{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:49:41Z","timestamp":1776836981979,"version":"3.51.2"},"reference-count":19,"publisher":"American Mathematical Society (AMS)","issue":"255","license":[{"start":{"date-parts":[[2007,5,3]],"date-time":"2007-05-03T00:00:00Z","timestamp":1178150400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    In the present paper, the modified Runge-Kutta method is constructed, and it is proved that the modified Runge-Kutta method preserves the order of accuracy of the original one. The necessary and sufficient conditions under which the modified Runge-Kutta methods with the variable mesh are asymptotically stable are given. As a result, the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"theta\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03b8\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\theta<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -methods with\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"one half less-than-or-equal-to theta less-than-or-equal-to 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mstyle displaystyle=\"false\" scriptlevel=\"0\">\n                              <mml:mfrac>\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:mfrac>\n                            <\/mml:mstyle>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mi>\n                              \u03b8\n                              \n                            <\/mml:mi>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\tfrac 12\\leq \\theta \\leq 1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , the odd stage Gauss-Legendre methods and the even stage Lobatto\n                    <roman>IIIA<\/roman>\n                    and\n                    <roman>IIIB<\/roman>\n                    methods are asymptotically stable. Some experiments are given.\n                  <\/p>","DOI":"10.1090\/s0025-5718-06-01844-8","type":"journal-article","created":{"date-parts":[[2006,5,24]],"date-time":"2006-05-24T14:43:01Z","timestamp":1148481781000},"page":"1201-1215","source":"Crossref","is-referenced-by-count":18,"title":["The stability of modified Runge-Kutta methods for the pantograph equation"],"prefix":"10.1090","volume":"75","author":[{"given":"M.","family":"Liu","sequence":"first","affiliation":[]},{"given":"Z.","family":"Yang","sequence":"additional","affiliation":[]},{"given":"Y.","family":"Xu","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2006,5,3]]},"reference":[{"issue":"4","key":"1","doi-asserted-by":"publisher","first-page":"529","DOI":"10.1093\/imanum\/22.4.529","article-title":"Preservation of superconvergence in the numerical integration of delay differential equations with proportional delay","volume":"22","author":"Bellen, Alfredo","year":"2002","journal-title":"IMA J. 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