Spectral Collocation Solutions to Problems on Unbounded Domains

1 Interpolation on Unbounded Intervals
1.1 Interpolation problem
1.2 Hermite and Sinc functions
1.2.1 Hermite functions
1.2.2 Hermite collocation
1.2.3 Sinc functions
1.2.4 Sinc collocation
1.3 Laguerre functions
1.3.1 The rate of convergence of polynomial Laguerre series
1.3.2 Laguerre collocation
1.3.3 Laguerre Gauss Radau collocation
1.4 Mapping techniques
1.4.1 Preconditioned differentiation
1.5 Miscellanies
1.5.1 The rate of convergence of eigenfunction expansion
1.5.2 Polynomial transforms
1.5.3 Boundary condition implementation

2  1D Problems on Unbounded Domains
2.1 Some TPBVP on the half line
2.1.1 Linear second order TPBVP
2.1.2 The Heun’s equation
2.1.3 Global solutions to a class of nonlinear second order TPBVP
2.1.4 Systems of TPBVP
2.1.5 Another boundary layer type problem
2.2 TPBVP on the real line
2.2.1 The order of approximation for SiC
2.2.2 SiC vs. HC for TPBVP on the real line

3 Eigenvalue Problems
3.1 Singular eigenvalue problems on the half line
3.1.1 ”Good” and ”bad” eigenvalues
3.1.2 Problems with parameter dependent boundary conditions
3.1.3 Schrodinger eigenvalue problems on the half line
3.1.4 A singular SturmLiouville problem with a complex potential
3.1.5 The Orr-Sommerfeld problem for boundary-layer flows
3.1.6 A fourth order singular eigenvalue problem
3.2 Singular eigenvalue problems on the real line
3.2.1 The eigenfunctions orthogonality as a check of the accuracy
3.2.2 Continuous spectra and numerical eigenvalues
3.3 Solving algebraic generalized eigenvalue problems

4 Problems Attached to PDEs
4.1 Multidimensional problems reductive to TPBVP
4.2 MoL for second order parabolic PDEs
4.2.1 The normality of D(2) H and D(2) Si matrices
4.2.2 The region of absolute stability of TR-BDF2 finite difference scheme
4.2.3 Unsteady diffusion equation on the half line
4.2.4 Viscous Burgers’ equation on the real line
4.3 Fischer’s equation
4.4 The BBM type equations
4.4.1 HC and SiC solutions to BBM
4.4.2 Conservation of the energy integral
4.5 An IVP for Fokker-Planck equation
4.6 The KdV equation
4.7 Linear Schroedinger equation
4.8 The NLS equation
4.8.1 General-power Schroedinger equation
4.8.2 Waveguide solutions to the cubic NLS equation
4.8.3 Blow-up self similar solutions to the cubic NLS equation
4.8.4 Radially symmetric solutions to NLS
4.8.5 An envelope soliton problem attached to NLS

5 MATLAB scripts
5.1 Boundary value problems
5.1.1 Blasius boundary value problem
5.1.2 A singular nonlinear TPBVP
5.2 Eigenvalue problems
5.2.1 A fourth-order eigenvalue problem on half line
5.2.2 An second order eigenvalue problem on the real line
5.3 Initial value problems

6 Concluding remarks and open problems

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