实用偏微分方程（英文版·原书第5版）
| 作者 |
| [美]理查德·哈伯曼（Richard Haberman） |
| 丛书名 |
| 华章数学原版精品系列 |
| 出版社 |
| 机械工业出版社 |
| ISBN |
| 9787111648611 |
| 简要 |
| 简介 |
| 内容简介书籍数学书籍 本书讨论偏微分方程在工程技术科学与自然科学中的应用，以傅里叶方法（傅里叶级数、傅里叶变换和拉普拉斯变换）作为讲授的主线，讲授的内容是高级工程数学、自然科学范畴的数学方法中非常重要的部分。 |
| 目录 |
| 第1章 热传导方程 1 1.1 引言 1 1.2 一维杆中热传导方程的推导 2 1.3 边界条件 11 1.4 平衡温度分布 14 1.4.1 给定温度 14 1.4.2 绝热边界 16 1.5 二维或三维热传导方程的推导 19 第2章 分离变量法 32 2.1 引言 32 2.2 线性性质 32 2.3 在有限端处具有零温度的热传导方程 35 2.3.1 概述 35 2.3.2 分离变量 35 2.3.3 时变常微分方程 37 2.3.4 边值问题 38 2.3.5 乘积解和叠加原理 43 2.3.6 正弦函数的正交性 46 2.3.7 实例 48 2.3.8 小结 50 2.4 有关热传导方程的例子：其他边值问题 55 2.4.1 绝热端杆中的热传导 55 2.4.2 细绝热圆环中的热传导 59 2.4.3 边值问题小结 64 2.5 拉普拉斯方程：求解和定性性质 67 2.5.1 矩形区域内的拉普拉斯方程 67 2.5.2 圆盘内的拉普拉斯方程 72 2.5.3 绕过圆柱体的流体流动（升力） 76 2.5.4 拉普拉斯方程的定性性质 79 第3章 傅里叶级数 86 3.1 引言 86 3.2 收敛定理 88 3.3 傅里叶余弦级数和傅里叶正弦级数 92 3.3.1 傅里叶正弦级数 92 3.3.2 傅里叶余弦级数 102 3.3.3 用正弦级数和余弦级数表示f(x) 105 3.3.4 偶部和奇部 106 3.3.5 连续傅里叶级数 107 3.4 傅里叶级数的逐项微分 112 3.5 傅里叶级数的逐项积分 123 3.6 傅里叶级数的复形式 127 第4章 波动方程：振动弦与振动膜 130 4.1 引言 130 4.2 弦振动方程的建立 130 4.3 边界条件 133 4.4 端点固定的振动弦 137 4.5 振动膜 143 4.6 电磁波与声波的反射与折射 145 4.6.1 斯涅耳折射定律 146 4.6.2 反射波与折射波的强度（振幅） 148 4.6.3 内部全反射 149 第5章 施图姆–刘维尔特征值问题 151 5.1 引言 151 5.2 例子 151 5.2.1 非均匀杆内的热流 151 5.2.2 圆对称热流 153 5.3 施图姆–刘维尔特征值问题 155 5.3.1 一般分类 155 5.3.2 正则施图姆–刘维尔特征值问题 156 5.3.3 定理的举例和说明 157 5.4 例子：非均匀杆中的无热源热流 163 5.5 自伴算子和施图姆–刘维尔特征值问题 167 5.6 瑞利商 184 5.7 例子：非均匀弦的振动 189 5.8 第三类边界条件 192 5.9 大特征值（渐近行为） 207 5.10 逼近性质 211 第6章 偏微分方程的有限差分数值法 217 6.1 引言 217 6.2 有限差分与截断泰勒级数 217 6.3 热传导方程 224 6.3.1 概述 224 6.3.2 偏差分方程 224 6.3.3 计算 226 6.3.4 傅里叶–冯·诺伊曼稳定性分析 228 6.3.5 偏差分方程的分离变量和常差分方程的解析解 235 6.3.6 矩阵记号 238 6.3.7 非齐次问题 242 6.3.8 其他数值格式 242 6.3.9 其他类型的边界条件 243 6.4 二维热传导方程 247 6.5 波动方程 250 6.6 拉普拉斯方程 253 6.7 有限元法 260 6.7.1 非正交函数逼近（偏微分方程的弱形式） 260 6.7.2 最简三角形有限元 263 第7章 高维偏微分方程 268 7.1 引言 268 7.2 时间变量的分离 269 7.2.1 振动膜：任意形状 269 7.2.2 热传导：任意区域 271 7.2.3 小结 272 7.3 振动矩形膜 272 7.4 特征值问题φ+φ= 0的定理叙述和说明 282 7.5 格林公式、自伴算子和多维特征值问题 287 7.6 瑞利商和拉普拉斯方程 293 7.6.1 瑞利商 293 7.6.2 依赖时间的热传导方程与拉普拉斯方程 294 7.7 振动圆形膜和贝塞尔函数 295 7.7.1 概述 295 7.7.2 分离变量 296 7.7.3 特征值问题（一维情形） 297 7.7.4 贝塞尔微分方程 299 7.7.5 奇异点和贝塞尔微分方程 299 7.7.6 贝塞尔函数及其渐近性质（在z=0附近） 301 7.7.7 涉及贝塞尔函数的特征值问题 302 7.7.8 振动圆形膜的初值问题 304 7.7.9 圆对称情形 305 7.8 贝塞尔函数的进一步讨论 312 7.8.1 贝塞尔函数的定性性质 312 7.8.2 特征值的渐近公式 313 7.8.3 贝塞尔函数的零点和结点曲线 314 7.8.4 贝塞尔函数的级数表示 316 7.9 圆柱体上的拉普拉斯方程 319 7.9.1 概述 319 7.9.2 分离变量 320 7.9.3 侧面及顶部或底部为零温度的情形 322 7.9.4 顶部和底部为零温度的情形 323 7.9.5 修正贝塞尔函数 326 7.10 球内的问题和勒让德多项式 330 7.10.1 概述 330 7.10.2 分离变量和一维特征值问题 330 7.10.3 连带勒让德函数和勒让德多项式 332 7.10.4 径向特征值问题 335 7.10.5 乘积解、振动模式和初值问题 335 7.10.6 球内部的拉普拉斯方程 336 第8章 非齐次问题 341 8.1 引言 341 8.2 有源热流与非齐次边界条件 341 8.3 带齐次边界条件的特征函数展开法（微分特征函数的级数） 347 8.4 利用格林公式的特征函数展开法（带或不带齐次边界条件） 353 8.5 受迫振动膜与共振 358 8.6 泊松方程 366 第9章 定常问题的格林函数 374 9.1 引言 374 9.2 一维热传导方程 374 9.3 常微分方程边值问题的格林函数 379 9.3.1 一维稳态热传导方程 379 9.3.2 参数变易法 379 9.3.3 格林函数的特征函数展开法 382 9.3.4 狄拉克δ函数及其与格林函数的关系 384 9.3.5 非齐次边界条件 391 9.3.6 小结 392 9.4 弗雷德霍姆择一性与广义格林函数 398 9.4.1 概述 398 9.4.2 弗雷德霍姆择一性 400 9.4.3 广义格林函数 402 9.5 泊松方程的格林函数 409 9.5.1 概述 409 9.5.2 多维狄拉克δ函数与格林函数 410 9.5.3 用特征函数展开法表示格林函数与弗雷德霍姆择一性 411 9.5.4 格林函数的直接解法（一维特征函数）（可选） 413 9.5.5 用格林函数解带非齐次边界条件的问题 415 9.5.6 无穷空间格林函数 416 9.5.7 用无穷空间格林函数得到有界区域的格林函数 419 9.5.8 用无穷空间格林函数求半无穷平面（y>0）的格林函数：像源法 