Spectral Collocation Solutions to Problems on Unbounded Domains

CONTENTS

Preface III
Contents VII
List of Figures IX
Acronyms XV

1. APPROXIMATION AND INTERPOLATION ON UNBOUNDED INTERVALS 1
1.1. Interpolation problem 1
1.2. Hermite and Sinc functions 2
1.2.1. Hermite functions 2
1.2.2. Hermite collocation 3
1.2.3. Sinc functions 5
1.2.4. Sinc collocation 6
1.3. Laguerre functions 6
1.3.1. The rate of convergence of polynominal Laguerre series 7
1.3.2. Laguerre collocation 8
1.3.3. Laguerre Gauss Radau collocation 9
1.4. Mapping techniques 11
1.4.1. Preconditioned differentiation 13
1.5. Miscellanies 14
1.5.1. The rate of convergence of eigeunfunction expansion 14
1.5.2. Polynominal transforms 15
1.5.3. Boundary condition implementation 16

2. 1D PROBLEMS ON UNBOUNDED DOMAINS 17
2.1. Some t. p. b. v. p. on the half line 18
2.1.1. Linear second order t. p. b. v. p 18
2.1.2. The Heun’s equation 21
2.1.3. Global solutions to a class of nonlinear second order t. p. b. v. p. 23
2.1.4. Systems of t. p. b. v. p. 31
2.1.5. Another boundary layer type problem 35
2.2. T. p. b. v. p. on the real line 42
2.2.1. The order of approximation for SiC 42
2.2.2. SiC vs. HC for t. p. b. v. p. on the real line 44

3. EIGENVALUE PROBLEMS 47
3.1. Singular eigenvalue problems on the half line 48
3.1.1. „Good” and „bad” eigenvalues 48
3.1.2. Problems with parameter dependent boundary conditions 51
3.1.3. Schrodinger eigenvalue problems on the half line 56
3.1.4. A singular SturmLiouville problem with a complex potential 64
3.1.5. The Orr-Sommerfeld problem for boundary-layer flows 65
3.1.6. A fourth order singular eigenvalue problem 69
3.2. Singular eigenvalue problems on the real line 71
3.2.1. The eigenfunctions orthogonality as a check of the accuracy 81
3.2.2. Continuous spectra and numerical eigenvalues 84
3.3. Solving algebraic generalized eigenvalue problems 86

4. PROBLEMS ATTACHED TO P. D. E. 87
4.1. Multidimensional problems reductive to t. p. b. v. p. 88
4.2. MoL for second order parabolic p. d. e. 90
4.2.1. The normality of D(2)H and D(2)Si matrices 90
4.2.2. The region of absolute stability of TR-BDF2 finite difference scheme 90
4.2.3. Unsteady diffusion equation on the half line 92
4.2.4. Viscous Burgers’ equation on the real line 94
4.3. Fischer’s equation 95
4.4. The BBM type equations 98
4.4.1. HC and SiC solutions to BBM 100
4.4.2. Conservation of the energy integral 104
4.5. An i. v. p. for Fokker-Planck equation 105
4.6. The KdV equation 107
4.7. Linear SchrOdinger equation 115
4.8. The NLS equation 115
4.8.1. General-power Schrodinger equation 115
4.8.2. Waveguide solutions to the cubic NLS equation 116
4.8.3. Blow-up self similar solutions to the cubic NLS equation 116
4.8.4. Radially symmetric solutions to NLS 119
4.8.5. An envelope soliton problem attached to NLS 122

5. MATLAB SCRIPTS 129
5.1. Boundary value problems 129
5.1.1. Blasius boundary value problem 129
5.1.2. A singular nonlinear t. p. b. v. p. 131
5.2. Eigenvalue problems 132
5.2.1. A fourth-order eigenvalue problem on half line 132
5.2.2. An second order eigenvalue problem on the real line 132
5.3. Initial value problems 133

6. CONCLUDING REMARKS AND OPEN PROBLEMS 135