工程师与科学家必备：详解4000线性偏微分方程手册

需积分: 9 19 浏览量更新于2024-07-18 1 收藏 29.2MB PDF 举报

《工程师与科学家手册：线性偏微分方程》是一本专门针对科技工作者的权威参考书籍，第二版由Andrei D. Polyanin和Vladimir E. Nazaikinskii共同编撰。该手册汇集了近4000个线性偏微分方程，包括一阶、二阶、三阶乃至高阶的单个和系统的方程，涵盖了各种类型，如椭圆型、双曲型、抛物型和混合型等。书中不仅提供了详尽的解析解，还包括符号计算（如Maple、Mathematica）和数值方法（如MATLAB R）的运用，帮助读者理解和解决实际问题。本书强调了新发现的线性方程、精确解、变换和构建解的有效公式，通过具体实例展示了书中方法的应用。它还深入探讨了边值问题和特征值问题，这些是解决许多工程和科学领域问题的关键步骤。值得注意的是，书中提及的MATLAB R是在作者许可下的使用，但The MathWorks并不对书中内容或练习的准确性提供保证，其使用并不意味着对该软件的特定教学方法或用途的支持。《工程师与科学家手册：线性偏微分方程》的第二版由CRC Press出版，隶属于Taylor & Francis Group，地址位于佛罗里达州博卡拉顿。版权信息表明，尽管书中信息来源于权威且受人尊敬的来源，但作者和出版社仍尽力确保数据和信息的可靠性，但无法对所有内容的准确性做出完全担保。此外，该书的电子版国际标准书号(ISBN)为1-4665-8149-4，版本日期为2015年11月19日。《工程师与科学家手册：线性偏微分方程》是一部极其丰富的资源，对于研究者、工程师以及对线性偏微分方程感兴趣的学者来说，它提供了一个全面的工具箱，涵盖了理论分析、数值模拟以及实践应用的方方面面，是现代科学技术领域不可或缺的一份指南。