420 9.5.9 圆的格林函数：像源法 423 9.6 扰动特征值问题 430 9.6.1 概述 430 9.6.2 数学例子 431 9.6.3 拟圆膜振动 432 9.7 小结 435 第10章 无穷域问题：偏微分方程的傅里叶变换解法 437 10.1 引言 437 10.2 无穷域上的热传导方程 437 10.3 傅里叶变换对 441 10.3.1 傅里叶级数恒等式的启示 441 10.3.2 傅里叶变换 442 10.3.3 高斯函数的傅里叶逆变换 443 10.4 傅里叶变换与热传导方程 450 10.4.1 热传导方程 450 10.4.2 傅里叶变换热传导方程：导数的变换 455 10.4.3 卷积定理 457 10.4.4 傅里叶变换性质小结 461 10.5 傅里叶正弦和余弦变换：半无穷区间上的热传导方程 463 10.5.1 概述 463 10.5.2 半无穷区间上的热传导方程Ⅰ 463 10.5.3 傅里叶正弦和余弦变换 465 10.5.4 导数的变换 466 10.5.5 半无穷区间上的热传导方程Ⅱ 467 10.5.6 傅里叶正弦和余弦变换表 469 10.6 应用变换求解的例子 473 10.6.1 无穷区间上的一维波动方程 473 10.6.2 半无穷带上的拉普拉斯方程 475 10.6.3 半平面上的拉普拉斯方程 479 10.6.4 四分之一平面上的拉普拉斯方程 482 10.6.5 平面上的热传导方程（二维傅里叶变换） 486 10.6.6 二重傅里叶变换表 490 10.7 散射和逆散射 495 第11章 波动方程和热传导方程的格林函数 499 11.1 引言 499 11.2 波动方程的格林函数 499 11.2.1 概述 499 11.2.2 格林公式 500 11.2.3 互反性 502 11.2.4 使用格林函数 504 11.2.5 波动方程的格林函数 506 11.2.6 格林函数的另一个微分方程 506 11.2.7 一维波动方程的无穷空间格林函数和达朗贝尔解 507 11.2.8 三维波动方程的无穷空间格林函数（惠更斯原理） 509 11.2.9 二维无穷空间格林函数 511 11.2.10 小结 511 11.3 热传导方程的格林函数 514 11.3.1 概述 514 11.3.2 热传导方程的非自伴特性 515 11.3.3 格林公式 516 11.3.4 伴随格林函数 517 11.3.5 互反性 518 11.3.6 用格林函数表示解 518 11.3.7 格林函数的另一个微分方程 520 11.3.8 扩散方程的无穷空间格林函数 521 11.3.9 热传导方程的格林函数（在半无穷域上） 522 11.3.10 热传导方程的格林函数（在有限区域上） 523 第12章 线性和拟线性波动方程的特征线法 527 12.1 引言 527 12.2 一阶波动方程的特征线 528 12.2.1 概述 528 12.2.2 一阶偏微分方程的特征线法 529 12.3 一维波动方程的特征线法 534 12.3.1 通解 534 12.3.2 初值问题（无穷区域） 536 12.3.3 达朗贝尔解 540 12.4 半无界弦和反射 543 12.5 定长振动弦的特征线法 548 12.6 拟线性偏微分方程的特征线法 552 12.6.1 特征线法 552 12.6.2 交通流量 553 12.6.3 特征线法(Q=0) 555 12.6.4 冲击波 558 12.6.5 拟线性举例 570 12.7 一阶非线性偏微分方程 575 12.7.1 由波动方程推导出的短时距方程 575 12.7.2 求解均匀介质中的短时距方程和反射波 576 12.7.3 一阶非线性偏微分方程 579 第13章 偏微分方程的拉普拉斯变换解法 581 13.1 引言 581 13.2 拉普拉斯变换的性质 581 13.2.1 概述 581 13.2.2 拉普拉斯变换的奇点 582 13.2.3 导数的变换 586 13.2.4 卷积定理 587 13.3 常微分方程初值问题的格林函数 591 13.4 波动方程的信号问题 593 13.5 有限长度振动弦的信号问题 597 13.6 波动方程及其格林函数 600 13.7 用复平面上的围线积分计算拉普拉斯逆变换 603 13.8 利用拉普拉斯变换求解波动方程（复变量） 608 第14章 色散波：缓变、稳定性、非线性性和扰动法 611 14.1 引言 611 14.2 色散波和群速度 612 14.2.1 行波和色散关系 612 14.2.2 群速度Ⅰ 615 14.3 波导 617 14.3.1 对f频率集中周期性源的响应 620 14.3.2 模式传播的格林函数 620 14.3.3 模式不传播的格林函数 621 14.3.4 设计思路 622 14.4 光纤 623 14.5 群速度Ⅱ和稳定相位法 627 14.5.1 稳定相位法 628 14.5.2 对线性色散波的应用 630 14.6 缓变色散波（群速度和焦散曲线） 634 14.6.1 色散偏微分方程的近似解 634 14.6.2 焦散曲线的形成 636 14.7 波包络方程（集中波数） 642 14.7.1 薛定谔方程 643 14.7.2 线性化KdV方程 645 14.7.3 非线性色散波：KdV方程 647 14.7.4 孤立子与逆散射 650 14.7.5 非线性薛定谔方程 652 14.8 稳定性和不稳定性 656 14.8.1 常微分方程和分歧理论简介 656 14.8.2 偏微分方程稳定平衡解的基本例子 663 14.8.3 偏微分方程的典型不稳定平衡点和模式形成 664 14.8.4 不适定问题 667 14.8.5 微不稳定色散波和线性化复金茨堡–朗道方程 668 14.8.6 非线性复金茨堡–朗道方程 670 14.8.7 长波的不稳定性 676 14.8.8 反应扩散方程的模式形成和图灵不稳定性 676 14.9 奇异扰动法：多尺度 683 14.9.1 常微分方程：弱非线性阻尼振子 683 14.9.2 常微分方程：缓变振子 686 14.9.3 固定空间域上的微不稳定偏微分方程 690 14.9.4 关于波动方程的缓变介质 692 14.9.5 缓变线性色散波（包括弱非线性作用） 695 14.10 奇异扰动法：匹配渐近展开的边界层法 700 14.10.1 常微分方程中的边界层 700 14.10.2 由对流支配的污染物扩散 705 参考文献 712 带*号习题的答案 716 Contents 1 Heat Equation 1 1.1Introduction .................................1 1.2 Derivation of the Conduction of Heat in a One-Dimensional Rod . . . . 2 1.3BoundaryConditions ............................11 1.4 EquilibriumTemperature Distribution ................... 14 1.4.1PrescribedTemperature .....................14 1.4.2InsulatedBoundaries .......................16 1.5 Derivation of the Heat Equation in Two or Three Dimensions ...... 