Accurate Spectral Collocation Solutions to 2nd-Order Sturm–Liouville Problems
Abstract
:1. Introduction
2. 2nd-Order Sturm–Liouville Eigenproblems
3. Chebfun vs. Conventional Spectral Collocation
3.1. Chebfun
3.2. Spectral Collocation Methods
3.3. The Drift of Eigenvalues
3.4. Preconditioning
4. Numerical Experiments
4.1. Hinged Ends or Simply Supported Boundary Conditions
4.1.1. The Viola’s Eigenproblem-Revisited
4.1.2. The Bénard Stability Problem
N=256; % order of approximation nu=-(1+4*(pi^2))-sqrt(1+4*(pi^2)); % parameter \nu [x,D]=chebdif(N,2); D2=D(2:N-1,2:N-1,2); % differentiation matrices I=eye(size(D2)); Z=zeros(size(D2)); A=[ 4*D2 -I Z; Z 4*D2 -I; -(nu^2+2*nu+2)*I (nu^2+4*nu+3)*I 4*D2-(2*nu+3)*I]; B=[Z Z Z; Z Z Z; I Z Z]; % block matrices in pencil k = 8; % number of computed eigs E=eigs( @(x)(A\(B*x)), size(A,1),k, ’SM’) % Arnoldi method
4.1.3. A Self-Adjoint Eighth-Order Problem
4.2. Clamped Boundary Conditions
4.2.1. A Fourth Order Problem with a Third Derivative Term
4.2.2. A Fourth Order Eigenproblem from Spherical Geometry
4.2.3. A Set of Sixth Order Eigenproblems
4.3. Problems with Mixed Boundary Conditions
4.3.1. The Free Lateral Vibration of a Uniform Clamped–Hinged Beam
4.3.2. A Fourth Order Eigenproblem with Higher Order Boundary Conditions
% Cantilevered beam in Euler-Bernouilli theory dom=[0,pi];x=chebfun(’x’,dom); % the domain L = chebop(dom); L.op = @(x,y) diff(y,4); % the operator L.lbc = @(y)[y; diff(y,1)]; % fixed b. c. L.rbc = @(y)[diff(y,2); diff(y,3)];% free b. c. [U,D]=eigs(L,40,’SM’); % first six eigs. % Sorted eigenpairs (eigenvalues and eigenvectors) D=diag(D); [t,o]=sort(D); D=D(o); disp((D.^(1/4))) U=U(:,o);
4.3.3. The Harmonic Oscillator and Its Second and Third Powers
% The sinc differentiation matrices [Weideman & Reddy] N=400;M=6;h=0.1; %Orders of approximation and differentiation and scaling factor [x, D] = sincdif(N, M, h); D1=D(:,:,1);D2=D(:,:,2);D6=D(:,:,6); % The cube of the "harmonic oscillator" operator L=-D6+D2*(3*diag(x.^2)*D2)+D1*(diag(8-3*(x.^4))*D1)+diag(x.^6-14*(x.^2)); % Finding eigenpairs of L [U,S]=eigs(L,250,0); S=diag(S); [t,o]=sort(S); S=S(o);
5. Conclusions and Open Problems
Funding
Acknowledgments
Conflicts of Interest
Abbreviations
ChC | Chebyshev collocation method |
ChT | Chebyshev tau method |
strategy to reduce a 2nd-order equation to a second order system | |
FCT | fast Chebyshev transform |
FD | finite difference method |
FE | finite element method |
MATSLISE | a MATLAB package for the numerical solution of SL and Schröedinger equations |
SiC | sinc spectral collocation |
SL | Sturm–Liouville |
SLEDGE | Sturm–Liouville estimates determined by global errors |
SLEUTH | Sturm–Liouville Eigenvalues using Theta Matrices |
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according to [14] | |||
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i | Chebfun | Preconditioned Chebfun | Solution to (11) |
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1 | |||
2 |
j | Chebfun | According to [12] |
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1 | ||
2 | ||
3 | ||
4 | ||
5 |
j | by Chebfun | Exact Solutions of (18) |
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1 | ||
2 | ||
3 | ||
4 |
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Gheorghiu, C.-I. Accurate Spectral Collocation Solutions to 2nd-Order Sturm–Liouville Problems. Symmetry 2021, 13, 385. https://doi.org/10.3390/sym13030385
Gheorghiu C-I. Accurate Spectral Collocation Solutions to 2nd-Order Sturm–Liouville Problems. Symmetry. 2021; 13(3):385. https://doi.org/10.3390/sym13030385
Chicago/Turabian StyleGheorghiu, Călin-Ioan. 2021. "Accurate Spectral Collocation Solutions to 2nd-Order Sturm–Liouville Problems" Symmetry 13, no. 3: 385. https://doi.org/10.3390/sym13030385