{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T04:13:17Z","timestamp":1776831197125,"version":"3.51.2"},"reference-count":30,"publisher":"American Mathematical Society (AMS)","issue":"250","license":[{"start":{"date-parts":[[2005,10,27]],"date-time":"2005-10-27T00:00:00Z","timestamp":1130371200000},"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                    We consider the Cauchy problem for the first and the second order differential equations in Banach and Hilbert spaces with an operator coefficient\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper A left-parenthesis t right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>A<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">A(t)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    depending on the parameter\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"t\">\n                        <mml:semantics>\n                          <mml:mi>t<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">t<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We develop discretization methods with high parallelism level and without accuracy saturation; i.e., the accuracy adapts automatically to the smoothness of the solution. For analytical solutions the rate of convergence is exponential. These results can be viewed as a development of parallel approximations of the operator exponential\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"e Superscript minus t upper A\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>e<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\n                                \u2212\n                                \n                              <\/mml:mo>\n                              <mml:mi>t<\/mml:mi>\n                              <mml:mi>A<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">e^{-tA}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and of the operator cosine family\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"cosine StartRoot upper A EndRoot t\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>cos<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:msqrt>\n                                <mml:mi>A<\/mml:mi>\n                              <\/mml:msqrt>\n                              <mml:mi>t<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\cos {\\sqrt {A} t}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with a constant operator\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper A\">\n                        <mml:semantics>\n                          <mml:mi>A<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">A<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    possessing exponential accuracy and based on the Sinc-quadrature approximations of the corresponding Dunford-Cauchy integral representations of solutions or the solution operators.\n                  <\/p>","DOI":"10.1090\/s0025-5718-04-01720-x","type":"journal-article","created":{"date-parts":[[2005,1,14]],"date-time":"2005-01-14T14:25:09Z","timestamp":1105712709000},"page":"555-583","source":"Crossref","is-referenced-by-count":6,"title":["Algorithms without accuracy saturation for evolution equations in Hilbert and Banach spaces"],"prefix":"10.1090","volume":"74","author":[{"given":"Ivan","family":"Gavrilyuk","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Volodymyr","family":"Makarov","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2004,10,27]]},"reference":[{"key":"1","volume-title":"{\\cyr Osnovy chislennogo analiza}","author":"Babenko, K. 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