{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T19:54:44Z","timestamp":1777751684662,"version":"3.51.4"},"reference-count":25,"publisher":"American Mathematical Society (AMS)","issue":"307","license":[{"start":{"date-parts":[[2018,2,15]],"date-time":"2018-02-15T00:00:00Z","timestamp":1518652800000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11571238"],"award-info":[{"award-number":["11571238"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11301343"],"award-info":[{"award-number":["11301343"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11571238"],"award-info":[{"award-number":["11571238"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11301343"],"award-info":[{"award-number":["11301343"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    In this paper, we present an\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"h p\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>h<\/mml:mi>\n                            <mml:mi>p<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">hp<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -version Legendre-Jacobi spectral collocation method for Volterra integro-differential equations with smooth and weakly singular kernels. We establish several new approximation results of the Legendre\/Jacobi polynomial interpolations for both smooth and singular functions. As applications of these approximation results, we derive\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"h p\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>h<\/mml:mi>\n                            <mml:mi>p<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">hp<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -version error bounds of the Legendre-Jacobi collocation method under the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H Superscript 1\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>H<\/mml:mi>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">H^1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -norm for the Volterra integro-differential equations with smooth solutions on arbitrary meshes and singular solutions on quasi-uniform meshes. We also show the exponential rates of convergence for singular solutions by using geometric time partitions and linearly increasing polynomial degrees. Numerical experiments are included to illustrate the theoretical results.\n                  <\/p>","DOI":"10.1090\/mcom\/3183","type":"journal-article","created":{"date-parts":[[2016,5,26]],"date-time":"2016-05-26T08:34:48Z","timestamp":1464251688000},"page":"2285-2324","source":"Crossref","is-referenced-by-count":51,"title":["An \u210e\ud835\udc5d-version Legendre-Jacobi spectral collocation method for Volterra integro-differential equations with smooth and weakly singular kernels"],"prefix":"10.1090","volume":"86","author":[{"given":"Zhong-qing","family":"Wang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yu-ling","family":"Guo","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Li-jun","family":"Yi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2017,2,15]]},"reference":[{"issue":"165","key":"1","doi-asserted-by":"publisher","first-page":"95","DOI":"10.2307\/2007561","article-title":"Implicit Runge-Kutta methods of optimal order for Volterra integro-differential equations","volume":"42","author":"Brunner, Hermann","year":"1984","journal-title":"Math. 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