## Wayne State University Dissertations

WSU Access

January 2011

Dissertation

Ph.D.

Mathematics

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.

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