19 2 Method of Separation of Variables 32 2.1Introduction .................................32 2.2 Linearity ................................... 32 2.3 Heat Equation with ZeroTemperatures atFinite Ends ......... 35 2.3.1Introduction ............................35 2.3.2SeparationofVariables ......................35 2.3.3 Time-Dependent OrdinaryDi.erential Equation ........ 37 2.3.4 BoundaryValue (Eigenvalue) Problem ............. 38 2.3.5 Product Solutions and the Principle of Superposition ..... 43 2.3.6OrthogonalityofSines ......................46 2.3.7 Formulation, Solution, and Interpretation of an Example . . . 48 2.3.8Summary.............................. 50 2.4 Worked Examples with the Heat Equation (Other Boundary ValueProblems) ...............................55 2.4.1 Heat Conduction in a Rod with Insulated Ends ........ 55 2.4.2 Heat Conduction in a Thin Insulated Circular Ring ...... 59 2.4.3 Summary of BoundaryValue Problems ............. 64 2.5 Laplace’s Equation: Solutions and Qualitative Properties ........ 67 2.5.1 Laplace’s Equation Inside a Rectangle ............. 67 2.5.2 Laplace’s Equation Inside a Circular Disk ........... 72 2.5.3 FluidFlow Outsidea Circular Cylinder (Lift) ......... 76 2.5.4 Qualitative Properties of Laplace’s Equation .......... 79 3 Fourier Series 86 3.1Introduction .................................86 3.2StatementofConvergenceTheorem ....................88 3.3 FourierCosineandSineSeries .......................92 3.3.1FourierSineSeries ........................92 3.3.2FourierCosineSeries .......................102 3.3.3 Representing f (x)by Both a Sine and Cosine Series . . . . . . 105 3.3.4EvenandOddParts .......................106 3.3.5ContinuousFourierSeries ....................107 3.4 Term-by-Term Di.erentiation ofFourier Series .............. 112 3.5 Term-By-Term Integration ofFourier Series ................ 123 3.6ComplexFormofFourierSeries ......................127 4 Wave Equation: Vibrating Strings and Membranes 130 4.1Introduction .................................130 4.2 DerivationofaVertically Vibrating String ................130 4.3BoundaryConditions ............................133 4.4 VibratingStringwithFixedEnds .....................137 4.5 VibratingMembrane............................. 143 4.6 Re.ection and Refraction of Electromagnetic (Light)andAcoustic(Sound)Waves .................... 145 4.6.1Snell’sLawofRefraction .....................146 4.6.2 Intensity (Amplitude) of Re.ected and RefractedWaves . . . 148 4.6.3TotalInternalRe.ection .....................149 5 Sturm–Liouville Eigenvalue Problems 151 5.1Introduction .................................151 5.2 Examples ................................... 151 5.2.1 HeatFlowinaNonuniformRod .................151 5.2.2 Circularly Symmetric HeatFlow .................153 5.3 Sturm–Liouville Eigenvalue Problems ...................155 5.3.1GeneralClassi.cation .......................155 5.3.2 Regular Sturm–Liouville Eigenvalue Problem .......... 156 5.3.3 Example and Illustration of Theorems ............. 157 5.4 Worked Example: Heat Flow in a Nonuniform Rodwithout Sources . . 