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Wayne State University Dissertations
Collocation, Geometric, Runge-Kutta, Supergeometric, Weakly Singular
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Spectral Collocation Method For Compact Integral Operators, Can Huang
Spectral Collocation Method For Compact Integral Operators, Can Huang
Wayne State University Dissertations
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 …
\mu<1.>