Methods of Applied Mathematics

CHAPTER 1 MATRICES，DETERMINANTS，AND LINEAR EQUATIONS 1

1．1．Introduction 1

1．2．Linear Equations．The Gauss-Jordan Reduction 1

1．3．Matrices 4

1．4．Determinants．Cramer＇s Rule 10

1．5．Special Matrices 13

1．6．The Inverse Matrix 15

1．7．Elementary Operations 18

1．8．Solvability of Sets of Linear Equations 21

1．9．Linear Vector Space 23

1．10．Linear Equations and Vector Space 27

1．11．Characteristic-value Problems 29

1．12．Orthogonalization of Vector Sets 34

1．13．Quadratic Forms 35

1．14．A Numerical Example 39

1．15．Equivalent Matrices and Transformations 42

1．16．Hermitian Matrices 42

1．17．Definite Forms 46

1．18．Discriminants and Invariants 49

1．19．Coordinate Transformations 53

1．20．Diagonalization of Symmetric Matrices 56

1．21．Multiple Characteristic Numbers 59

1．22．Functions of Symmetric Matrices 62

1．23．Numerical Solution of Characteristic-value Problems 68

1．24．Additional Techniques 70

1．25．Generalized Characteristic-value Problems 74

1．26．Characteristic Numbers of Nonsymmetric Matrices 80

1．27．A Physical Application 83

1．28．Function Space 87

1．29．Sturm-Liouville Problems 95

Problems 100

CHAPTER 2 CALCULUS OF VARIATIONS AND APPLICATIONS 120

2．1．Maxima and Minima 120

2．2．The Simplest Case 125

2．3．Illustrative Examples 128

2．4．The Variational Notation 130

2．5．The More General Case 134

2．6．Constraints and Lagrange Multipliers 139

2．7．Sturm-Liouville Problems 144

2．8．Hamilton＇s Principle 147

2．9．Lagrange＇s Equations 150

2．10．Generalized Dynamical Entities 155

2．11．Constraints in Dynamical Systems 162

2．12．Small Vibrations about Equilibrium．Normal Coordinates 168

2．13．Numerical Example 174

2．14．Variational Problems for Deformable Bodies 177

2．15．Useful Transformations 183

2．16．The Variational Problem for the Elastic Plate 185

2．17．The Ritz Method 187

2．18．A Semidirect Method 197

Problems 200

CHAPTER 3 DIFFERENCE EQUATIONS 227

3．1．Introduction 227

3．2．Difference Operators 230

3．3．Formulation of Difference Equations 233

3．4．Homogeneous Linear Difference Equations with Constant Coefficients 237

3．5．Particular Solutions of Nonhomogeneous Linear Equations 242

3．6．The Loaded String 249

3．7．Properties of Sums and Differences 257

3．8．Special Finite Sums 259

3．9．Characteristic-value Problems 267

3．10．Matrix Notation 270

3．11．The Vibrating Loaded String 272

3．12．Linear Equations with Variable Coefficients 275

3．13．Approximate Solution of Ordinary Differential Equations 277

3．14．The One-dimensional Heat-flow Equation 282

3．15．The Two-dimensional Heat-flow Equation 286

3．16．Laplace＇s Equation in Two Dimensions 292

3．17．Relaxation Methods and Laplace＇s Equation 295

3．18．Treatment of Boundary Conditions 301

3．19．Other Applications of Relaxation Methods 309

3．20．Convergence of Finite-difference Approximations 319

3．21．The One-dimensional Wave Equation 322

3．22．Instability 328

3．23．Stability Criteria 334

Problems 346

CHAPTER 4 INTEGRAL EQUATIONS 381

4．1．Introduction 381

4．2．Relations between Differential and Integral Equations 384

4．3．The Green＇s Function 388

4．4．Alternative Definition of Green＇s Function 394

4．5．Linear Equations in Cause and Effect．The Influence Function 401

4．6．Fredholm Equations with Separable Kernels 406

4．7．Illustrative Example 409

4．8．Hilbert-Schmidt Theory 411

4．9．Iterative Methods for Solving Equations of the Second Kind 421

4．10．The Neumann Series 429

4．11．Fredholm Theory 432

4．12．Singular Integral Equations 435

4．13．Special Devices 438

4．14．Iterative Approximations to Characteristic Functions 442

4．15．Approximation of Fredholm Equations by Sets of Algebraic Equations 444

4．16．Approximate Methods of Undetermined Coefficients 448

4．17．The Method of Collocation 450

4．18．The Method of Weighting Functions 451

4．19．The Method of Least Squares 452

4．20．Approximation of the Kernel 459

Problems 461

Appendix：The Crout Method for Solving Sets of Linear Algebraic Equations 503

Answers to Problems 509

Index 519