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Numerical Solution of Singular Eigenvalue

   Added On :  03.06.2016 02:23 pm
   For :  Numerical Methods

Numerical Solution of Singular Eigenvalue Problems for ODEs with a Focus on Problems Posed on Semi-Infinite Intervals - By R. Hammerling, O. Koch, Ch. Simon, and E. Weinm?ller.

Contents
1 Introduction 1
1.1 Mathematical Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Outline of the Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Eigenvalue Problems for Ordinary Differential Equations 5
2.1 Sturm-Liouville Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Example for a Sturm-Liouville Problem ? The Hydrogen Atom . . 8
2.2 Standard Codes for the Solution of Eigenvalue Problems in ODEs . . . . . 11
3 Problems Posed on Semi-Infinite Intervals 19
3.1 Transformation of the Independent Variable . . . . . . . . . . . . . . . . . 19
3.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4 Collocation Methods for the Computation of Eigenvalues and Eigenfunctions 25
4.1 Conversion of the Eigenvalue Problem to a ?Standard? BVP . . . . . . . . 25
4.2 Singular Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . 26
4.3 Polynomial Collocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.4 Convergence of Collocation Methods . . . . . . . . . . . . . . . . . . . . . 28
4.5 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.7 Implementation of the Eigenvalue Module . . . . . . . . . . . . . . . . . . 43
5 Matrix Methods 47
5.1 The Algebraic Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . . 47
5.2 A Matrix Method for a Singular First Order Differential EVP . . . . . . . 48
5.3 A Matrix Method for a Singular Second Order Differential EVP . . . . . . 54
5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.5 Matrix Method ? a Procedure for Obtaining Accurate Starting Profiles . . 63
5.6 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.6.1 Files for this Module . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.6.2 Getting started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.6.3 Graphical User Interface Tutorial . . . . . . . . . . . . . . . . . . . 77
5.6.4 Graphical User Interface Controls . . . . . . . . . . . . . . . . . . . 78
5.6.5 The evpfile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6 Conclusions 85

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