{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:52:09Z","timestamp":1776786729240,"version":"3.51.2"},"reference-count":41,"publisher":"American Mathematical Society (AMS)","issue":"260","license":[{"start":{"date-parts":[[2008,4,19]],"date-time":"2008-04-19T00:00:00Z","timestamp":1208563200000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>An exponentially convergent approximation to the solution of a nonlinear first order differential equation with an operator coefficient in Banach space is proposed. The algorithm is based on an equivalent Volterra integral equation including the operator exponential generated by the operator coefficient. The operator exponential is represented by a Dunford-Cauchy integral along a hyperbola enveloping the spectrum of the operator coefficient, and then the integrals involved are approximated using the Chebyshev interpolation and an appropriate Sinc quadrature. Numerical examples are given which confirm theoretical results.<\/p>","DOI":"10.1090\/s0025-5718-07-01987-4","type":"journal-article","created":{"date-parts":[[2007,7,26]],"date-time":"2007-07-26T07:46:03Z","timestamp":1185435963000},"page":"1895-1923","source":"Crossref","is-referenced-by-count":1,"title":["An exponentially convergent algorithm for nonlinear differential equations in Banach spaces"],"prefix":"10.1090","volume":"76","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":[[2007,4,19]]},"reference":[{"key":"1","isbn-type":"print","first-page":"40","article-title":"Representation and approximation of solutions of initial value problems for differential equations in Hilbert space based on the Cayley transform","author":"Arov, D. Z.","year":"1995","ISBN":"https:\/\/id.crossref.org\/isbn\/0582239613"},{"key":"2","series-title":"Operator Theory: Advances and Applications","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-8518-8","volume-title":"Well-posedness of parabolic difference equations","volume":"69","author":"Ashyralyev, A.","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/3764350245"},{"issue":"1","key":"3","first-page":"11","article-title":"Estimates for the stability of a general discretization method","volume":"309","author":"Bakaev, N. Yu.","year":"1989","journal-title":"Dokl. Akad. Nauk SSSR","ISSN":"https:\/\/id.crossref.org\/issn\/0002-3264","issn-type":"print"},{"key":"4","unstructured":"T.Ju. Bohonova, I.P. Gavrilyuk, V.L. Makarov and V. Vasylyk, Exponentially convergent Duhamel\u2019s like algorithms for differential equations with an operator coefficient possessing a variable domain in Banach space, Reports on Numerical Mathematics, Friedrich-Schiller-Universit\u00e4t Jena (http:\/\/www.minet.uni-jena.de\/Math-Net\/reports05\/reports.html #2005), 05-06 (2005), 1-25."},{"key":"5","series-title":"Springer Series in Computational Physics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-84108-8","volume-title":"Spectral methods in fluid dynamics","author":"Canuto, Claudio","year":"1988","ISBN":"https:\/\/id.crossref.org\/isbn\/0387173714"},{"key":"6","unstructured":"M.L. Fernandez, Ch. Lubich, C. Palencia and A. Sch\u00e4dle, Fast Runge-Kutta approximation of inhomogeneous parabolic equations, Numerische Mathematik 5, (2005), 1-17."},{"key":"7","unstructured":"H. Fujita, N. Saito,and T. Suzuki, Operator Theory and Numerical Methods, Elsevier, Heidelberg, 2001."},{"issue":"2","key":"8","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1006\/jmaa.1999.6430","article-title":"Strongly \ud835\udc43-positive operators and explicit representations of the solutions of initial value problems for second-order differential equations in Banach space","volume":"236","author":"Gavrilyuk, Ivan P.","year":"1999","journal-title":"J. Math. Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-247X","issn-type":"print"},{"key":"9","unstructured":"I.P. Gavrilyuk, Algorithms without accuracy saturation and exponential convergent algorithms for operator equations, Journal of Numerical and Applied Mathematics (ISSN 0868-6912) 236 (1999), 28\u201343."},{"issue":"1","key":"10","first-page":"25","article-title":"\u210b-matrix approximation for elliptic solution operators in cylinder domains","volume":"9","author":"Gavrilyuk, I. P.","year":"2001","journal-title":"East-West J. Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0928-0200","issn-type":"print"},{"issue":"1","key":"11","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1007\/s002110100360","article-title":"\u210b-matrix approximation for the operator exponential with applications","volume":"92","author":"Gavrilyuk, Ivan P.","year":"2002","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"247","key":"12","doi-asserted-by":"publisher","first-page":"1297","DOI":"10.1090\/S0025-5718-03-01590-4","article-title":"Data-sparse approximation to the operator-valued functions of elliptic operator","volume":"73","author":"Gavrilyuk, Ivan P.","year":"2004","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"2","key":"13","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1007\/s00607-004-0086-y","article-title":"Hierarchical tensor-product approximation to the inverse and related operators for high-dimensional elliptic problems","volume":"74","author":"Gavrilyuk, Ivan P.","year":"2005","journal-title":"Computing","ISSN":"https:\/\/id.crossref.org\/issn\/0010-485X","issn-type":"print"},{"issue":"250","key":"14","doi-asserted-by":"publisher","first-page":"681","DOI":"10.1090\/S0025-5718-04-01703-X","article-title":"Data-sparse approximation to a class of operator-valued functions","volume":"74","author":"Gavrilyuk, Ivan P.","year":"2005","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"2","key":"15","doi-asserted-by":"publisher","first-page":"495","DOI":"10.4171\/ZAA\/712","article-title":"Representation and approximation of the solution of an initial value problem for a first order differential equation in Banach spaces","volume":"15","author":"Gavrilyuk, I. P.","year":"1996","journal-title":"Z. Anal. Anwendungen","ISSN":"https:\/\/id.crossref.org\/issn\/0232-2064","issn-type":"print"},{"issue":"250","key":"16","doi-asserted-by":"publisher","first-page":"555","DOI":"10.1090\/S0025-5718-04-01720-X","article-title":"Algorithms without accuracy saturation for evolution equations in Hilbert and Banach spaces","volume":"74","author":"Gavrilyuk, Ivan P.","year":"2005","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"4","key":"17","doi-asserted-by":"publisher","first-page":"333","DOI":"10.2478\/cmam-2001-0022","article-title":"Exponentially convergent parallel discretization methods for the first order evolution equations","volume":"1","author":"Gavrilyuk, