Nonlinear Partial Differential Equations in Engineering: v. 1 (Mathematics in Science & Engineering Volume 18)

Front Cover......Page 1
Nonliner Partial Differential Equations in Engineering, Volume 18......Page 6
Copyright Page......Page 7
Contents......Page 10
Preface......Page 8
1.0 Introduction......Page 16
1.1 What is Nonlinearity?......Page 17
1.2 Equations from Diffusion Theory......Page 19
1.3 Equations from Fluid Mechanics......Page 23
1.4 Equations from Solid Mechanics......Page 25
1.5 Miscellaneous Examples......Page 28
References......Page 32
2.0 Introduction......Page 35
2.1 Transformations on Dependent Variables......Page 36
2.2 Transformations on Independent Variables......Page 46
2.3 Mixed Transformations......Page 50
2.4 The Unknown Function Approach......Page 62
2.5 General Solutions......Page 64
2.6 General Solutions of First-Order Equations......Page 65
2.7 General Solutions of Second-Order Equations......Page 73
2.8 Table of General Solutions......Page 80
References......Page 84
3.0 Introduction......Page 86
3.1 The Quasi-Linear System......Page 87
3.2 An Example of the Quasi-Linear Theory......Page 93
3.3 The Poisson-Euler-Darboux Equation......Page 99
3.4 Remarks on the PED Equation......Page 103
3.5 One-Dimensional Anisentropic Flows......Page 105
3.6 An Alternate Approach to Anisentropic Flow......Page 109
3.7 General Solution for Anisentropic Flow......Page 115
3.8 Vibration of a Nonlinear String......Page 118
3.10 Direct Separation of Variables......Page 124
3.11 Other Solutions Obtained by Ad Hoc Assumptions......Page 132
References......Page 136
4.1 An Ad Hoc Solution from Magneto-Gas Dynamics......Page 138
4.2 The Utility of Lagrangian Coordinates......Page 141
4.3 Similarity Variables......Page 148
4.4 Similarity via One-Parameter Groups......Page 150
4.5 Extensions of the Similarity Procedure......Page 156
4.6 Similarity via Separation of Variables......Page 159
4.7 Similarity and Conservation Laws......Page 165
4.8 General Comments on Transformation Groups......Page 171
4.9 Similarity Applied to Moving Boundary Problems......Page 173
4.10 Similarity Considerations in Three Dimensions......Page 177
4.11 General Discussion of Similarity......Page 181
4.12 Integral Equation Methods......Page 182
4.13 The Hdograph......Page 186
4.14 Simple Examples of Hodograph Application......Page 188
4.15 The Hodograph in More Complicated Problems......Page 192
4.16 Utilization of the General Solutions of Chapter 2......Page 195
4.17 Similar Solutions in Heat and Mass Transfer......Page 198
4.18 Similarity Integrals in Compressible Gases......Page 201
4.19 Some Disjoint Remarks......Page 205
References......Page 207
5.0 Introduction......Page 210
5.1 Perturbation Concepts......Page 211
5.2 Regular Perturbations in Vibration Theory......Page 212
5.3 Perturbation and Plasma Oscillations......Page 213
5.4 Perturbation in Elasticity......Page 219
5.5 Other Applications......Page 222
5.6 Perturbation about Exact Solutions......Page 223
5.7 The Singular Perturbation Problem......Page 226
5.8 Singular Perturbations in Viscous Flow......Page 230
5.9 The “Inner-Outer’’ Expansion (a Motivation)......Page 234
5.10 The Inner and Outer Expansions......Page 237
5.11 Examples......Page 241
5.12 Higher Approximations for Flow past a Sphere......Page 246
5.13 Asymptotic Approximations......Page 252
5.14 Asymptotic Solutions in Diffusion with Reaction......Page 255
5.15 Weighted Residual Methods: General Discussion......Page 258
5.16 Examples of the Use of Weighted Residual Methods......Page 264
5.17 Comments on the Methods of Weighted Residuals......Page 276
5.18 Mathematical Problems of Approximate Methods......Page 277
References......Page 282
6.1 Integral Methods in Fluid Mechanics......Page 286
6.2 Nonlinear Boundary Conditions......Page 293
6.3 Integral Equations and Boundary Layer Theory......Page 295
6.4 Iterative Solutions for Δ2u = bu2......Page 299
6.5 The Maximum Operation......Page 302
6.6 Equations of Elliptic Type and the Maximum Operation......Page 304
6.7 Other Applications of the Maximum Operation......Page 307
6.8 Series Expansions......Page 310
6.9 Goertler’s Series......Page 314
6.10 Series Solutions in Elasticity......Page 316
6.11 “Traveling Wave” Solutions by Series......Page 320
References......Page 327
7.0 Introduction......Page 330
7.1 Terminology and Computational Molecules......Page 331
7.2 Explicit Methods for Parabolic Systems......Page 335
7.3 Some Nonlinear Examples......Page 339
7.4 Alternate Explicit Methods......Page 341
7.6 Singularities......Page 345
7.7 A Treatment of Singularities (Example)......Page 349
7.8 Implicit Procedures......Page 353
7.9 A Second-Order Method for Lu = f(x, t, u)......Page 358
7.10 Predictor Corrector Methods......Page 360
7.11 Traveling Wave Solutions......Page 363
7.12 Finite Differences Applied to the Boundary Layer Equations......Page 364
7.13 Other Nonlinear Parabolic Examples......Page 370
7.14 Finite Difference Formula for Elliptic Equations in Two Dimensions......Page 380
7.15 Linear Elliptic Equations......Page 385
7.16 Methods of Solution of Au = v......Page 388
7.17 Point Iterative Methods......Page 390
7.18 Block Iterative Methods......Page 399
7.19 Examples of Nonlinear Elliptic Equations......Page 404
7.20 Singularities......Page 426
7.21 Method of Characteristics......Page 431
7.22 The Supersonic Nozzle......Page 438
7.23 Properties of Hyperbolic Systems......Page 441
7.24 One-Dimensional Isentropic Flow......Page 447
7.25 Method of Characteristics: Numerical Computation......Page 450
7.26 Finite Difference Methods: General Discussion......Page 452
7.27 Explicit Methods......Page 453
7.28 Explicit Methods in Nonlinear Second-Order Systems......Page 455
7.29 Implicit Methods for Second-Order Equations......Page 458
7.30 “Hybrid” Methods for a Nonlinear First-Order System......Page 460
7.31 Finite Difference Schemes in One-Dimensional Flow......Page 463
7.32 Conservation Equations......Page 468
7.33 Intetfaces......Page 469
7.34 Shocks......Page 471
7.35 Additional Methods......Page 476
7.37 Hydrodynamic Flow and Radiation Diffusion......Page 477
7.38 Nonlinear Vibrations of a Moving Threadline......Page 479
References......Page 482
8.0 Introduction......Page 489
8.1 Well-Posed Problems......Page 490
8.2 Existence and Uniqueness in Viscous Incompressible Flow......Page 493
8.3 Existence and Uniqueness in Boundary Layer Theory......Page 497
8.4 Existence and Uniqueness in Quasi-Linear Parabolic Equations......Page 501
8.5 Uniqueness Questions for Quasi-Linear Elliptic Equations......Page 502
References......Page 504
A.1 Basic Definitions......Page 506
A.2 Groups of Transformations......Page 507
Author Index......Page 510
Subject Index......Page 516