163 5.5 Self-AdjointOperators and Sturm–Liouville Eigenvalue Problems . . . . 167 5.6RayleighQuotient .............................. 184 5.7 Worked Example: Vibrations of a Nonuniform String ........... 189 5.8 Boundary Conditionsofthe ThirdKind ..................192 5.9 Large Eigenvalues (Asymptotic Behavior) ................. 207 5.10ApproximationProperties .......................... 211 6 Finite Di.erence Numerical Methods for Partial Di.erential Equations 217 6.1Introduction .................................217 6.2 Finite Di.erences andTruncatedTaylor Series .............. 217 6.3HeatEquation ................................224 6.3.1 Introduction ............................224 6.3.2 APartial Di.erenceEquation ..................224 6.3.3 Computations........................... 226 6.3.4 Fourier–von Neumann Stability Analysis ............ 228 6.3.5 Separation ofVariables forPartial Di.erence Equations and Analytic Solutions of OrdinaryDi.erence Equations . . . 235 6.3.6 MatrixNotation .......................... 238 6.3.7 NonhomogeneousProblems ...................242 6.3.8 OtherNumericalSchemes ....................242 6.3.9 OtherTypes of Boundary Conditions .............. 243 6.4Two-DimensionalHeatEquation ......................247 6.5WaveEquation ................................250 6.6Laplace’sEquation ..............................253 6.7 FiniteElementMethod ........................... 260 6.7.1 Approximation with NonorthogonalFunctions (WeakForm of thePartial Di.erential Equation) ....... 260 6.7.2 The SimplestTriangular Finite Elements ............ 263 7 Higher-Dimensional Partial Di.erential Equations 268 7.1Introduction .................................268 7.2SeparationoftheTimeVariable ......................269 7.2.1 VibratingMembrane:AnyShape ................269 7.2.2 HeatConduction:AnyRegion ..................271 7.2.3 Summary.............................. 272 7.3 VibratingRectangularMembrane .....................272 7.4 Statements and Illustrations of Theorems for the Eigenvalue Problem .2φ +λφ =0 ...........................282 7.5 Green’sFormula, Self-AdjointOperators, and Multidimensional EigenvalueProblems............................. 287 7.6 Rayleigh Quotient and Laplace’s Equation ................ 293 7.6.1 RayleighQuotient ......................... 293 7.6.2 Time-Dependent Heat Equation and Laplace’s Equation . . . 294 7.7 Vibrating Circular Membrane and BesselFunctions ........... 295 7.7.1 Introduction ............................295 7.7.2 SeparationofVariables ......................296 7.7.3 Eigenvalue Problems (One-Dimensional) ............ 297 7.7.4 Bessel’s Di.erentialEquation ..................299 7.7.5 SingularPoints and Bessel’s Di.erential Equation ....... 299 7.7.6 BesselFunctions and Their Asymptotic Properties (Near z =0) ............................301 7.7.7 Eigenvalue Problem Involving BesselFunctions ........ 302 7.7.8 InitialValue Problem fora Vibrating Circular Membrane. . . 304 7.7.9 CircularlySymmetricCase ....................305 7.8 MoreonBesselFunctions .......................... 312 7.8.1 Qualitative Properties of BesselFunctions ........... 