《METHODS OF APPLIED MATHEMATICS》

CHAPTER 1MATRICES,DETERMINANTS,AND LINEAR EQUATINS1

1．1．Introduction1

1．2．Linear Equations．The Gauss-Jordan Reduction1

1．3．Matrices4

1．4．Determinants．Cramer＇s Rule10

1．5．Special Matrices13

1．6．The Inverse Matrix15

1．7．Elementary Operations18

1．8．Solvability of Sets of Linear Equations21

1．9．Linear Vector Space23

1．10．Linear Equations and Vector Space27

1．11．Characteristic-value Problems29

1．12．Orthogonalization of Vector Sets34

1．13．Quadratic Forms35

1．14．A Numerical Example39

1．15．Equivalent Matrices and Transformations42

1．16．Hermitian Matrices42

1．17．Definite Forms46

1．18．Discriminants and Invariants49

1．19．Coordinate Transformations53

1．20．Diagonalization of Symmetric Matrices56

1．21．Multiple Characteristic Numbers59

1．22．Functions of Symmetric Matrices62

1．23．Numerical Solution of Characteristic-value Problems68

1．24．Additional Techniques70

1．25．Generalized Characteristic-value Problems74

1．26．Characteristic Numbers of Nonsymmetric Matrices80

1．27．A Physical Application83

1．28．Function Space87

1．29．Sturm-Liouville Problems95

Problems100

CHAPTER 2CALCULUS OF VARIATIONS AND APPLICATIONS120

2．1．Maxima and Minima120

2．2．The Simplest Case125

2．3．Illustrative Examples128

2．4．The Variational Notation130

2．5．The More General Case134

2．6．Constraints and Lagrange Multipliers139

2．7．Sturm-Liouville Problems144

2．8．Hamilton＇s Principle147

2．9．Lagrange＇s Equations150

2．10．Generalized Dynamical Entities155

2．11．Constraints in Dynamical Systems162

2．12．Small Vibrations about Equilibrium．Normal Coordinates168

2．13．Numerical Example174

2．14．Variational Problems for Deformable Bodies177

2．15．Useful Transformations183

2．16．The Variational Problem for the Elastic Plate185

2．17．The Ritz Method187

2．18．A Semidirect Method197

Problems200

CHAPTER 3DIFFERENCE EQUATIONS227

3．1．Introduction227

3．2．Difference Operators230

3．3．Formulation of Difference Equations233

3．4．Homogeneous Linear Difference Equations with Constant Coefficients237

3．5．Particular Solutions of Nonhomogeneous Linear Equations242

3．6．The Loaded String249

3．7．Properties of Sums and Differences257

3．8．Special Finite Sums259

3．9．Characteristic-value Problems267

3．10．Matrix Notation270

3．11．The Vibrating Loaded String272

3．12．Linear Equations with Variable Coefficients275

3．13．Approximate Solution of Ordinary Differential Equations277

3．14．The One-dimensional Heat-flow Equation282

3．15．The Two-dimensional Heat-flow Equation286

3．16．Laplace＇s Equation in Two Dimensions292

3．17．Relaxation Methods and Laplace＇s Equation295

3．18．Treatment of Boundary Conditions301

3．19．Other Applications of Relaxation Methods309

3．20．Convergence of Finite-difference Approximations319

3．21．The One-dimensional Wave Equation322

3．22．Instability328

3．23．Stability Criteria334

Problems346

CHAPTER 4INTEGRAL EQUATIONS381

4．1．Introduction381

4．2．Relations between Differential and Integral Equations384

4．3．The Green＇s Function388

4．4．Alternative Definition of Green＇s Function394

4．5．Linear Equations in Cause and Effect．The Influence Function401

4．6．Fredholm Equations with Separable Kernels406

4．7．Illustrative Example409

4．8．Hilbert-Schmidt Theory411

4．9．Iterative Methods for Solving Equations of the Second Kind421

4．10．The Neumann Series429

4．11．Fredholm Theory432

4．12．Singular Integral Equations435

4．13．Special Devices438

4．14．Iterative Approximations to Characteristic Functions442

4．15．Approximation of Fredholm Equations by Sets of Algebraic Equations444

4．16．Approximate Methods of Undetermined Coefficients448

4．17．The Method of Collocation450

4．18．The Method of Weighting Functions451

4．19．The Method of Least Squares452

4．20．Approximation of the Kernel459

Problems461

Appendix:The Crout Method for Solving Sets of Linear Algebraic Equations503

Answers to Problems509

Index519