Spectral Collocation Solutions to Problems on Unbounded Domains

 Contents:

1. Interpolation on Unbounded Itervals
   1. Iterpolation problem
   2. Hermite and Sinc functions
      1. Hermite functions
      2. Hermite collocation
      3. Sinc functions
      4. Sinc collocation
   3. Laguerre functions

1.3.1. The rate of convergence of polynominal 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 eigeunfunction expansion

1.5.2. Polynominal transforms

1.5.3. Boundary condition implementation

2. 1D Problems on Unbounded Domains
   1. Some t. p. b. v. p. on the half line
      1. Linear second order t. p. b. v. p
      2. The Heun’s equation
      3. Global solutions to a class of nonlinear second order  t. p. b. v. p.
      4. Systems of t. p. b. v. p.
      5. Another boundary layer type problem
   2. T. p. b. v. p. on the real line
      1. The order of approximation for SiC
      2. SiC vs. HC for t. p. b. v. p. on the real line
3. Eigenvalue Problems
   1. Singular eigenvalue problems on the half line
      1. ”Good” and ”bad” eigenvalues
      2. Problems with parameter dependent boundary conditions
      3. Schrödinger eigenvalue problems on the half line
      4. A singular SturmLiouville problem with a complex potential
      5. The Orr-Sommerfeld problem for boundary-layer flows
      6. A fourth order singular eigenvalue problem
   2. Singular eigenvalue problems on the reali line
      1. The eigenfunctions orthogonality as a check of the accuracy
      2. Continuous spectra and numerical eigenvalues
   3. Solving algebraic generalized eigenvalue problems
4. Problems Attached to P. D. E.
   1. Multidimensional problems reductive to t. p. b. v. p.
   2. MoL for second order parabolic p. d. e.
      1. The normality of D⁽²⁾_H  and D⁽²⁾_Si matrices
      2. The region of absolute stability of TR-BDF2 finite difference scheme
      3. Unsteady diffusion equation on the half line
      4. Viscous Burgers’ equation on the real line
   3. Fischer’s equation
   4. The BBM type equations
      1. HC and SiC solutions to BBm
      2. Conservation of the energy intergral
   5. An i. v. p. for Fokker-Planck equation
   6. The KdV equation
   7. Linear Schrödinger equation
   8. The NLS equation
      1. General-power Schrödinger equation
      2. Waveguide solutions to the cubic NLS equation
      3. Blow-up self similar solutions to the cubic NLS equation
      4. Radially symmetric solutions to NLS
      5. An envelope soliton problem attached to NLS
5. MATLAB scripts
   1. Boundary value problems
      1. Blasius boundary value problem
      2. A singular nonlinear t. p. b. v. p.
   2. Eigenvalue problems
      1.  A fourth-order eigenvalue problem on half time
      2. An second order eigenvalue problem on the real line
   3. Initial value problems
6. Concluding  remarks and open problems