xiv CONTENTS

12.15 Nondissipative Thermoelasticity Equations (the Green–Naghdi Model) . . . . . 1103

12.15.1 Vector Form of the Nondissipative Thermoelasticity Equations . . . . 1103

12.15.2 Decompositions of the Nondissipative Thermoelasticity Equations

with f = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104

12.15.3 Decompositions of Thermoelasticity Equations with f 6= 0 . . . . . . . . 1105

12.16 Viscoelasticity Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106

12.16.1 Vector Form of Viscoelasticity Equations . . . . . . . . . . . . . . . . . . . . . . . 1106

12.16.2 Decompositions of Viscoelasticity Equations with f = 0 . . . . . . . . . . 1106

12.16.3 Various Forms of Decompositions for Viscoelasticity Equations with

f6= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1108

12.17 Maxwell Equations (Electromagnetic Field Equations) . . . . . . . . . . . . . . . . . . . 1108

12.17.1 Maxwell Equations in a Medium and Constitutive Relations . . . . . . 1108

12.17.2 Some Transformations and Solutions of the Maxwell Equations . . . 1109

12.18 Vector Equations of General Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1110

12.18.1 Vector Equations Containing O perators div and ∇ . . . . . . . . . . . . . . . 1110

12.18.2 Decompositions of the Homogeneous Vector Equation . . . . . . . . . . . 1111

12.18.3 Decompositions of the Nonhomogeneous Vector Equation . . . . . . . . 1112

12.18.4 Vector Equations Containing More General Operators . . . . . . . . . . . . 1113

12.19 General Systems Involving Vector and Scalar Equations: Part I . . . . . . . . . . . . 1114

12.19.1 Systems Containing Operators div and ∇ . . . . . . . . . . . . . . . . . . . . . . . 1114

12.19.2 Decompositions of Systems with Homogeneous Vector Equation . . 1115

12.19.3 Decompositions of Systems with Nonhomogeneous Vector

Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115

12.19.4 Equations for u and p. Reduction to One Vector Equation . . . . . . . . . 1116

12.19.5 Systems Containing More General Operators . . . . . . . . . . . . . . . . . . . 1117

12.20 General Systems Involving Vector and Scalar Equations: Part II . . . . . . . . . . . 1118

12.20.1 Class of Systems Considered . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1118

12.20.2 Asymmetric Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1118

12.20.3 Symmetric Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1119

Part II Analytical Methods 1121

13 Methods for First-Order Linear PDEs 1123

13.1 Linear PDEs with Two Independent Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1123

13.1.1 Special First-Order Linear PDEs with Two Independent Variables . . . 1123

13.1.2 General First-Order Linear PDE with Two Independent Variables . . . . 1126

13.2 First-Order Linear PDEs with Three or More Independent Variables . . . . . . . . 1129

13.2.1 Characteristic System. General Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 1129

13.2.2 Cauchy Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1131

14 S econd-Order Linear PDEs: Classiﬁcation, Problems, Particular Solutions 1133

14.1 Classiﬁcation of Second-Order Linear Partial Differential Equations . . . . . . . . 1133

14.1.1 Equations with Two Independent Variables . . . . . . . . . . . . . . . . . . . . . . . 1133

14.1.2 Equations with Many Independent Variables . . . . . . . . . . . . . . . . . . . . . . 1138

14.2 Basic Problems of Mathematical Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1140

14.2.1 Initial and Boundary Conditions. Cauchy Problem. Boundary Value

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1140

14.2.2 First, Second, Third, and Mixed Boundary Value Problems . . . . . . . . . 1142

CONTENTS xv

14.3 Properties and Particular Solutions of Linear Equations . . . . . . . . . . . . . . . . . . . . 1144

14.3.1 Homogeneous Linear Equations. Basic Properties of Particular

Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1144

14.3.2 Separable Solutions. Solutions in the Form of Inﬁnite Series . . . . . . . . 1147

14.3.3 Nonhomogeneous Linear Equations and Their Properties . . . . . . . . . . . 1150

14.3.4 General Solutions of Some Hyperbolic Equations . . . . . . . . . . . . . . . . . . 1150

15 S ep aration of Variables and Integral Transform Methods 1153

15.1 Separation of Variables (Fourier Method) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1153

15.1.1 Description of Separation of Variables. General Stage of Solution . . . 1153

15.1.2 Problems for Parabolic Equations: Final Stage of Solution . . . . . . . . . . 1157

15.1.3 Problems for Hyperbolic Equations: Final Stage of Solution . . . . . . . . 1159

15.1.4 Solution of Boundary Value Problems for Elliptic Equations . . . . . . . . 1160

15.1.5 Solution of Boundary Value Problems for Higher-Order Equations . . . 1163

15.2 Integral Transform Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165

15.2.1 Laplace Transform and Its Application in Mathematical Physics . . . . . 1165

15.2.2 Fourier Transform and Its Application in Mathematical Physics . . . . . 1170

15.2.3 Fourier Sine and Cosine Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1173

15.2.4 Mellin, Hankel, and Other Integral Transforms . . . . . . . . . . . . . . . . . . . . 1177

16 Cauchy Problem. Fundamental Solutions 1181

16.1 Dirac Delta Function. Fundamental Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1181

16.1.1 Dirac Delta Function and Its Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 1181

16.1.2 Fundamental Solutions. Constructing Particular Solutions . . . . . . . . . . 1182

16.2 Representation of the Solution of the Cauchy Problem via the Fundamental

Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1185

16.2.1 Cauchy Problem for Ordinary Differential Equations . . . . . . . . . . . . . . . 1185

16.2.2 Cauchy Problem for Parabolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . 1187

16.2.3 Cauchy Problem for Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . . . . 1190

16.2.4 Higher-O rder Linear PDEs. Generalized Cauchy P roblem . . . . . . . . . . 1193

17 Boundary Value Problems. Green’s Fu nction 1199

17.1 Boundary Value Problems for Parabolic Equations with One Space Variable.

Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1199

17.1.1 Representation of Solutions via the Green’s Function . . . . . . . . . . . . . . 1199

17.1.2 Problems for Equation s(x)

∂w

∂t

∂

∂x



p(x)

∂w

∂x



−q(x)w +Φ(x, t) . . . 1202

17.2 Boundary Value Problems for Hyperbolic Equations with One Space Variable.

Green’s Function. Goursat Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205