312 7.8.2 AsymptoticFormulas for the Eigenvalues ............ 313 7.8.3 Zeros of BesselFunctions and Nodal Curves .......... 314 7.8.4 Series Representation of BesselFunctions ............ 316 7.9 Laplace’s Equationina Circular Cylinder .................319 7.9.1 Introduction ............................319 7.9.2 SeparationofVariables ......................320 7.9.3 ZeroTemperature on the Lateral Sides and on the Bottom orTop ............................... 322 7.9.4 ZeroTemperature on theTop and Bottom ........... 323 7.9.5 Modi.edBesselFunctions ....................326 7.10 Spherical Problems and LegendrePolynomials .............. 330 7.10.1 Introduction ............................330 7.10.2 Separation ofVariables and One-Dimensional Eigenvalue Problems.............................. 330 7.10.3 Associated LegendreFunctions and LegendrePolynomials . . 332 7.10.4 RadialEigenvalueProblems ...................335 7.10.5 Product Solutions, Modes of Vibration, and the Initial ValueProblem........................... 335 7.10.6 Laplace’s Equation Inside a Spherical Cavity .......... 336 8 Nonhomogeneous Problems 341 8.1 Introduction .................................341 8.2 Heat Flow with Sources and Nonhomogeneous BoundaryConditions ............................341 8.3 Method of Eigenfunction Expansion with Homogeneous Boundary Conditions (Di.erentiating Series of Eigenfunctions) ........... 347 8.4 Method of Eigenfunction Expansion Using Green’sFormula (With or Without Homogeneous Boundary Conditions) ......... 353 8.5 Forced Vibrating Membranes and Resonance ............... 358 8.6 Poisson’sEquation ..............................366 9 Green’s Functions for Time-Independent Problems 374 9.1 Introduction .................................374 9.2 One-Dimensional Heat Equation (Optional) ................ 374 9.3 Green’sFunctions for BoundaryValue Problems for Ordinary Di.erentialEquations ............................379 9.3.1 One-Dimensional Steady-State Heat Equation ......... 379 9.3.2 The MethodofVariation ofParameters (Optional) ...... 379 9.3.3 The Methodof Eigenfunction Expansion for Green’s Functions .............................382 9.3.4 TheDirac DeltaFunction and Its Relationship to Green’s Functions .............................384 9.3.5 Nonhomogeneous Boundary Conditions ............. 391 9.3.6 Limitof Time-Dependent Problem ...............392 9.4 Fredholm Alternative and Generalized Green’sFunctions ........ 398 9.4.1 Introduction ............................398 9.4.2 FredholmAlternative ....................... 400 9.4.3 Generalized Green’sFunctions ..................402 9.5 Green’sFunctions forPoisson’s Equation ................. 409 9.5.1 Introduction ............................409 9.5.2 Multidimensional DiracDeltaFunction and Green’s Functions .............................410 9.5.3 Green’sFunctionsby the Method of Eigenfunction Expansion and theFredholm Alternative ............ 411 9.5.4 Direct Solution of Green’sFunctions (One-Dimensional Eigenfunctions)(Optional) ....................413 9.5.5 Using Green’sFunctions for Problems with Nonhomogeneous BoundaryConditions .......................415 9.5.6 In.nite Space Green’sFunctions .................416 9.5.7 Green’sFunctions for Bounded Domains Using In.nite Space Green’sFunctions ......................... 