{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,5]],"date-time":"2025-11-05T11:16:39Z","timestamp":1762341399281,"version":"3.40.5"},"reference-count":24,"publisher":"Walter de Gruyter GmbH","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2021,1,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We represent the solution <jats:inline-formula id=\"j_cmam-2019-0120_ineq_9999\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>u<\/m:mi>\n                              <m:mo>\u2062<\/m:mo>\n                              <m:mrow>\n                                 <m:mo stretchy=\"false\">(<\/m:mo>\n                                 <m:mi>t<\/m:mi>\n                                 <m:mo stretchy=\"false\">)<\/m:mo>\n                              <\/m:mrow>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2019-0120_eq_0352.png\"\/>\n                        <jats:tex-math>{u(t)}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> of an initial value problem (IVP)\nfor the first-order differential equation with an operator coefficient\nas a series using the Cayley transform of the corresponding operator coefficient and the Laguerre polynomials.\nIn the case of a boundary value problem (BVP) for the second-order differential equation\nwith an operator coefficient, we represent its solution using the Cayley transform and the Meixner-type polynomials.\nThe approximate solution is the truncated sum of <jats:italic>N<\/jats:italic> (the discretization parameter) summands.\nWe give the error estimate of these approximations\ndepending on <jats:italic>N<\/jats:italic> and the distance of <jats:italic>t<\/jats:italic> to the initial point of the time interval\nor of the spatial argument <jats:italic>x<\/jats:italic> to the boundary of the spatial domain.<\/jats:p>","DOI":"10.1515\/cmam-2019-0120","type":"journal-article","created":{"date-parts":[[2020,2,11]],"date-time":"2020-02-11T09:01:53Z","timestamp":1581411713000},"page":"53-68","source":"Crossref","is-referenced-by-count":10,"title":["Weighted Estimates of the Cayley Transform Method for Abstract Differential Equations"],"prefix":"10.1515","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3115-9690","authenticated-orcid":false,"given":"Ivan P.","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 L.","family":"Makarov","sequence":"additional","affiliation":[{"name":"Department of Numerical Mathematics , Institute of Mathematics of the National Academy of Sciences of Ukraine , 3 Tereshchenkivska Str., 01024 Kyiv , Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8810-7464","authenticated-orcid":false,"given":"Nataliya V.","family":"Mayko","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Theoretical Radiophysics , Taras Shevchenko National University of Kyiv , 64\/13 Volodymyrska Str., 01601 Kyiv , Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2020,2,7]]},"reference":[{"key":"2023033111222051177_j_cmam-2019-0120_ref_001","doi-asserted-by":"crossref","unstructured":"D. Z.  Arov and I. P.  Gavrilyuk,\nA method for solving initial value problems for linear differential equations in Hilbert space based on the Cayley transform,\nNumer. Funct. Anal. Optim. 14 (1993), no. 5\u20136, 459\u2013473.","DOI":"10.1080\/01630569308816534"},{"key":"2023033111222051177_j_cmam-2019-0120_ref_002","unstructured":"D. Z.  Arov, I. P.  Gavrilyuk and V. L.  Makarov,\nRepresentation and approximation of solutions of initial value problems for differential equations in Hilbert space based on the Cayley transform,\nElliptic and Parabolic Problems (Pont-\u00e0-Mousson 1994),\nPitman Res. Notes Math. Ser. 325,\nLongman Scientific & Technical, Harlow (1995), 40\u201350."},{"key":"2023033111222051177_j_cmam-2019-0120_ref_003","unstructured":"K. I.  Babenko,\nFundamentals of Numerical Analysis,\n\u201cNauka\u201d, Moscow, 1986."