17.2.1 Representation of Solutions via the Green’s Function . . . . . . . . . . . . . . 1205

17.2.2 Problems for Equation s(x)

∂

∂t

∂

∂x



p(x)

∂w

∂x



−q(x)w +Φ(x, t) . . . 1207

17.2.3 Problems for Equation

∂

∂t

+ a(t)

∂w

∂t

b(t)



∂

∂x



p(x)

∂w

∂x



−q(x)w



+Φ(x, t) . . . . . . . . . . . . . . . . . . . . . . . . . . . 1208

17.2.4 Generalized Cauchy Problem with Initial Conditions Set along a

Curve. Riemann Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1210

17.2.5 Goursat Problem (a Problem with Initial Data on Characteristics) . . . . 1212

17.3 Boundary Value Problems for Elliptic Equations with Two Space Variables . . 1214

17.3.1 Problems and the Green’s Functions for Equation

a(x)

∂

∂x

∂

∂y

+b(x)

∂w

∂x

+c(x)w = −Φ(x, y) . . . . . . . . . . . . . . . . . . . . 1214

17.3.2 Representation of Solutions of Boundary Value Problems via Green’s

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216

xvi CONTENTS

17.4 Boundary Value Problems with Many Space Variables. Green’s Function . . . . 1218

17.4.1 Problems for Parabolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1218

17.4.2 Problems for Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1220

17.4.3 Problems for Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1221

17.4.4 Comparison of the Solution Structures for Boundary Value Problems

for Equations of Various Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1222

17.5 Construction of the Green’s Functions. G eneral Formulas and Relations . . . . . 1223

17.5.1 Green’s Functions of Boundary Value Problems for Equations of

Various Types in Bounded Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1223

17.5.2 Green’s Functions Admitting Incomplete Separation of Variables . . . . 1224

17.5.3 Construction of Green’s Functions via Fundamental Solutions . . . . . . . 1227

18 Duhamel’s Principles. Some Transformations 1233

18.1 Duhamel’s Principles in Nonstationary Problems . . . . . . . . . . . . . . . . . . . . . . . . . 1233

18.1.1 Problems for Homogeneous Linear Equations . . . . . . . . . . . . . . . . . . . . . 1233

18.1.2 Problems for Nonhomogeneous Linear Equations . . . . . . . . . . . . . . . . . . 1235

18.2 Transformations Simplifying Initial and Boundary Conditions . . . . . . . . . . . . . . 1237

18.2.1 Transformations That Lead to Homogeneous B oundary Conditions . . 1237

18.2.2 Transformations That Lead to Homogeneous Initial and Boundary

Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1238

19 S ystems of Linear Coupled PDEs. Decomposition Methods 1239

19.1 Asymmetric and Symmetric Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1239

19.1.1 Asymmetric Decomposition. Order of Decomposition . . . . . . . . . . . . . . 1239

19.1.2 Symmetric Decomposition. Invariant Transformations . . . . . . . . . . . . . 1242

19.2 First-Order Decompositions. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1244

19.2.1 Systems of Linear PDEs without Mass Forces (f = 0) . . . . . . . . . . . . . . 1244

19.2.2 Systems of Linear PDEs with Mass Forces . . . . . . . . . . . . . . . . . . . . . . . . 1249

19.3 Higher-Order Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1251

19.3.1 Decomposition of Systems Consisting of One Vector Equation . . . . . . 1251

19.3.2 Decomposition of Systems Consisting of a Vector Equation and a

Scalar Equation (the First Approach) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1252

19.3.3 Decomposition of Systems Consisting of a Vector Equation and a

Scalar Equation (the Second Approach) . . . . . . . . . . . . . . . . . . . . . . . . . . 1253

