Differential Equations of Applied Mathematics

Contnets ix
Preface v
@=12
1. Finite Systems 1
...1.1. Vectors and Linear Spaces 1
...1.2. Matrices and Linear Transformations 12
...1.3. Ordinary Differential Systems 19
...1.4. Finite Mechanical Systems 26
...1.5. Lagrange\'s Equations and Hamilton\'s Principle 28
...1.6. Systems with Constant Coefficients 35
...1.7. The Response Matrix and Distributions 39
...1.8. Two-Point Boundary Conditions 44
2. Distributions and Waves 49
...2.1. Equations in Two Independent Variables 49
...2.2. Wave Motion of a String 53
...2.3. Reflection of Waves 59
...2.4. Theory of Distributions 66
...2.5. Applications to the Initial Problem 73
...2.6. The Separation of Variables 81
...2.7. Fourier Series 87
3. Parabolic Equations and Fourier Integrals 95
...3.1. The Heat Flow Equation 95
...3.2. Heat Flow on a Finite Interval 100
...3.3. Fourier Integral Transforms 108
...3.4. Diffusion on an Infinite Interval 113
...3.5. Semi-Infinite Intervals 118
...3.6. Fourier Transforms of Distributions 124
...3.7. Finite Difference Calculations 129
4. Laplace\'s Equation and Complex Variables 133
...4.1. Mathematical and Physical Applications 133
...4.2. Boundary Value Problems for Harmonic Functions 136
...4.3. Circular Harmonics 141
...4.4. Rectangular Harmonics 148
...4.5. Half-Plane Problems 153
...4.6. Complex Integrals 159
...4.7. Fourier and Laplace Transforms 167
...4.8. The Finite Difference Laplace Equation 173
5. Equations of Motion 180
...5.1. Vibrations of a Membrane 180
...5.2. Lateral Vibration of Rods and Plates 187
...5.3. Integral Theorems and Vector Calculus 194
...5.4. Equations of Motion of an Elastic Solid 201
...5.5. Motion of a Fluid 206
...5.6. Equations of the Electromagnetic Field 214
...5.7. Equations of Quantum Mechanics 220
6. General Theory of Eigenvalues and Eigenfunctions 228
...6.1. The Minimum Problem 228
...6.2. Sequences of Eigenvalues and Eigenfunctions 233
...6.3. Variational Properties of Eigenvalues and Eigenfunctions 239
...6.4. Eigenfunction Expansions 244
...6.5. The Rayleigh-Ritz Approximation Method 250
...6.6. On the Separation of Variables 254
...6.7. Series Expansions and Integral Transforms 260
7. Green\'s Functions 264
...7.1. Inverses of Differential Operators 264
...7.2. Examples of Green\'s Functions 270
...7.3. The Neumann and Robin Functions 277
...7.4. Differential and Integral Equations 283
...7.5. Source Functions for Parabolic Equations 288
...7.6. Convergence of Series of Distributions 292
8. Cylindrical Eigenfunctions 298
...8.1. Bessel Functions 298
...8.2. Eigenfunctions for Finite Regions 306
...8.3. The Fourier-Bessel Series 310
...8.4. The Green\'s Function 314
...8.5. Functions of Large Argument 317
...8.6. Diffraction by a Cylinder 323
...8.7. Modified Bessel Functions 327
...8.8. The Hankel and Weber Formulas 333
9. Spherical Eigenfunctions 341
...9.1. Legendre Functions 341
...9.2. Eigenfunctions of the Spherical Surface 345
...9.3. Eigenfunctions for the Solid Sphere 351
...9.4. Diffraction by a Sphere: Addition Theorem 356
...9.5. Interior and Exterior Expansions 362
...9.6. Functions of Nonintegral Order 367
10. Wave Propagation in Space 372
...10.1. Characteristic Surfaces 372
...10.2. Source Function for the Wave Equation 378
...10.3. Applications. Huyghens\' Premise 385
...10.4. Electromagnetic and Elastic Waves 390
...10.5. Wave Fronts and Rays 397
...10.6. Reflection and Diffraction 401
Tables 411
Bibliography 417
Index 419