{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T17:14:44Z","timestamp":1776791684634,"version":"3.51.2"},"reference-count":8,"publisher":"American Mathematical Society (AMS)","issue":"272","license":[{"start":{"date-parts":[[2011,4,29]],"date-time":"2011-04-29T00:00:00Z","timestamp":1304035200000},"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 second-order linear differential equations\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"phi left-parenthesis x right-parenthesis y plus f left-parenthesis x right-parenthesis y Superscript prime Baseline plus g left-parenthesis x right-parenthesis y equals h left-parenthesis x right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03c6\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mi>y<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>y<\/mml:mi>\n                              <mml:mo>\u2032<\/mml:mo>\n                            <\/mml:msup>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>g<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mi>y<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>h<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\varphi (x)y+f(x)y\u2019+g(x)y=h(x)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in the interval\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis negative 1 comma 1 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(-1,1)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with Dirichlet, Neumann or mixed Dirichlet-Neumann boundary conditions. We consider\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"phi left-parenthesis x right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03c6\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\varphi (x)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"f left-parenthesis x right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">f(x)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"g left-parenthesis x right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>g<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">g(x)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"h left-parenthesis x right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>h<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">h(x)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    analytic in a Cassini disk with foci at\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"x equals plus-or-minus 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mo>\n                              \u00b1\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">x=\\pm 1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    containing the interval\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis negative 1 comma 1 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(-1,1)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . The two-point Taylor expansion of the solution\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"y left-parenthesis x right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>y<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">y(x)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    at the extreme points\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"plus-or-minus 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo>\n                              \u00b1\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\pm 1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is used to give a criterion for the existence and uniqueness of solution of the boundary value problem. This method is constructive and provides the two-point Taylor approximation of the solution(s) when it exists.\n                  <\/p>","DOI":"10.1090\/s0025-5718-10-02370-7","type":"journal-article","created":{"date-parts":[[2010,7,6]],"date-time":"2010-07-06T17:17:12Z","timestamp":1278436632000},"page":"2103-2115","source":"Crossref","is-referenced-by-count":6,"title":["Two-point Taylor expansions and one-dimensional boundary value problems"],"prefix":"10.1090","volume":"79","author":[{"given":"Jos\u00e9","family":"L\u00f3pez","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ester","family":"Sinus\u00eda","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2010,4,29]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1016\/S0096-3003(02)00290-4","article-title":"Taylor polynomial solutions of linear differential equations","volume":"142","author":"Ke\u015fan, Cenk","year":"2003","journal-title":"Appl. Math. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0096-3003","issn-type":"print"},{"key":"2","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511755293","volume-title":"Differential equations","author":"King, A. C.","year":"2003","ISBN":"https:\/\/id.crossref.org\/isbn\/0521816580"},{"key":"3","first-page":"167","article-title":"Parabolic equations","author":"Lax, P. D.","year":"1954"},{"issue":"4","key":"4","doi-asserted-by":"publisher","first-page":"297","DOI":"10.1111\/1467-9590.00225","article-title":"Two-point Taylor expansions of analytic functions","volume":"109","author":"L\u00f3pez, Jos\u00e9 L.","year":"2002","journal-title":"Stud. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-2526","issn-type":"print"},{"issue":"2","key":"5","doi-asserted-by":"publisher","first-page":"519","DOI":"10.1016\/j.amc.2008.11.015","article-title":"Multi-point Taylor approximations in one-dimensional linear boundary value problems","volume":"207","author":"L\u00f3pez, Jos\u00e9 L.","year":"2009","journal-title":"Appl. Math. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0096-3003","issn-type":"print"},{"issue":"3","key":"6","doi-asserted-by":"publisher","first-page":"463","DOI":"10.1137\/S0036144597315341","article-title":"On the asymptotic and numerical solution of linear ordinary differential equations","volume":"40","author":"Olde Daalhuis, A. B.","year":"1998","journal-title":"SIAM Rev.","ISSN":"https:\/\/id.crossref.org\/issn\/1095-7200","issn-type":"print"},{"key":"7","doi-asserted-by":"crossref","unstructured":"M. Sezer, A method for the approximate solution of the second-order linear differential equations in terms of Taylor polynomials, Int. J. Math. Edu. Sci. Technol. 27 (1996), 821\u2013834.","DOI":"10.1080\/0020739960270606"},{"key":"8","unstructured":"I. Stakgold, Green\u2019s functions and boundary value problems, Wiley and Sons, New York, 1988."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2010-79-272\/S0025-5718-10-02370-7\/S0025-5718-10-02370-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2010-79-272\/S0025-5718-10-02370-7\/S0025-5718-10-02370-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:34:38Z","timestamp":1776789278000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2010-79-272\/S0025-5718-10-02370-7\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,4,29]]},"references-count":8,"journal-issue":{"issue":"272","published-print":{"date-parts":[[2010,10]]}},"alternative-id":["S0025-5718-10-02370-7"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-10-02370-7","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2010,4,29]]}}}