20 S ome Asymptotic Results and Formulas 1255

20.1 Regular Perturbation Theory Formulas for the Eigenvalues . . . . . . . . . . . . . . . . . 1256

20.1.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1256

20.1.2 Formulas for the Coefﬁcients of the E xpansion . . . . . . . . . . . . . . . . . . . . 1256

20.2 Singular Perturbation T heory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1257

20.2.1 Cauchy Problem for the Schr¨odinger Equation . . . . . . . . . . . . . . . . . . . . 1257

20.2.2 Stationary Phase Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1261

20.2.3 Fourier Transform with a Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1264

21 Elements of Theory of Generalized F unctions 1265

21.1 Generalized Functions of One Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265

21.1.1 Important Terminological Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265

21.1.2 Test Function Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265

21.1.3 Distribution Space. Dirac Delta Function . . . . . . . . . . . . . . . . . . . . . . . . . 1266

21.1.4 Derivatives of Distributions. Some Formulas . . . . . . . . . . . . . . . . . . . . . . 1267

21.1.5 Operations on Distributions and Some Transformations . . . . . . . . . . . . 1269

CONTENTS xvii

21.1.6 Tempered Distributions and Fourier Transform . . . . . . . . . . . . . . . . . . . . 1270

21.1.7 Generalized Solutions of Linear Ordinary Differential Equations . . . . 1271

21.2 Generalized Functions of Several Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1271

21.2.1 Some Deﬁnitions. Partial Derivatives. Direct Product. Linear

Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1271

21.2.2 Dirac Delta Function. Generalized Solutions of Linear PDEs . . . . . . . . 1273

Part III Symbolic and Numerical Solutions with Maple, Mathematica,

and MATLAB



1275

22 Linear Partial Differential Equations with Maple 1277

22.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1277

22.1.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1277

22.1.2 Brief Introduction to Maple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1279

22.1.3 Maple Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1280

22.2 Analytical Solutions and Their Visualizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1282

22.2.1 Constructing Analytical Solutions in Terms of Predeﬁned Functions . 1282

22.2.2 Constructing General Solutions via the Method of Characteristics . . . . 1289

22.2.3 Constructing General Solutions via Transformations to Canonical

Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1291

22.2.4 Constructing Analytical Solutions of C auchy P roblems . . . . . . . . . . . . . 1293

22.2.5 Constructing Analytical Solutions of B oundary Value Problems . . . . . 1297

22.2.6 Constructing Analytical Solutions of Initial-Boundary Value Problems 1298

22.2.7 Constructing Analytical Solutions of Systems of Linear PDEs . . . . . . . 1299

22.3 Analytical Solutions of Mathematical Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 1301

22.3.1 Constructing Separable Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1301

22.3.2 Constructing Analytical Solutions via Integral Transform Methods . . . 1305

22.3.3 Constructing Analytical Solutions in Terms of Green’s Functions . . . . 1306

22.4 Numerical Solutions and Their Visualizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1309

22.4.1 Constructing Numerical Solutions in Terms of Predeﬁned Functions . 1310

22.4.2 Numerical Methods Embedded in Maple . . . . . . . . . . . . . . . . . . . . . . . . . 1312

22.4.3 Numerical Solutions of Initial-Boundary Value Problems . . . . . . . . . . . 1318

22.4.4 Numerical Solutions of Boundary Value Problems . . . . . . . . . . . . . . . . . 1322

22.4.5 Numerical Solutions of Cauchy Problems . . . . . . . . . . . . . . . . . . . . . . . . 1323

22.4.6 Numerical Solutions of Systems of Linear PDEs . . . . . . . . . . . . . . . . . . 1325

23 Linear Partial Differential Equations with Mathematica 1327

23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1327

23.1.1 Some Notational Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1327

23.1.2 Brief Introduction to Mathematica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1327

23.1.3 Mathematica Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1329

23.1.4 Dynamic Computation and Visualization in Mathematica Notebook . . 1332

23.2 Analytical Solutions and Their Visualizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1333

23.2.1 Constructing Analytical Solutions in Terms of Predeﬁned Functions . 1333

23.2.2 Constructing General Solutions via the Method of Characteristics . . . . 1335

23.2.3 Constructing General Solutions via Conversion to Canonical Forms . . 1338

23.2.4 Constructing Analytical Solutions of C auchy P roblems . . . . . . . . . . . . . 1340

