Partial differential equations in mechanics

Cover......Page 1
Title page......Page 3
Copyright page......Page 4
Dedication......Page 5
Preface......Page 7
Acknowledgements......Page 13
Contents......Page 15
1.2 Dot or scalar product......Page 21
1.3 Cross or vector product......Page 22
1.5 Results involving derivatives......Page 23
1.6.1 The gradient of a scalar field......Page 24
1.6.3 The Laplacian of a scalar or vector field......Page 26
1.7 Divergence of a vector field: an application......Page 27
1.8 Divergence or Green's theorem......Page 29
1.9 Green's theorem in two dimensions......Page 33
1.10 Orthogonal curvilinear coordinates......Page 34
1.11 Gradient and Laplacian in orthogonal curvilinear coordinates......Page 37
1.12 Integral transforms......Page 40
1.12.1 Laplace transform......Page 41
1.12.2 Fourier transforms......Page 61
1.12.3 Hankel transforms......Page 77
1.13 PROBLEM SET 1......Page 83
2. General concepts in partial differential equations......Page 91
2.1 Fundamental concepts......Page 92
2.1.1 The order of a partial differential equation......Page 93
2.1.2 The linearity of a partial differential equation......Page 94
2.1.3 Homogeneity of a partial differential equation......Page 96
2.2.1 Boundary conditions......Page 97
2.2.2 Initial conditions......Page 100
2.2.3 Well-posed problems......Page 101
2.3 PROBLEM SET 2......Page 102
3.1 General concepts......Page 107
3.2 Examples involving first-order equations......Page 110
3.3 Advective transport in reactor column......Page 120
3.3.1 Governing equation - one dimensional case......Page 121
3.3.2 Governing equation - generalized formulation......Page 125
3.4 A heat exchanger problem......Page 134
3.5 PROBLEM SET 3......Page 136
4. Partial differential equations of the second-order......Page 141
4.1 Classification of second-order partial differential equations......Page 142
4.2 Reduction to canonical forms......Page 145
4.3 Applications of the procedures......Page 152
4.4 Classification of second-order pdes for n independent variables......Page 160
4.5 PROBLEM SET 4......Page 168
5. Laplace's equation......Page 171
5.1.1 Irrotational flow in fluid mechanics......Page 172
5.1.2 Flow of fluids in porous media......Page 179
5.2.1 Boundary conditions for fluid flow......Page 186
5.2.2 Boundary conditions for porous media flow......Page 187
5.2.3 Boundary conditions for heat conduction......Page 188
5.3 Generalized results......Page 189
5.4 Methods of solution of Laplace's equation......Page 197
5.4.1 Direct solution procedure......Page 198
5.4.2 Separation of variables method - Cartesian coordinates......Page 201
5.4.3 Separation of variables method - plane polar coordinates......Page 215
5.5 Integral transform solution of Laplace's equation......Page 224
5.6 Line source within a half-plane region......Page 230
5.7 Uniqueness theorem......Page 234
5.8 A maximum principle......Page 238
5.9 PROBLEM SET 5......Page 240
6.1 Derivation of the diffusion equation......Page 255
6.1.1 Heat conduction in solids......Page 256
6.1.2 Pressure transients in porous media......Page 260
6.1.3 Chemical mass transport in porous media......Page 264
6.1.4 Drying of porous solids......Page 268
6.1.5 Thermal oxidation of silicon......Page 270
6.1.6 Motion of a plate on a viscous fluid......Page 272
6.2.1 Dirichlet-type boundary condition......Page 274
6.2.2 Neumann-type boundary conditions......Page 275
6.2.3 Combined boundary conditions......Page 276
6.2.5 Initial conditions......Page 277
6.2.6 Change in dependent variable for homogeneous initial conditions......Page 279
6.3.1 Direct solution procedure......Page 282
6.3.2 Trial function approach......Page 284
6.3.3 Separation of variables method - Cartesian coordinates......Page 286
6.3.4 Separation of variables method - plane polar coordinates......Page 312
6.4.1 Reduction to Helmholtz equation......Page 321
6.4.2 Product solutions for the diffusion equation......Page 322
6.4.3 Sturm-Liouville problems......Page 326
6.5 Separation of variables method for spatially two-dimensional problems......Page 332
6.5.1 Spatially two-dimensional problems - Cartesian coordinates......Page 333
6.5.2 Spatially two-dimensional problems - plane polar coordinates......Page 343
6.5.3 Prodnct solutions and solntions for infinite domains......Page 357
6.6 Uniqueness theorem......Page 365
6.7 A maximum principle......Page 369
6.8 PROBLEM SET 6......Page 373
7. The wave equation......Page 389
7.1 Wave motion in strings......Page 391
7.1.1 Harmonic waves......Page 394
7.1.2 d'Alembert's solution......Page 398
7.1.3 Fourier analysis of the stretched string......Page 405
7.1.4 Reflection and transmission at boundaries......Page 406
7.1.5 Energy in a string......Page 413
7.1.6 Forced motion of a semi-infinite string......Page 417
7.1.7 Forced motion of an infinite string......Page 422
7.2.1 Waves in a stretched finite string......Page 435
7.2.2 Vibrations of a stretched finite string: trial function approach......Page 438
7.2.3 Vibrations of a stretched finite string - variables separable solution......Page 442
7.2.4 Vibrations of a stretched string: variable boundary conditions......Page 451
7.2.5 Forced vibration of a stretched finite string......Page 457
7.3.1 Elastically supported string......Page 466
7.3.2 Energy dissipation and damping in a stretched string......Page 471
7.4 Waves and vibrations in stretched membranes......Page 473
7.4.1 Equation of motion for a stretched membrane......Page 474
7.4.2 Plane wave motion in a stretched infinite membrane......Page 480
7.4.3 Free vibrations of a stretched membrane of infinite extent......Page 483
7.4.4 Symmetric free vibrations of the stretched membrane......Page 489
7.4.5 Green's function for the vibration of a stretched membrane......Page 493
7.5.1 Vibrations of a stretched square membrane......Page 495
7.5.2 Free vibrations of a stretched rectangular membrane......Page 502
7.5.3 Forced vibrations of a stretched rectangular membrane......Page 509
7.5.4 Free vibrations of a stretched circular membrane......Page 512
7.5.5 Hankel transform analysis of free vibrations of a stretched circular membrane......Page 519
7.5.6 Hankel transform analysis of forced vibrations of a stretched circular membrane......Page 521
7.5.7 Vibrations of a circular membrane-general formulation......Page 525
7.6 Wave motion and vibrations in membranes: non-classical effects......Page 531
7.7 Wave equation for ptoblems in solid mechanics......Page 532
7.7.1 Longitudinal wave motion in a slender elastic rod......Page 533
7.7.2 Torsional waves in a slender circular elastic rod......Page 542
7.8 Shallow water waves......Page 545
7.9 Uniqueness theorem......Page 551
7.10 PROBLEM SET 7......Page 561
Bibliography......Page 577
Index......Page 607