Solution techniques for elementary partial differential equations, second edition (1)

Solution Techniques for Elementary Partial Differential Equations Second Edition K10569_FM.indd 4/28/10 9:50:09 AM K10569_FM.indd 4/28/10 9:50:09 AM Solution Techniques for Elementary Partial Differential Equations Second Edition Christian Constanda University of Tulsa Oklahoma K10569_FM.indd 4/28/10 9:50:09 AM Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2010 by Taylor and Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S Government works Printed in the United States of America on acid-free paper 10 International Standard Book Number-13: 978-1-4398-1140-5 (Ebook-PDF) This book contains information obtained from authentic and highly regarded sources Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint Except as permitted under U.S Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers For permission to photocopy or use material electronically from this work, please access www.copyright com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400 CCC is a not-for-profit organization that provides licenses and registration for a variety of users For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com For Lia Contents Foreword Preface to the Second Edition Preface to the First Edition xi xiii xv Chapter Ordinary Differential Equations: Brief Review 1.1 First-Order Equations 1.2 Homogeneous Linear Equations with Constant Coefficients 1.3 Nonhomogeneous Linear Equations with Constant Coefficients 1.4 Cauchy–Euler Equations 1.5 Functions and Operators Exercises Chapter Fourier Series 11 2.1 The Full Fourier Series 2.2 Fourier Sine Series 11 17 2.3 Fourier Cosine Series 2.4 Convergence and Differentiation 21 23 Exercises 24 Chapter Sturm–Liouville Problems 27 3.1 Regular Sturm–Liouville Problems 27 3.2 Other Problems 39 3.3 Bessel Functions 3.4 Legendre Polynomials 41 47 3.5 Spherical Harmonics Exercises 50 54 Chapter Some Fundamental Equations of Mathematical Physics 59 4.1 The Heat Equation 59 4.2 The Laplace Equation 4.3 The Wave Equation 67 73 4.4 Other Equations Exercises 78 81 viii Chapter The Method of Separation of Variables 5.1 The Heat Equation 5.2 The Wave Equation 83 83 95 5.3 The Laplace Equation 5.4 Other Equations 101 109 5.5 Equations with More than Two Variables Exercises 113 124 Chapter Linear Nonhomogeneous