Mathematical Methods for Engineers and Scientists 3: Fourier Analysis, Partial Differential Equations and Variational Methods

Copyright\r......Page 4
Preface......Page 6
Table of Contents\r......Page 8
Part I: Fourier Analysis\r......Page 14
1.1.1 Orthogonality of Trigonotric Functions......Page 16
1.1.2 The Fourier Coefficients......Page 18
1.1.3 Expansion of Functions in Fourier Series......Page 19
1.2.1 Dirichlet Conditions......Page 22
1.2.2 Fourier Series and Delta Function......Page 23
1.3.1 Change of Interval......Page 26
1.3.2 Fourier Series of Even and Odd Functions......Page 34
1.4 Fourier Series of Nonperiodic Functions in Limited Range\r......Page 37
1.5 Complex Fourier Series......Page 42
1.6 The Method of Jumps......Page 45
1.7.1 Parseval’s Theorem......Page 50
1.7.2 Sums of Reciprocal Powers of Integers......Page 52
1.7.3 Integration of Fourier Series......Page 55
1.7.4 Differentiation of Fourier Series......Page 56
1.8.1 Differential Equation with Boundary Conditions......Page 58
1.8.2 Periodically Driven Oscillator......Page 62
2.1 Fourier Integral as a Limit of a Fourier Series......Page 74
2.1.1 Fourier Cosine and Sine Integrals......Page 78
2.1.2 Fourier Cosine and Sine Transforms......Page 80
2.3 The Fourier Transform......Page 85
2.4.1 Orthogonality......Page 92
2.4.2 Fourier Transforms Involving Delta Functions......Page 93
2.4.3 Three-Dimensional Fourier Transform Pair......Page 94
2.5.2 Gaussian Function......Page 98
2.5.3 Exponentially Decaying Function......Page 100
2.6.1 Symmetry Property......Page 101
2.6.2 Linearity, Shifting, Scaling......Page 102
2.6.3 Transform of Derivatives......Page 104
2.6.5 Parseval’s Theorem......Page 105
2.7.1 Mathematical Operation of Convolution......Page 107
2.7.2 Convolution Theorems......Page 109
2.8 Fourier Transform and Differential Equations......Page 112
2.9 The Uncertainty of Waves......Page 116
Part II: Sturm–Liouville Theory and Special Functions\r......Page 122
3.1.1 Vector Space......Page 124
3.1.2 Inner Product and Orthogonality......Page 126
3.1.3 Orthogonal Functions......Page 129
3.2 Generalized Fourier Series......Page 134
3.3.1 Adjoint and Self-adjoint (Hermitian) Operators......Page 136
3.3.2 Properties of Hermitian Operators......Page 138
3.4.1 Sturm–Liouville Equations......Page 143
3.4.2 Boundary Conditions of Sturm–Liouville Problems......Page 145
3.4.3 Regular Sturm–Liouville Problems......Page 146
3.4.4 Periodic Sturm–Liouville Problems......Page 154
3.4.5 Singular Sturm–Liouville Problems......Page 155
3.5.1 Green’s Function and Inhomogeneous Differential Equation......Page 162
3.5.2 Green’s Function and Delta Function......Page 163
4: Bessel and Legendre Functions\r......Page 176
4.1.1 Power Series Solution of Differential Equation......Page 177
4.1.2 Classifying Singular Points......Page 179
4.1.3 Frobenius Series......Page 180
4.2 Bessel Functions......Page 184
4.2.1 Bessel Functions Jn(x) of Integer Order......Page 185
4.2.2 Zeros of the Bessel Functions......Page 187
4.2.3 Gamma Function......Page 188
4.2.4 Bessel Function of Noninteger Order......Page 190
4.2.6 Neumann Functions and Hankel Functions......Page 192
4.3.1 Recurrence Relations......Page 195
4.3.2 Generating Function of Bessel Functions......Page 198
4.3.3 Integral Representation......Page 199
4.4.1 Boundary Conditions of Bessel’s Equation......Page 200
4.4.2 Orthogonality of Bessel Functions......Page 201
4.4.3 Normalization of Bessel Functions......Page 202
4.5.1 Modified Bessel Functions......Page 204
4.5.2 Spherical Bessel Functions......Page 205
4.6.1 Series Solution of Legendre Equation......Page 209
4.6.2 Legendre Polynomials......Page 213
4.6.3 Legendre Functions of the Second Kind......Page 215
4.7.1 Rodrigues’ Formula......Page 217
4.7.2 Generating Function of Legendre Polynomials......Page 219