419 9.5.8 Green’sFunctions fora Semi-In.nite Plane(y> 0) Using In.nite Space Green’sFunctions: The Method of Images . . . 420 9.5.9 Green’sFunctions fora Circle: The Methodof Images . . . . 423 9.6 PerturbedEigenvalueProblems.......................430 9.6.1 Introduction ............................430 9.6.2 MathematicalExample ......................431 9.6.3 Vibrating Nearly Circular Membrane .............. 432 9.7Summary ...................................435 10 In.nite Domain Problems: Fourier Transform Solutions of Partial Di.erential Equations 437 10.1Introduction .................................437 10.2 HeatEquationonan In.niteDomain ...................437 10.3FourierTransformPair ........................... 441 10.3.1 Motivation fromFourier Series Identity ............. 441 10.3.2 FourierTransform ......................... 442 10.3.3 InverseFourierTransformofa Gaussian ............443 10.4 FourierTransformandthe Heat Equation .................450 10.4.1 HeatEquation ........................... 450 10.4.2 FourierTransforming the Heat Equation:Transforms ofDerivatives ........................... 455 10.4.3 ConvolutionTheorem .......................457 10.4.4 Summary of Properties of theFourierTransform ....... 461 10.5 Fourier Sine and CosineTransforms: The Heat Equation onSemi-In.niteIntervals ..........................463 10.5.1 Introduction ............................463 10.5.2 Heat Equationona Semi-In.niteIntervalI ...........463 10.5.3 Fourier Sine and CosineTransforms ............... 465 10.5.4 TransformsofDerivatives ....................466 10.5.5 Heat Equationona Semi-In.niteIntervalII .......... 467 10.5.6 Tables ofFourier Sine and CosineTransforms ......... 469 10.6Worked ExamplesUsingTransforms ....................473 10.6.1 One-DimensionalWave Equation on an In.nite Interval . . . 473 10.6.2 Laplace’s Equation in a Semi-In.nite Strip ........... 475 10.6.3 Laplace’s Equationina Half-Plane ...............479 10.6.4 Laplace’s Equation in a Quarter-Plane ............. 482 10.6.5 Heat Equationina Plane(Two-DimensionalFourier Transforms) ............................486 10.6.6 Table ofDoubleFourierTransforms ............... 490 10.7ScatteringandInverseScattering ......................495 11 Green’s Functions for Wave and Heat Equations 499 11.1Introduction .................................499 11.2 Green’sFunctions for theWave Equation ................. 499 11.2.1 Introduction ............................499 11.2.2 Green’sFormula for theWave Equation ............ 500 11.2.3 Reciprocity .............................502 11.2.4 UsingtheGreen’sFunction ...................504 11.2.5 Green’sFunction for theWave Equation ............ 506 11.2.6 Alternate Di.erential Equation for the Green’sFunction . . . 506 11.2.7 In.nite Space Green’sFunction for the One-Dimensional Wave Equation and d’Alembert’s Solution ........... 507 11.2.8 In.nite Space Green’sFunction for the Three-Dimensional Wave Equation (Huygens’ Principle) .............. 509 11.2.9 Two-Dimensional In.nite Space Green’sFunction ....... 