Problems 131 6.1 Equilibrium Solutions 131 6.2 Nonhomogeneous Problems Exercises 136 140 Chapter The Method of Eigenfunction Expansion 143 7.1 The Heat Equation 7.2 The Wave Equation 143 149 7.3 The Laplace Equation 7.4 Other Equations 152 155 Exercises Chapter The Fourier Transformations 159 165 8.1 The Full Fourier Transformation 8.2 The Fourier Sine and Cosine Transformations 165 172 8.3 Other Applications Exercises 179 181 Chapter The Laplace Transformation 9.1 Definition and Properties 9.2 Applications Exercises 187 187 192 202 Chapter 10 The Method of Green’s Functions 205 10.1 The Heat Equation 10.2 The Laplace Equation 205 213 10.3 The Wave Equation Exercises 217 223 ix Chapter 11 General Second-Order Linear Partial Differential Equations with Two Independent Variables 11.1 The Canonical Form 11.2 Hyperbolic Equations 227 227 231 11.3 Parabolic Equations 235 11.4 Elliptic Equations Exercises 238 239 Chapter 12 The Method of Characteristics 241 12.1 First-Order Linear Equations 241 12.2 First-Order Quasilinear Equations 12.3 The One-Dimensional Wave Equation 248 249 12.4 Other Hyperbolic Equations 256 Exercises 260 Chapter 13 Perturbation and Asymptotic Methods 263 13.1 Asymptotic Series 13.2 Regular Perturbation Problems 263 266 13.3 Singular Perturbation Problems Exercises 274 280 Chapter 14 Complex Variable Methods 285 14.1 Elliptic Equations 14.2 Systems of Equations 285 291 Exercises 294 Answers to Odd-Numbered Exercises 297 Appendix 313 Bibliography 319 Index 321 ANSWERS (43) u(x, y) = (1/π) (45) u(x, y) = (4/π) 307 ∞ (ω + 4ω)−1 {csch ω[−2(ω + 4) sinh(ωx) ∞ + 2ω sinh(ω(x − 1))] + 2(ω + 4)x} sin(ωy) dω ω −3 sin (ω/2) × {−1 + csch ω[sinh(ωx) + (2ω − 1) sinh(ω(x − 1))]} sin(ωy) dω (47) u(x, y) = (x2 − y)e−y (49) u(x, y) = (y + 1)ex−2y (51) u(x, y) = (2/π) (53) u(x, y) = (2/π) ∞ (ω + ω )−1 [(1 − ω ) csch ω sinh(ω(x − 1)) ∞ + (ω + 1)(1 − x)] cos(ωy) dω ω −3 (ω − sin ω) ×{csch ω[sinh(ωx)−sinh(ω(x−1))]−1} cos(ωy) dω (55) u(x, y) = (2xy + y )e−2y (57) u(x, y) = (x2 + x − 1)e−y CHAPTER (1) 3/(s2 + 2s + 10) − 72/s5 (3) (s − 4)/(s2 + 4s + 8) − 2((s2 + 2s + 2)/s3 )e−s (5) et [2 cos(5t) + (3/5) sin(5t)] √ √ (7) (3/(2 2π))t−3/2 e−1/(8t) + erfc(1/(2 2t )) √ (9) u(x, t) = t + erfc(x/(2 t )) + e−4t sin(2x) (11) u(x, t) = e−3t + 2t(x − 1)2 H(1 − x) (13) u(x, t) = (2t − 1) cos x (15) u(x, t) = e−x [sin(2t) − cos(2t)] √ (17) u(x, t) = − e−3t + erfc(x/(4 t )) (19) u(x, t) = (1/2)[t2 − 2t − (x2 − 2xt + t2 + 4x − 4t)H(t − x)] (21) u(x, t) = (1/9)(e−3t − 6t − 1) − (1/2)(x − 2t)2 H(t − x/2) (23) u(x, t) = (2 − e−2t )e−x (25) u(x, t) = 4e−t/2 − sin(2x) (27) u(x, t) = t2 + t − 2H(t − x/2) sin(t − x/2) √ (29) u(x, t) = −2ex erfc(x/(2 t )) (31) u(x, t) = (3t + 1) sin x (33) u(x, t) = − 2(1 + t)e−t + (1 + t − x)(e−x + 2e−t )H(t − x) (35) u(x, t) = (t − 2)e−2x 308 ANSWERS CHAPTER 10 (1) G(x, ξ) = x(1 − ξ)/2, ξ(1 − x)/2, (3) G(x, ξ) = x, ξ, (5) G(x, ξ) = x/2, ξ/2, (7) G(x, ξ) = (1 − ξ)/3, (1 − x)/3, (9) G(x, ξ) = − ξ, − x, x ≤ ξ, u(x) = (1/12)(12 − 37x + 3x2 − 2x3 ) x > ξ, x ≤ ξ, u(x) = + 5x − 2x2 x > ξ, ∞ (11) G(x, t; ξ, τ ) = n=1 ∞ u(x, t) = ∞ n=1 x ≤ ξ, u(x) = x > ξ, 2e−2n e−n π (t−τ ) 59/8 − 2x − x2 , − 9x/2 + 3x2 /2, t + 2t − 1) sin(πx) − e−8π t sin(2πx) π (t−τ )/4 sin(nπx/2) sin(nπξ/2), n=1 π t/4 ∞ n=1 2e−n ∞ π (t−τ ) ∞ (17) G(x, t; ξ, τ ) = n=1 e n=1 ∞ n=1 2e −(2n−1)2 π (t−τ )/4 −(2n−1)2 π t/8 π t/8 ]} sin((2n−1)πx/4) −[8/((2n − 1)3 π )]{(−1)n (2n − 1)2 π − [2 − (2n − 1)2 π − 2e(2n−1) (21) G(x, y; ξ, η) = ∞ ∞ m=1 n=1 2 π t/4 ] sin((2n − 1)π/4)} −(2n−1)2 π t/4 cos((2n−1)πx/2) −[8/((m + 4n )π )] sin(nπx) sin(mπy/2) × sin(nπξ) sin(mπη/2), [(−1)n − 1]2[n(n2 + 1)π ]−1 sin(nπx) sin(πy) n=1 (23) G(x, y; ξ, η) = u(x, y) = cos(nπx) cos((2n − 1)πx/2) cos((2n − 1)πξ/2), ×e ∞ π2 t × {(2n − 1)4 π − 8[8 + ((2n − 1)2 π t − 8)e(2n−1) (19) G(x, t; ξ, τ ) = ∞ ]e−n sin((2n − 1)πx/4) sin((2n − 1)πξ/4), ×e u(x, y) = π2 t (4/((2n − 1)5 π )) ∞ sin(nπx/2) [(−1)n − 1](2/(n6 π )) n=1 −(2n−1)2 π (t−τ )/8 n=1 π t/4 cos(nπx) cos(nπξ), × [2 + n4 π + 2(n2 π t − 1)en u(x, t) = − 1) ×[1+(−1)n −2 cos(nπ/2)]}e−n u(x, t) = 1/2 + t2 /2 + ∞ ≤ x ≤ 1/2, 1/2 < x ≤ sin(nπx) sin(nπξ), (2/(n3 π )){[1 + (−1)n ]n2 π + 4(en (15) G(x, t; ξ, τ ) = + u(x, t) = ≤ x ≤ 1, < x ≤ x ≤ ξ, u(x) = (1/3)(20 − 9x + x2 ) x > ξ, u(x, t) = (1/(4π ))(e−2π (13) G(x, t; ξ, τ ) = x2 /4 + 7x/2 − 2, −x2 /2 + 5x − 11/4, x ≤ ξ, u(x) = x > ξ, ∞ ∞ m=1 n=1 −[16/((4m2 + 4n2 − 4m + 1)π )] ×sin(nπx) sin((2m−1)πy/2) sin(nπξ) sin((2m−1)πη/2), (−1) 8[n(4n2 + 9)π ]−1 sin(nπx) sin(3πy/2) n=1 n ANSWERS (25) G(x, y; ξ, η) = u(x, y) = ∞ n=1 (27) G(x, y; ξ, η) = u(x, y) = ∞ ∞ ∞ m=0 n=1 −[32/((16m2 + 4n2 − 4n + 1)π )] ×sin((2n−1)πx/4) cos(mπy) sin((2n−1)πξ/4) cos(mπη), −128[(2n−1)(4n2 −4n+145)π ]−1 sin((2n−1)πx/4) cos(3πy) ∞ ∞ m=1 n=1 ∞ m=1 n=1 (29) G(x, y; ξ, η) = 309 ∞ −(4/((m2 + n2 )π )) × sin(nπx) sin(mπy) sin(nπξ) sin(mπη), −[(−1)m − 1][(−1)n − 1]4[mn(m2 + n2 )π ]−1 × sin(nπx) sin(mπy) ∞ m=1 n=1 −[8/((2m2 + 2n2 − 2m − 2n + 1)π )] ×cos((2n−1)πx/2) sin((2m−1)πy/2) cos((2n−1)πξ/2) sin((2m−1)πη/2), u(x, y) = ∞ ∞ m=1 n=1 32[2 + (−1)n (2n − 1)π] × [(2m − 1)(2n − 1)2 (2m2 + 2n2 − 2m − 2n + 1)π ]−1 × cos((2n − 1)πx/2) sin((2m − 1)πy/2) (31) u(−1, 2) = −1/3 (37) u(−3, 4) = 8/3 (33) u(1, 2) = −27/32 (35) u(3, 2) = 1/3 (39) u(4, 2) = 133/24 CHAPTER 11 (1) {(x, y) : −1 < x < 3}: elliptic; {(x, y) : x < −1 or x > 3}: hyperbolic; {(x, y) : x = −1 or x = 3}: parabolic (3) {(x, y) : |x| < 2|y|}: elliptic; {(x, y) : |x| > 2|y|}: hyperbolic; {(x, y) : |x| = 2|y|}: parabolic (5) {(x, y) : (x + 1)2 + (y − 2)2 < 1}: elliptic; {(x, y) : (x + 1)2 + (y − 2)2 > 1}: hyperbolic; {(x, y) : (x + 1)2 + (y − 2)2 = 1}: parabolic (7) vrs = 2r, u(x, y) = (3x + y)2 (x + 2y) + ϕ(3x + y) + ψ(x + 2y) (9) vrs = 2e2s , u(x, y) = (x + 3y)e−2x+2y + ϕ(x + 3y) + ψ(y − x) (11) vrs + 3vs = 0, u(x, y) = ϕ(3y − 2x)e−3(x+2y) + ψ(x + 2y) (13) vrs − 2vs = −r, u(x, y) = (1/4)(y − x) + (1/2)(2x + 3y)(y − x) + ϕ(2x + 3y) + ψ(y − x)e4x+6y (15) 2vrs −vs = 2−r−2s, u(x, y) = (4x−3y)(3x−y)+ϕ(2y−x)+ψ(y−3x)ey−x/2 (17) vss + 4v = 1, u(x, y) = 1/4 + ϕ(y − 4x) cos(2y) + ψ(y − 4x) sin(2y) (19) vss − 3vs + 2v = −2rs, u(x, y) = (1/2)(2y + 3)(3x − 5y) + ϕ(5y − 3x)ey + ψ(5y − 3x)e2y (21) vss −vs −2v = r+s, u(x, y) = (1/4)(1+2x−6y)+ϕ(2y−x)e−y +ψ(2y−x)e2y 310 ANSWERS (23) wαα + wββ + 5wα − 2wβ = (25) wαα + wββ + 4wα − wβ − 3w = −α − β (27) Parabolic; vss − vs = −3r, u(x, y) = − 12x + 8y − (1 − 9x + 6y)ey − 9xy + 6y (29) Hyperbolic; vrs = 2r+4s, u(x, y) = (3y−10x)(8x2 −6xy+y +1)+3y−10x (31) Hyperbolic; 2(2r − s + 1)vrs − 2vs + sv = s − r (33) Parabolic; s2 vss + (2r/s − r)vr + 2vs = s − ln(r/s) (35) Hyperbolic; 2(2r − s − 16)vrs − 2vs + (2r + 2)v = rs CHAPTER 12 (1) x = 2t + x0 , u(x, t) = − x + 2t + t2 /2 (3) x = t2 + t + x0 , u(x, t) = e2t sin(x − t2 − t) (5) x = x0 et , u(x, t) = x2 e−2t − e−t + (7) x = −3t + x0 , u(x, t) = −xt − 3t2 /2 + t + e−2(x+3t) (9) x = 4t + x0 , u(x, t) = [x − 4t + + cos(2x − 8t)]et − x − t − (11) x = t/2 + x0 , (t − t0 )/2, (13) x = 2t + x0 , 2(t − t0 ), x ≥ t/2, u(x, t) = x < t/2, x2 t − 2xt2 + 4t3 /3 + x − 2t, x3 /6 + x2 /4 − xt + t2 + 1, t + x0 , x ≥ t, t − t0 , x < t, u(x, t) = x ≥ t/2, x < t/2 x ≥ 2t, x < 2t, u(x, t) = (15) x = ex+t/2 , e2x , x > 2t, x < 2t [x − t + + cos(x − t)]et − x − 1, 2(t − x + 1)ex − x − 1, x ≥ t, x < t (17) x = 2y + x0 , u(x, y) = (x − 2y + 1)e(x+y−1)/3 (19) x = y/2 + x0 , u(x, y) = (1/4)[(2x − y + 1)e(2x+3y−2)/2 + 2y + 1] (21) x = −2y + x0 , u(x, y) = −(1/9)(2x2 + 8xy + 8y + 7x + 14y − 31)e(y−x+2)/3 (23) x = (2t + 1)x0 − 3t2 − t, u(x, t) = (3t2 + 4t − x + 1)/(2t + 1) (25) x = t2 + t + (t + 1)x0 , u(x, t) = (x + t + 1)/(t + 1) (27) x = t + x0 , u(x, t) = −1/(ex−t + t2 ) (29) u(x, t) = x2 t + t3 /3 + 2x + t + (31) u(x, t) = (1/2)et−2x + (1/2)e−t−2x − x2 t − t3 /12 + xt +x+y −1 ANSWERS 311 (33) u(x, t) = x2 + 2xt + t2 + t (35) u(x, t) = x2 + 4xt − 11t2 − x − 2t (37) u(x, t) = (39) u(x, t) = (41) u(x, t) = (43) u(x, t) = 8xt, ≤ x < t, 2x2 + 4xt + 2t2 − x + t, x ≥ t −6xt + (1/3) cos x cos(3t), 2 − x − 9t + (1/3) cos x sin(3t), 2x /9 + 2xt/3 + x, 2xt − 2t + x + 1, x > 3t ≤ x < t, x ≥ t x + 2xt + 3t, x > 3t ≤ x < 3t, x + 2xt + 3x, ≤ x < 3t, CHAPTER 13 (1) u(r) = (1/4)(3 + r ) + (1/16)ε(29 − 12r − r ) + O(ε2 ) (3) u(r) = −3 + (13/2)ε(r − 1) + O(ε2 ) (5) u(x, t) = e−t sin x − εte−t (cos x + sin x) + O(ε2 ) (7) u(x, t) = e−2t−x + (1/2)ε(1 − e−2t + 6te−2t−x ) + O(ε2 ) (9) u(x, t) = x − 3t + 2εt(x − 3t) + O(ε2 ) (11) u(x, t) = (2 + 2t − x)et − + ε[(xt − 2t2 − t + 2)et − 2] + O(ε2 ) (13) u(x, t) = ex−2t + t2 − εtex−2t + O(ε2 ) (15) u(x, y) = xy + εy(ex − 2x) + O(ε2 ) (17) u(x, y) = −2x2 + εy cos x + O(ε2 ) (19) u(x, y) = 3xy/2 − x2 y/2 +(1/24)ε(−15x+18x3 −3x4 −14xy +18x2 y −4x3 y)+O(ε2 ) (21) u(x, y) = 6x + 2y + 3εyex + O(ε2 ) (23) u(x, y) = 2x − y − e−x + 2εxe−x + O(ε2 ) (25) u(x, y) = x(sin y − cos y) + (1/2)εx[y cos y + (y − 1) sin y] + O(ε2 ) (27) u(x, y) = sin(2x)[2 cos(3y) − sin(3y) − 1] −(1/18)ε sin(2x)[(3y +2) cos(3y)+(6y −1) sin(3y)−2]+O(ε2 ) (29) u(x, y) = e−2x+2y − (3/16)εe−2x [e−2y + (4y − 1)e2y ] + O(ε2 ) (31) u(x, y) = ex+2y − εex [ey + (y − 1)e2y ] + O(ε2 ) (33) u(x, y) = (2x − 1)e2y + (3/16)ε(2x − 1)[(1 − 4y)e2y − e−2y ] + O(ε2 ) 312 ANSWERS (35) uc (x, y) = 2ye1−x + 2ε(xy − 2x − y + 2)e1−x + [1 − x + (1 − 2e)y − (1 − 2e)xy + 2εe(y − 2)]e−x/ε + O(ε2 ) (37) uc (x, y) = 2y + + 2εx + (3xy + 3x − 9y − − 4ε)e−(2−x)/ε + O(ε2 ) (39) uc (x, y) = (1 − y)ex−1 + εy(1 − x)ex−1 + (exy + xy + ey − x + y − − εy)e−1−x/ε + O(ε2 ) (41) uc (x, y) = 2x − y − + (x2 + xy − 2x − y + 3)e−2y/ε + (4xy + y + 2x + y + 2)e−x/ε + O(ε2 ) (43) uc (x, y) = 2x2 + 4xy + 2y − + 12εy + (−2x2 + 4xy − 7x + 3y − − 12ε)e−(1−y)/(2ε) + (2xy − y + − 12εy)e−x/ε + O(ε2 ) (45) uc (x, y) = x2 − 4xy + 4y + 4x − 8y + + 34ε(1 − y) − (x2 + 4xy + 3x + 6y + + 34ε)e−y/(4ε) +[−4xy−4y +3x+6y−1+34ε(y−1)]e−2x/ε +O(ε2 ) CHAPTER 14 (1) u(x, y) = x2 − 4y + (3) u(x, y) = −2x2 + 3xy + y + y − (5) u(x, y) = 3x2 − 4xy − y + 2x − 3y (7) u(x, y) = 2x2 − xy + y (9) u(x, y) = x2 + 4xy − 2y − y (11) u(x, y) = x2 − 3xy + 3y + 2x − (13) u(x, y) = x2 − 2xy + 3y + x − 2y + (15) u1 (x, y) = −x2 + y + 2x, u2 (x, y) = 2xy + x (17) u1 (x, y) = 4xy − x + 2, u2 (x, y) = (1/5)(−14x2 − 4y + 14) (19) u1 (x, y) = 2x2 − 2xy − 2y + 1, u2 (x, y) = (1/7)(x2 − 28xy + y + 7x − 1) (21) u1 (x, y) = (1/7)(3x2 + 14xy + 3y + 7y + 11), u2 (x, y) = (1/7)(27x2 − 14xy − 15y − 6) (23) u1 (x, y) = (1/3)(x2 − 6xy − 5y + 2), u2 (x, y) = (1/3)(−8x2 + 4y + 3y + 5) Appendix A1 Useful Integrals For all m, n = 1, 2, , L nπx dx = 0; cos L L sin −L nπx dx = 0; L sin mπx nπx sin dx = L L 0, n = m, L/2, n = m; cos mπx nπx cos dx = L L 0, n = m, L/2, n = m; sin mπx nπx cos dx = 0; L L sin (2m − 1)πx (2n − 1)πx sin dx = 2L 2L 0, n = m, L/2, n = m; cos (2m − 1)πx (2n − 1)πx cos dx = 2L 2L 0, n = m, L/2, n = m L L −L L L L For all real numbers a, b, c, and p = 0, (ax2 + bx + c) cos(px) dx = 1 (2ax + b) cos(px) + p2 (ax2 + bx + c) − 2a sin(px) + const; p p (ax2 + bx + c) sin(px) dx 1 p (ax2 + bx + c) − 2a cos(px) + (2ax + b) sin(px) + const; p p ax e eax cos(px) dx = a cos(px) + p sin(px) + const; a + p2 eax eax sin(px) dx = − p cos(px) + a sin(px) + const a + p2 =− 314 APPENDIX A2 Table of Fourier Transforms f (x) = F −1 [F ](x) F (ω) = F [f ](ω) f (x) −iωF (ω) f (x) −ω F (ω) f (ax + b) (a > 0) (f ∗ g)(x) δ(x) √ 2π eiax f (x) F (ω + a) e−a xe−a x2 e−a 10 x2 + a2 (a > 0) 11 x2 x + a2 (a > 0) −i(b/a)ω e F (ω/a) a F (ω)G(ω) 2 √ e−ω /(4a ) 2a x2 x2 i 2 √ ωe−ω /(4a ) 2a (a > 0) x2 2 √ (2a2 − ω )e−ω /(4a ) a5 (a > 0) π −a|ω| e a −i 1, |x| ≤ a 0, |x| > a 12 H(a − |x|) = 13 xH(a − |x|) = x, 0, |x| ≤ a |x| > a 14 e−a|x| 15 e−(x+b) 16 erf(ax) /(4a) + e−(x−b) /(4a) π ωe−a|ω| 2a sin(aω) π ω i sin(aω) − aω cos(aω) π ω2 a 2 π a + ω2 √ 2 2ae−aω cos(bω) i −ω2 /(4a2 ) e πω APPENDIX 315 A3 Table of Fourier Sine Transforms f (x) = FS−1 [F ](x) F (ω) = FS [f ](ω) f (x) −ωFC [f ](ω) f (x) f (ax) (a > 0) F (ω/a) a f (ax) cos(bx) (a, b > 0) ω+b F 2a a e−ax xe−ax x2 e−ax x x2 + a2 ωf (0) − ω F (ω) π +F ω−b a πω ω π a2 + ω (a > 0) 2aω 2 π (a + ω )2 (a > 0) (a > 0) 10 H(a − x) = 11 xH(a − |x|) = π −aω e 1, ≤ x ≤ a, 0, x > a x, |x| ≤ a 0, |x| > a 12 erfc(ax) (a > 0) 13 xe−a 3a2 ω − ω π (a2 + ω )3 x2 14 tan−1 (x/a) (a > 0) 1 − cos(aω) πω sin(aω) − aω cos(aω) π ω2 2 − e−ω /(4a ) πω 1 2 √ ωe−ω /(4a ) a 2 π −aω e ω 316 APPENDIX A4 Table of Fourier Cosine Transforms f (x) = FC−1 [F ](x) F (ω) = FC [f ](ω) f (x) − f (0) + ωFS [f ](ω) π f (x) − f (0) − ω F (ω) π f (ax) (a > 0) F (ω/a) a f (ax) cos(bx) (a, b > 0) ω+b F 2a a e−ax xe−ax x2 e−ax e−a x2 + a2 10 (a2 + x2 )3 (a > 0) π (a2 ω + 3aω + 3)e−aω 8a5 11 x2 (a2 + x2 )3 (a > 0) π (−a2 ω + aω + 1)e−aω 8a3 12 x4 (a2 + x2 )3 (a > 0) π 2 (a ω − 5aω + 3)e−aω 8a 2 a2 − ω π (a2 + ω )2 (a > 0) (a > 0) 2 a3 − 3aω π (a2 + ω )3 1 −ω2 /(4a2 ) √ e |a| x2 π −aω e a (a > 0) 1, ≤ x ≤ a, 0, x > a (1/b)e−bx cosh(ab), (1/b)e ω−b a a 2 π a + ω2 (a > 0) 13 H(a − x) = 14 +F −ab x ≥ a, cosh(bx), x < a (a, b > 0) sin(aω) πω cos(aω) π b2 + ω APPENDIX 317 A5 Table of Laplace Transforms f (t) = L−1 [F ](t) F (s) = L[f ](s) f (n) (t) sn F (s) − sn−1 f (0) − · · · (nth derivative) − f (n−1) (0) H(t − a)f (t − a) e−as F (s) eat f (t) F (s − a) (f ∗ g)(t) F (s)G(s) 1 s tn eat sin(at) cos(at) (n positive integer) (s > 0) n! sn+1 (s > a) s−a a (s > 0) s2 + a s (s > 0) s + a2 a (s > |a|) s − a2 s (s > |a|) s2 − a 10 sinh(at) 11 cosh(at) 12 δ(t − a) (a ≥ 0) e−as √ 13 ea t erfc a t √ s+a s 14 (s > 0) (a > 0) a √ t−3/2 e−a /(4t) π 15 erfc a √ t 16 −a t −a2 /(4t) e + π (a > 0) −a√s e s (a > 0) √ s e−a a a2 + t) erfc √ t (a > 0) −a√s e s2 318 APPENDIX A6 Second-Order Linear Equations If Auxx + Buxy + Cuyy + Dux + Euy + F u = G, if new variables r = r(x, y), s = s(x, y) are defined by the characteristic equations B− dy = dx √ B − 4AC , 2A dy B+ = dx √ B − 4AC , 2A and if u(x, y) = u x(r, s), y(r, s) = v(r, s), then ¯ rr + Bv ¯ rs + Cv ¯ ss + Dv ¯ r + Ev ¯ s + F¯ v = G, ¯ Av where A¯ = A(rx )2 + Brx ry + C(ry )2 , ¯ = 2Arx sx + B(rx sy + ry sx ) + 2Cry sy , B C¯ = A(sx )2 + Bsx sy + C(sy )2 , ¯ = Arxx + Brxy + Cryy + Drx + Ery , D ¯ = Asxx + Bsxy + Csyy + Dsx + Esy , E F¯ = F, ¯ = G G Bibliography L.C Andrews, Elementary Partial Differential Equations with Boundary Value Problems, Academic Press, Orlando, FL, 1986 J.W Brown and R.V Churchill, Fourier