Spectral collocation method for compact integral operators

<p>We propose and analyze a spectral collocation method for integral</p> <p>equations with compact kernels, e.g. piecewise smooth kernels and</p> <p>weakly singular kernels of the form \frac{1}{|t-s|^\mu}, </p> <p>0<μ<1. We prove that 1) for integral equations, the convergence</p> <p>rate depends on the smoothness of true solutions $y(t)$. If $y(t)$</p> <p>satisfies condition (R): \|y^(k)\|_{L^\infty[0,T]}\leq</p> <p>ck!R^-k}, we obtain a geometric rate of convergence; if $y(t)$</p> <p>satisfies condition (M): \|y^(k)\|_{L^\infty[0,T]}\leq cM^k ,</p> <p>we obtain supergeometric rate of convergence for both Volterra</p> <p>equations and Fredholm equations and related integro differential</p> <p>equations; 2) for eigenvalue problems, the convergence rate depends</p> <p>on the smoothness of eigenfunctions. The same convergence rate for</p> <p>the largest modulus eigenvalue approximation can be obtained.</p> <p>Moreover, the convergence rate doubles for positive compact</p> <p>operators. Our numerical experiments confirm our theoretical</p> <p>results.</p>

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Open Access Dissertation

Discipline thesis:degree_discipline

Mathematics

Year dc:date.available

2011

Author dc:creator

• Huang, Can

Contributors dc:contributor

• Zhimin Zhang

• Collocation, Geometric, Runge-Kutta, Supergeometric, Weakly Singular
• Mathematics

Repository record dc:identifier

https://digitalcommons.wayne.edu/oa_dissertations/352

OAI identifier oai:identifier

oai:digitalcommons.wayne.edu:oa_dissertations-1351