511 11.2.10 Summary.............................. 511 11.3 Green’sFunctionsfortheHeat Equation .................514 11.3.1 Introduction ............................514 11.3.2 Non-Self-Adjoint Nature of the Heat Equation ......... 515 11.3.3 Green’sFormula for the Heat Equation ............. 516 11.3.4 AdjointGreen’sFunction .....................517 11.3.5 Reciprocity .............................518 11.3.6 Representation of the Solution Using Green’sFunctions . . . . 518 11.3.7 Alternate Di.erential Equation for the Green’sFunction . . . 520 11.3.8 In.nite Space Green’sFunction for the Di.usion Equation . . 521 11.3.9 Green’sFunction for the Heat Equation (Semi-In.niteDomain) ......................522 11.3.10 Green’sFunction for the Heat Equation (onaFiniteRegion) .......................523 12 The Method of Characteristics for Linear and Quasilinear Wave Equations 527 12.1 Introduction .................................527 12.2 Characteristics for First-OrderWave Equations .............. 528 12.2.1 Introduction ............................528 12.2.2 Method of Characteristics for First-OrderPartial Di.erential Equations .............................529 12.3 Method of Characteristics for the One-DimensionalWave Equation . . 534 12.3.1 GeneralSolution ......................... 534 12.3.2 InitialValue Problem (In.nite Domain) ............ 536 12.3.3 D’Alembert’sSolution ......................540 12.4 Semi-In.nite Stringsand Re.ections ....................543 12.5 Method of Characteristics for a Vibrating String ofFixedLength ............................... 548 12.6 The Method of Characteristics for QuasilinearPartial Di.erential Equations...................................552 12.6.1 MethodofCharacteristics ....................552 12.6.2 Tra.cFlow ............................553 12.6.3 Method of Characteristics(Q =0) ............... 555 12.6.4 ShockWaves............................ 558 12.6.5 QuasilinearExample .......................570 12.7 First-Order NonlinearPartial Di.erential Equations ........... 575 12.7.1 Eikonal Equation Derived from theWave Equation ...... 575 12.7.2 Solving the Eikonal Equation in UniformMedia andRe.ectedWaves .......................576 12.7.3 First-Order NonlinearPartial Di.erential Equations ...... 579 13 Laplace Transform Solution of Partial Di.erential Equations 581 13.1 Introduction .................................581 13.2 Propertiesofthe LaplaceTransform ....................581 13.2.1 Introduction ............................581 13.2.2 Singularities of the LaplaceTransform ............. 582 13.2.3 TransformsofDerivatives ....................586 13.2.4 ConvolutionTheorem .......................587 13.3 Green’sFunctions for InitialValue Problems for Ordinary Di.erentialEquations ............................591 13.4 A Signal ProblemfortheWave Equation .................593 13.5 A Signal Problemfora Vibrating StringofFinite Length ........597 13.6 TheWave Equation and Its Green’sFunction ............... 600 13.7 Inversion of LaplaceTransforms Using Contour Integrals intheComplexPlane ............................603 13.8 Solving theWave Equation Using LaplaceTransforms (withComplexVariables) .......................... 