Spectral Methods for Differential Problems

I First Part

1 Chebyshev polynomials
1.1 General properties
1.2 Fourier and Chebyshev Series
1.2.1 The trigonometric Fourier series
1.2.2 The Chebyshev series
1.2.3 Discrete least square approximation
1.2.4 Chebyshev discrete least square approximation
1.2.5 Orthogonal polynomials least square approximation
1.2.6 Orthogonal polynomials and Gauss-type quadrature formulas
1.3 Chebyshev projection
1.4 Chebyshev interpolation
1.4.1 Collocation derivative operator
1.5 Problems

2 Spectral methods for ODEs
2.1 The idea behind the spectral methods
2.2 General formulation for linear problems
2.3 Tau-spectral method
2.4 Collocation spectralmethods (pseudospectral)
2.4.1 A class of nonlinear boundary value problems
2.5 Spectral-Galerkin methods
2.6 Problems

3 Spectral methods for PDEs
3.1 Parabolic problems
3.2 Conservative PDEs
3.3 Hyperbolic problems
3.4 Problems

4 Efficient implementation
4.1 Second order Dirichlet problems for ODEs
4.2 Third and fourth order Dirichlet problems for ODEs
4.3 Problems

5 Eigenvalue problems
5.1 Standard eigenvalue problems
5.2 Theoretical analysis of a model problem
5.3 Non-standard eigenvalue problems
5.4 Problems

II Second Part

6 Non-normality of spectral approximation
6.1 A scalarmeasure of non-normality
6.2 A CG method with different trial and test basis functions
6.3 Numerical experiments
6.3.1 Second order problems
6.3.2 Fourth order problems
6.3.3 Complex Schrodinger operators

7 Concluding remarks

8 Appendix
8.1 Lagrangian and Hermite interpolation
8.2 Sobolev spaces
8.2.1 The Spaces Cm¡Ω¢, m ≥ 0
8.2.2 The Lebesgue Integral and Spaces Lp (a, b) , 1 ≤ p ≤ ∞
8.2.3 Infinite Differentiable Functions and Distributions
8.2.4 Sobolev Spaces and Sobolev Norms
8.2.5 TheWeighted Spaces
8.3 MATLAB codes

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