Series and Boundary Value Problems, 6th ed., McGraw–Hill, New York, NY, 2000 P DuChateau and D Zachmann, Applied Partial Differential Equations, Harper & Row, New York, NY, 1989 D.G Duffy, Solutions of Partial Differential Equations, TAB Books, Blue Ridge Summit, PA, 1986 S.J Farlow, Partial Differential Equations for Scientists and Engineers, Dover, New York, NY, 1993 P.R Garabedian, Partial Differential Equations, 2nd ed., American Mathematical Society, Providence, RI, 1999 K.E Gustafson, Introduction to Partial Differential Equations and Hilbert Space Methods, 3rd ed., Dover, Mineola, NY, 1999 R Haberman, Elementary Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, 3rd ed., Prentice Hall, Upper Saddle River, NJ, 1998 M.K Keane, A Very Applied First Course in Partial Differential Equations, Prentice Hall, Upper Saddle River, NJ, 2002 J Kevorkian and J.D Cole, Perturbation Methods in Applied Mathematics, Springer, New York, NY, 1985 L.D Kovach, Boundary–Value Problems, Addison–Wesley, Reading, MA, 1984 J.A Leach and D.J Needham, Matched Asymptotic Expansions in Reaction–Diffusion Theory, Springer, London, 2004 T Myint-U and L Debnath, Linear Partial Differential Equations for Scientists and Engineers, 4th ed., Birkhă auser, Boston, MA, 2007 320 BIBLIOGRAPHY A.H Nayfeh, Introduction to Perturbation Techniques, Wiley, New York, NY, 1981 I Stakgold, Green’s Functions and Boundary Value Problems, Wiley–Interscience, New York, NY, 1979 D Vvedensky, Partial Differential Equations with Mathematica, Addison– Wesley, Reading, MA, 1993 F.Y.M Wan, Mathematical Models and Their Analysis, Harper & Row, New York, NY, 1989 E Zauderer, Partial Differential Equations of Applied Mathematics, 3rd ed., Wiley–Interscience, New York, NY, 2006 .. .Solution Techniques for Elementary Partial Differential Equations Second Edition K10569_FM.indd 4/28/10 9:50:09 AM K10569_FM.indd 4/28/10 9:50:09 AM Solution Techniques for Elementary Partial. .. the classroom Such is the case with Solution Techniques for Elementary Partial Differential Equations by Christian Constanda The author, a skilled classroom performer with considerable experience,... fundamentals of partial differential equations, I have made a concession when it comes to exam- xiv PREFACE TO THE SECOND EDITION ples and exercises involving special functions, transcendental equations,