4.7.3 Recurrence Relations......Page 221
4.7.4 Orthogonality and Normalization of Legendre Polynomials......Page 224
4.8.1 Associated Legendre Polynomials......Page 225
4.8.2 Orthogonality and Normalization of Associated Legendre Functions\r......Page 227
4.8.3 Spherical Harmonics......Page 230
4.9 Resources on Special Functions......Page 231
Part III: Partial Differential Equations\r......Page 240
5: Partial Differential Equations in Cartesian Coordinates\r......Page 242
5.1.1 The Governing Equation of a Vibrating String......Page 243
5.1.2 Separation of Variables......Page 245
5.1.3 Standing Wave......Page 251
5.1.4 Traveling Wave......Page 255
5.1.5 Nonhomogeneous Wave Equations......Page 261
5.1.6 D’Alembert’s Solution of Wave Equations......Page 265
5.2.1 The Governing Equation of a Vibrating Membrane......Page 274
5.2.2 Vibration of a Rectangular Membrane......Page 275
5.3 Three-Dimensional Wave Equations......Page 280
5.3.1 Plane Wave......Page 281
5.3.2 Particle Wave in a Rectangular Box......Page 283
5.4 Equation of Heat Conduction......Page 285
5.5 One-Dimensional Diffusion Equations......Page 287
5.5.1 Temperature Distributions with Specified Values at the Boundaries\r......Page 288
5.5.2 Problems Involving Insulated Boundaries......Page 291
5.5.3 Heat Exchange at the Boundary......Page 293
5.6 Two-Dimensional Diffusion Equations: Heat Transfer in a Rectangular Plate\r......Page 297
5.7 Laplace’s Equations......Page 299
5.7.1 Two-Dimensional Laplace’s Equation: Steady-State Temperature in a Rectangular Plate\r......Page 300
5.7.2 Three-Dimensional Laplace’s Equation: Steady-State Temperature in a Rectangular Parallelepiped\r......Page 302
5.8 Helmholtz’s Equations......Page 304
6: Partial Differential Equations with Curved Boundries\r......Page 314
6.1 The Laplacian......Page 315
6.2.1 Laplace’s Equation in Polar Coordinates......Page 317
6.2.2 Poisson’s Integral Formula......Page 325
6.3 Two-Dimensional Helmholtz’s Equations in Polar Coordinates\r......Page 328
6.3.1 Vibration of a Drumhead: Two-Dimensional Wave Equation in Polar Coordinates\r......Page 329
6.3.2 Heat Conduction in a Disk: Two-Dimensional Diffusion Equation in Polar Coordinates\r......Page 335
6.3.3 Laplace’s Equations in Cylindrical Coordinates......Page 339
6.3.4 Helmholtz’s Equations in Cylindrical Coordinates......Page 344
6.4.1 Laplace’s Equations in Spherical Coordinates......Page 347
6.4.2 Helmholtz’s Equations in Spherical Coordinates......Page 358
6.4.3 Wave Equations in Spherical Coordinates......Page 359
6.5 Poisson’s Equations......Page 362
6.5.1 Poisson’s Equation and Green’s Function......Page 364
6.5.2 Green’s Function for Boundary Value Problems......Page 368
Part IV: Variational Methods\r......Page 378
7: Calculus of Variation\r......Page 380
7.1.1 Stationary Value of a Functional......Page 381
7.1.2 Fundamental Theorem of Variational Calculus......Page 383
7.1.3 Variational Notation......Page 385
7.1.4 Special Cases......Page 386
7.2 Constrained Variation......Page 390
7.3.1 The Brachistochrone Problem......Page 393
7.3.2 Isoperimetric Problems......Page 397
7.3.3 The Catenary......Page 399
7.3.4 Minimum Surface of Revolution......Page 404
7.3.5 Fermat’s Principle......Page 407
7.4.1 Functionals with Higher Derivatives......Page 410
7.4.2 Several Dependent Variables......Page 412
7.4.3 Several Independent Variables......Page 414
7.5.1 Variational Formulation of Sturm–Liouville Problems......Page 416
7.5.2 Variational Calculations of Eigenvalues and Eigenfunctions......Page 418
7.6 Rayleigh–Ritz Methods for Partial Differential Equations\r......Page 423
7.6.1 Laplace’s Equation......Page 424
7.6.2 Poisson’s Equation......Page 428
7.6.3 Helmholtz’s Equation......Page 430
7.7 Hamilton’s Principle......Page 433
References......Page 444
Index......Page 446