Asymptotic analysis of differential equations / Roscoe B White.

Format:

Book

Author/Creator:

White, R. B.

Language:

English

Subjects (All):

Differential equations--Asymptotic theory.

Differential equations.

Physical Description:

xviii, 286 pages : illustrations ; 24 cm

Place of Publication:

London : Imperial College Press ; Singapore ; Hackensack, NJ : Distributed by World Scientific, [2005]

Summary:

An essential graduate level text on the asymptotic analysis of ordinary differential equations, this book covers all the important methods including dominant balance, the use of divergent asymptotic series, phase integral methods, asymptotic evaluation of integrals, and boundary layer analysis. The construction of integral solutions and the use of analytic continuation are used in conjunction with the asymptotic analysis, to show the interrelatedness of these methods. Some of the functions of classical analysis are used as examples, to provide an introduction to their analytic and asymptotic properties, and to give derivations of some of the important identities satisfied by them. There is no attempt to give a complete presentation of all these functions. The emphasis is on the various techniques of analysis: obtaining asymptotic limits, connecting different asymptotic solutions, and obtaining integral representation.

Contents:

1 Dominant Balance 1

1.2 Solutions Using Kruskal-Newton Graphs 3

1.2.1 Third order 3

1.2.2 Non polynomial form 5

1.2.3 Higher order 7

1.2.4 Hidden points 8

2 Exact Solutions 11

2.2 Constant Coefficients 13

2.3 Inhomogeneous Linear Equations 14

2.4 The Fredholm Alternative 16

2.5 The Diffusion Equation 18

2.6 Exact Solutions to Nonlinear Equations 19

2.7 Phase Plane Analysis 19

3 Complex Variables 25

3.1 Analyticity 25

3.2 Cauchy Integral Theorem 26

3.3 Series Representation 28

3.4 The Residue Theorem 29

3.5 Analytic Continuation 31

3.6 Inverse Functions 32

3.7 Problems 34

4 Local Approximate Solutions 37

4.2 Classification 38

4.2.1 Ordinary point 39

4.2.2 Regular singular point 39

4.2.3 Irregular singular point 42

4.3 Asymptotic Series 45

4.3.1 Properties 46

4.3.2 Truncation: A series about x = 0 48

4.3.3 Truncation: A series about x = [infinity] 50

4.3.4 Truncation: A series about x = 0 51

4.3.5 Asymptotic oscillation 55

4.4 Construction of Asymptotic Series 55

4.4.1 The error function 56

4.5 Origin of the Divergence 58

4.6 Improving Series Convergence 60

4.6.1 Shanks transformation 60

4.6.2 Euler summation 61

4.6.3 Borel summation 62

5 Phase Integral Methods 67

5.2 Connection Formulae: Isolated Zero 71

5.3 Derivation of Stokes Constants 73

5.4 Rules for Continuation 76

5.5 Causality 77

5.6 Bound States and Instabilities 78

5.7 Scattering 81

5.8 Eigenvalue Problems 85

5.9 The Budden Problem 86

5.10 The Error Function 88

6 Perturbation Theory 91

6.2 Eigenvalues of a Hermitian Matrix 93

6.3 Broken Symmetry Due to Tunneling 97

7 Asymptotic Evaluation of Integrals 103

7.2 End Point 105

7.3 Saddle Point 108

8 The Euler Gamma Function 115

8.2 The Stirling Approximation 117

8.3 The Euler-Mascheroni Constant 119

8.4 Sine Product Identity 120

8.5 Continuation of [Gamma](z) 122

8.6 Asymptotic [Gamma](z) 122

8.7 Euler Product for [Gamma] 125

8.8 Integral Representation for 1/[Gamma](z) 126

8.9 [Gamma](nx) 128

8.10 The Euler Beta Function 128

9 Integral Solutions 131

9.1 Constructing Integral Solutions 131

9.1.1 Integration by parts 132

9.1.2 Finding a discrete difference equation 134

9.1.3 Construction from a series 135

9.2 Causal Solutions 136

10 Expansion in Basis Functions 141

10.1 Legendre Functions 142

10.1.1 Local analysis 142

10.1.2 Euler integral representation 143

10.1.3 Recurrence relations 145

10.1.4 Laplace integral representation 147

10.1.5 Generating function 149

10.1.6 Legendre polynomials 150

10.2 Orthogonal Polynomials 151

10.3 Wavelets 154

10.3.2 Scaling function 154

10.3.3 Wavelet basis construction 158

10.3.4 Determining the expansion coefficients 160

10.3.6 Time-frequency analysis 166

11 Airy 171

11.1 WKB Analysis 172

11.2 Fourier Laplace Integral Representation 174

11.3 Asymptotic Limits 176

11.3.1 Large negative z 176

11.3.2 Large positive z 178

11.3.3 Small [vertical bar]z[vertical bar] 180

11.4 Mellin Integral Representation 181

11.5 Matching Local Solutions 183

11.6 The Wronskian 185

12 Bessel 187

12.1 Local Analysis 187

12.1.1 Local at zero 187

12.1.2 Analytic continuation in v 189

12.1.3 Local at infinity 189

12.2 WKB Analysis 190

12.3 Integral Representations 191

12.3.1 Fourier Laplace representation 191

12.3.2 The Hankel integrals 192

12.3.3 Asymptotic limits 193

12.3.4 Relation to Bessel and Neumann functions 195

12.3.5 The Wronskian 197

12.3.6 Sommerfeld integral representation 198

12.3.7 Mellin integral representation 201

12.4 Generating Function 202

12.5 Matching Local Solutions 202

12.6 Imaginary Argument 203

13 Weber-Hermite 207

13.1 Local Analysis at Infinity 208

13.2 Local Analysis at Zero 209

13.3 WKB Analysis 210

13.4 Euler Integral Representation 212

13.5 Fourier-Laplace Integral Representation I 213

13.6 Orthogonality 217

13.7 Fourier-Laplace Integral Representation II 218

13.8 The Wronskian 221

13.9 Mellin Integral Representation 222

14 Whittaker and Watson 225

14.1 Local Analysis at Infinity 226

14.2 Local Analysis at Zero 227

14.3 Euler Integral Representation 227

14.4 Relation Between M[subscript lambda,mu](z) and W[subscript lambda,mu](z) 230

14.5 Fourier Laplace Integral Representation 231

14.6 Mellin Integral Representation 232

14.7 Special Cases 235

14.7.1 The error function 235

14.7.2 The logarithmic integral function 235

15 Inhomogeneous Differential Equations 237

15.1 The Driven Oscillator 238

15.2 The Struve Equation 238

15.2.1 Local analysis at zero 239

15.2.2 Local analysis at infinity 239

15.2.3 Integral representation 240

15.3 Resistive Reconnection 240

15.4 Resistive Internal Kink 241

15.5 A Causal Inhomogeneous Problem 242

16 The Riemann Zeta Function 249

16.2 [zeta](s) and [zeta](1 - s) 251

16.3 The Euler Product for [zeta](s) 253

16.4 Distribution of Prime Numbers 254

16.5 Public Key Codes 258

16.6 Stirling Revisited 259

17 Boundary Layer Problems 263

17.2 Layer Location 264

17.3 Layer at Left Boundary 267

17.4 Layer in Domain Center 269

17.5 Layer at Right Boundary 271

17.6 Nested Boundary Layers 272

Appendix Lagrange's Theorem 277.

Notes:

Includes bibliographical references (pages 279-281) and index.

ISBN:

1860945872

1860946127

OCLC:

66901304