Access Type
Open Access Dissertation
Date of Award
January 2011
Degree Type
Dissertation
Degree Name
Ph.D.
Department
Mathematics
First Advisor
Zhimin Zhang
Abstract
We propose and analyze a spectral collocation method for integral
equations with compact kernels, e.g. piecewise smooth kernels and
weakly singular kernels of the form $\frac{1}{|t-s|^\mu}, \;
0<\mu<1. $ We prove that 1) for integral equations, the convergence
rate depends on the smoothness of true solutions $y(t)$. If $y(t)$
satisfies condition (R): $\|y^{(k)}\|_{L^\infty[0,T]}\leq
ck!R^{-k}$}, we obtain a geometric rate of convergence; if $y(t)$
satisfies condition (M): $\|y^{(k)}\|_{L^{\infty}[0,T]}\leq cM^k $,
we obtain supergeometric rate of convergence for both Volterra
equations and Fredholm equations and related integro differential
equations; 2) for eigenvalue problems, the convergence rate depends
on the smoothness of eigenfunctions. The same convergence rate for
the largest modulus eigenvalue approximation can be obtained.
Moreover, the convergence rate doubles for positive compact
operators. Our numerical experiments confirm our theoretical
results.
Recommended Citation
Huang, Can, "Spectral collocation method for compact integral operators" (2011). Wayne State University Dissertations. 352.
https://digitalcommons.wayne.edu/oa_dissertations/352