},{"key":"2023033111222051177_j_cmam-2019-0120_ref_004","unstructured":"A.  Erd\u00e9lyi, W.  Magnus, F.  Oberhettinger and F. G.  Tricomi,\nHigher Transcendental Functions. Vol. II,\nMcGraw-Hill Book, New York, 1988."},{"key":"2023033111222051177_j_cmam-2019-0120_ref_005","unstructured":"E. F.  Galba,\nThe order of exactness of a difference scheme for the Poisson equation with mixed boundary conditions,\nOptimization of Software Algorithms,\nAkad. Nauk Ukrain. SSR Inst. Kibernet., Kiev (1985), 30\u201334, 65."},{"key":"2023033111222051177_j_cmam-2019-0120_ref_006","doi-asserted-by":"crossref","unstructured":"I. P.  Gavrilyuk,\nSuper exponentially convergent approximation to the solution of the Schr\u00f6dinger equation in abstract setting,\nComput. Methods Appl. Math. 10 (2010), no. 4, 345\u2013358.","DOI":"10.2478\/cmam-2010-0020"},{"key":"2023033111222051177_j_cmam-2019-0120_ref_007","doi-asserted-by":"crossref","unstructured":"I. P.  Gavrilyuk, W.  Hackbusch and B. N.  Khoromskij,\nData-sparse approximation to a class of operator-valued functions,\nMath. Comp. 74 (2005), no. 250, 681\u2013708.","DOI":"10.1090\/S0025-5718-04-01703-X"},{"key":"2023033111222051177_j_cmam-2019-0120_ref_008","doi-asserted-by":"crossref","unstructured":"I. P.  Gavrilyuk, W.  Hackbusch and B. N.  Khoromskij,\nHierarchical tensor-product approximation to the inverse and related operators for high-dimensional elliptic problems,\nComputing 74 (2005), no. 2, 131\u2013157.","DOI":"10.1007\/s00607-004-0086-y"},{"key":"2023033111222051177_j_cmam-2019-0120_ref_009","doi-asserted-by":"crossref","unstructured":"I. P.  Gavrilyuk and B. N.  Khoromskij,\nQuantized-TT-Cayley transform for computing the dynamics and the spectrum of high-dimensional Hamiltonians,\nComput. Methods Appl. Math. 11 (2011), no. 3, 273\u2013290.","DOI":"10.2478\/cmam-2011-0015"},{"key":"2023033111222051177_j_cmam-2019-0120_ref_010","doi-asserted-by":"crossref","unstructured":"I. P.  Gavrilyuk and V. L.  Makarov,\nExplicit and approximate solutions of second order elliptic differential equations in Hilbert and Banach spaces,\nNumer. Funct. Anal. Optim. 20 (1999), no. 7\u20138, 695\u2013715.","DOI":"10.1080\/01630569908816919"},{"key":"2023033111222051177_j_cmam-2019-0120_ref_011","unstructured":"I. P.  Gavrilyuk and V. L.  Makarov,\nStrongly positive operators and numerical algorithms without accuracy saturation,\nPublishing House of the Institute of Mathematics of NAN of Ukraine, Kiev, 2004."},{"key":"2023033111222051177_j_cmam-2019-0120_ref_012","doi-asserted-by":"crossref","unstructured":"I. P.  Gavrilyuk, V. L.  Makarov and N. V.  Mayko,\nWeighted estimates for boundary value problems with fractional derivatives,\nComput. Methods Appl. Math. (2019), 10.1515\/cmam-2018-0305.","DOI":"10.1515\/cmam-2018-0305"},{"key":"2023033111222051177_j_cmam-2019-0120_ref_013","unstructured":"P.  Hartman,\nOrdinary Differential Equations,\nJohn Wiley & Sons, New York, 1964."},{"key":"2023033111222051177_j_cmam-2019-0120_ref_014","doi-asserted-by":"crossref","unstructured":"B. N.  Khoromskij and J. M.  Melenk,\nBoundary concentrated finite element methods,\nSIAM J. Numer. Anal. 41 (2003), no. 1, 1\u201336.","DOI":"10.1137\/S0036142901391852"},{"key":"2023033111222051177_j_cmam-2019-0120_ref_015","unstructured":"V. L.  Makarov,\nOn a priori estimates of difference schemes giving an account of the boundary effect,\nC. R. Acad. Bulgare Sci. 42 (1989), no. 5, 41\u201344."},{"key":"2023033111222051177_j_cmam-2019-0120_ref_016","doi-asserted-by":"crossref","unstructured":"V. L.  Makarov,\nMeixner polynomials and their properties,\nDopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki (2019), no. 7, 3\u20138.","DOI":"10.15407\/dopovidi2019.07.003"},{"key":"2023033111222051177_j_cmam-2019-0120_ref_017","doi-asserted-by":"crossref","unstructured":"V. L.  Makarov and L. I.  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