routine to verify that c01-math-0007.

7. One way to show that the transformation is one to one is to evaluate the Jacobian

Finally, solve c01-math-0009, c01-math-0010 for c01-math-0011 and c01-math-0012 to obtain the inverse transformation

8. With c01-math-0014, chain rule differentiations yield:

and, by continuing these chain rule differentiations and using the product rule,

Now collect terms to obtain

This, coupled with the fact that c01-math-0018 transforms to some function c01-math-0019, yields the conclusion.

9. From the solution of problem 8, the transformed equation is hyperbolic if c01-math-0020 because in that case we can choose c01-math-0021 and c01-math-0022 to make the coefficients of c01-math-0023 and c01-math-0024 vanish. This is done by choosing c01-math-0025 and c01-math-0026 to be the distinct roots of

which are the same quadratic equation. For example, we could choose

If c01-math-0029, use the transformation

Now chain rule differentiations yield

We do not need c01-math-0032, because c01-math-0033 in this case. Now we obtain

yielding a hyperbolic canonical form

of the given partial differential equation.

10. In this case suppose c01-math-0036. Now let

Now

Then

with two terms on the next to last line vanishing because c01-math-0040. This gives the canonical form

for the original partial differential equation when c01-math-0042.

11. Suppose now that c01-math-0043. Let the roots of c01-math-0044 be c01-math-0045. Let

Proceeding as in the preceding two problems, we find that

Now we need some information about c01-math-0048 and c01-math-0049. Because of the way c01-math-0050 was chosen,

This gives us

Then

In this case,

and we obtain the canonical form

for this case.

12. The diffusion equation is parabolic and the wave equation is hyperbolic.

14. c01-math-0056, so the equation is hyperbolic. With

the canonical form is

16. With c01-math-0059, and c01-math-0060, c01-math-0061, so the equation is elliptic. Solve c01-math-0062 to get c01-math-0063. Thus use the transformation

to obtain the canonical form

1.2 Fourier Series

2. c01-math-0066 is the Fourier series of c01-math-0067 on c01-math-0068. This converges to c01-math-0069 for c01-math-0070.

4. The Fourier series of c01-math-0071 on c01-math-0072 is

converging to c01-math-0074 for c01-math-0075. Figure 1.1 compares a graph of c01-math-0076 with the fifth partial sum of the series.

6. The Fourier series is

Figure 1.2 compares a graph of the function with the fifth partial sum of the series.

8. The Fourier series converges to

10. The series converges to

12. The series converges to

14. Multiply by c01-math-0083 to obtain

Integrate term by term:

Then

Upon division by c01-math-0087, this yields Parseval's equation.

16. The cosine series is

converging to c01-math-0089 for c01-math-0090, to c01-math-0091 for c01-math-0092, and to c01-math-0093 at c01-math-0094. Figure 1.3 compares the function to the 100th partial sum of this cosine expansion. The sine series is

converging to c01-math-0096 at the end points and at c01-math-0097, and to the function for c01-math-0098 and c01-math-0099. Figure 1.4 is the 100th partial sum of this sine series.

Figure 1.1 c01-math-0078 and the 5th partial sum of the Fourier series in Problem 4.

Figure 1.2 c01-math-0079 and the 5th partial sum of the Fourier series in Problem 6.

Figure 1.3 c01-math-0100 and the 100th partial sum of the cosine series in Problem 16.

Figure 1.4 c01-math-0101 and the 100th partial sum of the sine expansion in Problem 16.

18. The cosine expansion is

This converges to c01-math-0104 on c01-math-0105. Figure 1.5 compares the function with the 10th partial sum of this cosine series. The sine expansion is

converging to c01-math-0107 at c01-math-0108 and c01-math-0109, and to c01-math-0110 for c01-math-0111. Figure 1.6 compares the function with the 50th partial sum of this sine expansion.

Figure 1.5 c01-math-0102 and the 10th partial sum of the cosine series in Problem 18.

Figure 1.6 c01-math-0112 and the 50th partial sum of the sine expansion in Problem 18.

20. The cosine expansion is

converging to c01-math-0115 for c01-math-0116. Figure 1.7 shows the function and the 10th partial sum of this series.

Figure 1.7 c01-math-0113 and the 10th partial sum of the cosine series in Problem 20.

The sine expansion is

This series converges to c01-math-0118 at c01-math-0119 and at c01-math-0120, and to c01-math-0121 for c01-math-0122. Figure 1.8 shows the 50th partial sum.

Figure 1.8 c01-math-0123 and the 50th partial sum of the sine expansion in Problem 20.

22. The cosine expansion is

converging to c01-math-0125 on c01-math-0126. Figure 1.9 shows graphs of the function and the 10th partial sum of this cosine series. The sine series is

converging to c01-math-0128 on c01-math-0129. The function and the 10th partial sum of this sine series are shown in Figure 1.10.

Figure 1.9 c01-math-0130 and the 10th partial sum of the cosine series in Problem 22.

Figure 1.10 c01-math-0131 and the 50th partial sum of the sine expansion in Problem 22.

23. Expand c01-math-0132 in a cosine series on c01-math-0133:

Since c01-math-0135 if c01-math-0136 is odd, we need only to retain the even positive integers in the sum. Replace c01-math-0137 with c01-math-0138 to write

Now choose c01-math-0140.

1.3 Two Eigenvalue Problems

2. Eigenvalues and eigenfunctions are

4. Eigenvalues and eigenfunctions are

where c01-math-0147 is the c01-math-0148th positive root (in increasing order) of the equation c01-math-0149.

1.4 A Proof of the Convergence Theorem

The Fourier series of c01-math-0150 on c01-math-0151 is

Figures 1.11–1.14 show the function and the c01-math-0153th partial sum for c01-math-0154, respectively.

Figure 1.11 c01-math-0141 and the 10th partial sum.

Figure 1.12 c01-math-0142 and the 25th partial sum.

Figure 1.13 c01-math-0143 and the 50th partial sum.

Figure 1.14 c01-math-0144 and the 100th partial sum.

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!