608 14 Dispersive Waves: Slow Variations, Stability, Nonlinearity, and Perturbation Methods 611 14.1 Introduction .................................611 14.2 DispersiveWavesand GroupVelocity ................... 612 14.2.1 TravelingWaves and the Dispersion Relation.......... 612 14.2.2 GroupVelocityI ......................... 615 14.3 WaveGuides .................................617 14.3.1 Response to ConcentratedPeriodic Sources withFrequency ωf ........................620 14.3.2 Green’sFunction If Mode Propagates .............. 620 14.3.3 Green’sFunction If Mode Does Not Propagate ......... 621 14.3.4 DesignConsiderations ......................622 14.4 FiberOptics .................................623 14.5 GroupVelocityII and the Methodof Stationary Phase ......... 627 14.5.1 MethodofStationaryPhase ...................628 14.5.2 Application to LinearDispersiveWaves ............. 630 14.6 SlowlyVaryingDispersiveWaves (GroupVelocity and Caustics) . . . . 634 14.6.1 Approximate Solutions ofDispersivePartial Di.erentialEquations .......................634 14.6.2 FormationofaCaustic ......................636 14.7 Wave Envelope Equations (ConcentratedWave Number) ........ 642 14.7.1 Schr¨odingerEquation .......................643 14.7.2 Linearized Korteweg–deVries Equation ............. 645 14.7.3 NonlinearDispersiveWaves: Korteweg–deVries Equation . . 647 14.7.4 SolitonsandInverse Scattering .................650 14.7.5 Nonlinear Schr¨odinger Equation ................. 652 14.8 StabilityandInstability ........................... 656 14.8.1 Brief OrdinaryDi.erential Equations and Bifurcation Theory ...............................656 14.8.2 Elementary Exampleofa Stable Equilibrium foraPartial Di.erentialEquation .......................663 14.8.3 Typical Unstable Equilibrium foraPartial Di.erential EquationandPatternFormation ................664 14.8.4 Ill-posedProblems ........................667 14.8.5 Slightly Unstable DispersiveWaves and the Linearized Complex Ginzburg–Landau Equation .............. 668 14.8.6 Nonlinear Complex Ginzburg–Landau Equation ........ 670 14.8.7 Long-WaveInstabilities ......................676 14.8.8 PatternFormation for Reaction–Di.usion Equations and the TuringInstability......................... 676 14.9 SingularPerturbation Methods: Multiple Scales ............. 683 14.9.1 Ordinary Di.erential Equation:Weakly Nonlinearly DampedOscillator ........................683 14.9.2 Ordinary Di.erential Equation: SlowlyVarying Oscillator . . 686 14.9.3 Slightly UnstablePartial Di.erential Equation on Fixed SpatialDomain .......................... 690 14.9.4 SlowlyVarying Medium for theWave Equation ........ 692 14.9.5 SlowlyVarying LinearDispersiveWaves (IncludingWeak NonlinearE.ects) .........................695 14.10 SingularPerturbation Methods: Boundary Layers Methodof Matched AsymptoticExpansions ... ........................ 700 14.10.1 Boundary Layer in an OrdinaryDi.erential Equation ..... 700 14.10.2 Di.usionofaPollutantDominatedbyConvection .......705 Bibliography 712 Answers to Starred Exercises 716 |
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