23.2.5 Constructing Analytical Solutions of B oundary Value Problems . . . . . 1342

23.2.6 Constructing Analytical Solutions of Initial-Boundary Value Problems 1344

23.2.7 Constructing Analytical Solutions of Systems of Linear PDEs . . . . . . . 1345

xviii CONTENTS

23.3 Analytical Solutions of Mathematical Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 1347

23.3.1 Constructing Separable Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1347

23.3.2 Constructing Analytical Solutions via Integral Transform Methods . . . 1350

23.3.3 Constructing Analytical Solutions in Terms of Green’s Functions . . . . 1352

23.4 Numerical Solutions and Their Visualizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1356

23.4.1 Constructing Numerical Solutions in Terms of Predeﬁned Functions . 1356

23.4.2 Numerical Methods Embedded in Mathematica . . . . . . . . . . . . . . . . . . . 1358

23.4.3 Numerical Solutions of Initial-Boundary Value Problems . . . . . . . . . . . 1359

23.4.4 Numerical Solutions of Boundary Value Problems . . . . . . . . . . . . . . . . . 1363

23.4.5 Numerical Solutions of Cauchy Problems . . . . . . . . . . . . . . . . . . . . . . . . 1364

23.4.6 Numerical Solutions of Systems of Linear PDEs . . . . . . . . . . . . . . . . . . 1365

24 Linear Partial Differential Equations with MATLA B



1367

24.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1367

24.1.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1367

24.1.2 Brief Introduction to MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1368

24.1.3 MATLAB Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1371

24.2 Numerical Solutions of Linear PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1374

24.2.1 Constructing Numerical Solutions via Predeﬁned Functions . . . . . . . . . 1375

24.2.2 Numerical Methods Embedded in MATLAB . . . . . . . . . . . . . . . . . . . . . . 1383

24.2.3 Numerical Solutions of Cauchy Problems . . . . . . . . . . . . . . . . . . . . . . . . 1383

24.2.4 Numerical Solutions of Initial-Boundary Value Problems . . . . . . . . . . . 1385

24.2.5 Numerical Solutions of Boundary Value Problems . . . . . . . . . . . . . . . . . 1388

24.3 Constructing Finite Difference Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . 1392

24.3.1 Explicit Finite Difference Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1392

24.3.2 Implicit Finite Difference Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1395

24.4 Numerical Solutions of Systems of Linear PDEs . . . . . . . . . . . . . . . . . . . . . . . . . 1396

24.4.1 Linear Systems of 1D PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1396

24.4.2 Linear Systems of 2D PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1399

Part IV Tables and Supplements 1403

25 Elementary Functions and Their Properties 1405

25.1 Power, Exponential, and Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 1405

25.1.1 Properties of the Power Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1405

25.1.2 Properties of the Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 1405

25.1.3 Properties of the Logarithmic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 1406

25.2 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1407

25.2.1 Simplest Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1407

25.2.2 Reduction Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1408

25.2.3 Relations between Trigonometric Functions of Single Argument . . . . . 1408

25.2.4 Addition and Subtraction of Trigonometric Functions . . . . . . . . . . . . . . 1408

25.2.5 Products of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1409

25.2.6 Powers of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1409

25.2.7 Addition Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1409

25.2.8 Trigonometric Functions of Multiple Arguments . . . . . . . . . . . . . . . . . . 1410

25.2.9 Trigonometric Functions of Half Argument . . . . . . . . . . . . . . . . . . . . . . . 1410

25.2.10 Differentiation Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1410

25.2.11 Integration Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1410

25.2.12 Expansion in Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1411

剩余1622页未读，继续阅读

wang1062807258

粉丝: 13
资源: 272

工程师与科学家必备：详解4000线性偏微分方程手册

Handbook of Nonlinear Partial Differential Equations, 2ed

Handbook of Linear Partial Differential Equations for Engineers

handbook of linear partial differential equations

Handbook of Exponential and Related Distributions for Engineers and Scientists

Handbook of Mathematics for Engineers and Scientists

数学手册Handbook of Mathematics for Engineers and Scientists

Exact Solutions of a Class of Partial Differential Equations

Handbook of Differential Equations

Handbook of Differential Equations 3rd .D. Zwillinger.1997

Handbook_of_integral_equations-CRC_Press

最新资源