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.
","abstract_html":"<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 cMk ,</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>","abstract_has_math":true,"creators":["Huang, Can"],"institution":null,"degree_name":"Ph.D.","degree_level":"Open Access Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Zhimin Zhang"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-01-01T08:00:00Z","date_published":"2011-01-01T08:00:00Z","updated_at":"2026-07-24T05:58:57Z","subjects":["Collocation, Geometric, Runge-Kutta, Supergeometric, Weakly Singular","Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://digitalcommons.wayne.edu/oa_dissertations/352","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Zhimin Zhang"]},{"key":"dc:creator","label":"Author","values":["Huang, Can"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2011-01-01T08:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Open Access Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Collocation, Geometric, Runge-Kutta, Supergeometric, Weakly Singular","Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://digitalcommons.wayne.edu/oa_dissertations/352"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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.
"]},{"key":"dc:title","label":"Title","values":["Spectral collocation method for compact integral operators"]}]}],"canonical_facts":{"dc:contributor":["Zhimin Zhang"],"dc:creator":["Huang, Can"],"dc:date.available":["2011-01-01T08:00:00Z"],"dc:description.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.
"],"dc:identifier":["https://digitalcommons.wayne.edu/oa_dissertations/352"],"dc:subject":["Collocation, Geometric, Runge-Kutta, Supergeometric, Weakly Singular","Mathematics"],"dc:title":["Spectral collocation method for compact integral operators"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Open Access Dissertation"],"thesis:degree_name":["Ph.D."]},"updated_at":"2026-07-24T05:58:57Z"}