{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,7]],"date-time":"2025-10-07T12:03:11Z","timestamp":1759838591108,"version":"3.40.5"},"reference-count":23,"publisher":"Walter de Gruyter GmbH","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2018,1,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>A new algorithm for eigenvalue problems for the fractional Jacobi-type ODE is proposed.\nThe algorithm is based on piecewise approximation of the coefficients of the differential equation with subsequent recursive procedure adapted from some homotopy considerations.\nAs a result, the eigenvalue problem (which is in fact nonlinear) is replaced by a sequence of linear boundary value problems (besides the first one) with a singular linear operator called the exact functional discrete scheme (EFDS).\nA finite subsequence of <jats:italic>m<\/jats:italic> terms, called truncated functional discrete scheme (TFDS), is the basis for our algorithm.\nThe approach provides super-exponential convergence rate as <jats:inline-formula id=\"j_cmam-2017-0010_ineq_9999_w2aab3b7b1b1b6b1aab1c14b1b3Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>m<\/m:mi>\n                              <m:mo>\u2192<\/m:mo>\n                              <m:mi mathvariant=\"normal\">\u221e<\/m:mi>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" content-type=\"j_cmam-2017-0010_ineq_9999\" xlink:href=\"graphic\/j_cmam-2017-0010_eq_mi291.png\"\/>\n                        <jats:tex-math>{m\\to\\infty}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>. The eigenpairs can be computed in parallel for all given indexes.\nThe algorithm is based on some recurrence procedures including the basic arithmetical operations with the coefficients of some expansions only.\nThis is an exact symbolic algorithm (ESA) for <jats:inline-formula id=\"j_cmam-2017-0010_ineq_9998_w2aab3b7b1b1b6b1aab1c14b1b5Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>m<\/m:mi>\n                              <m:mo>=<\/m:mo>\n                              <m:mi mathvariant=\"normal\">\u221e<\/m:mi>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" content-type=\"j_cmam-2017-0010_ineq_9998\" xlink:href=\"graphic\/j_cmam-2017-0010_eq_mi286.png\"\/>\n                        <jats:tex-math>{m=\\infty}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> and a truncated symbolic algorithm (TSA) for a finite <jats:italic>m<\/jats:italic>.\nNumerical examples are presented to support the theory.<\/jats:p>","DOI":"10.1515\/cmam-2017-0010","type":"journal-article","created":{"date-parts":[[2017,6,20]],"date-time":"2017-06-20T10:01:30Z","timestamp":1497952890000},"page":"21-32","source":"Crossref","is-referenced-by-count":5,"title":["Super-Exponentially Convergent Parallel Algorithm for a Fractional Eigenvalue Problem of Jacobi-Type"],"prefix":"10.1515","volume":"18","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3115-9690","authenticated-orcid":false,"given":"Ivan","family":"Gavrilyuk","sequence":"first","affiliation":[{"name":"University of Cooperative Education Gera-Eisenach , Am Wartenberg 2, 99817 Eisenach , Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4883-6574","authenticated-orcid":false,"given":"Volodymyr","family":"Makarov","sequence":"additional","affiliation":[{"name":"Institute of Mathematics of National Academy of Sciences of Ukraine , Department of Numerical Mathematics, 3 Tereshchenkivs\u2019ka Str., 01004 Kyiv -4 , Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3497-7077","authenticated-orcid":false,"given":"Nataliia","family":"Romaniuk","sequence":"additional","affiliation":[{"name":"Institute of Mathematics of National Academy of Sciences of Ukraine , Department of Numerical Mathematics, 3 Tereshchenkivs\u2019ka Str., 01004 Kyiv -4 , Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2017,6,17]]},"reference":[{"key":"2023033115122809376_j_cmam-2017-0010_ref_001_w2aab3b7b1b1b6b1ab2b1b1Aa","doi-asserted-by":"crossref","unstructured":"G. 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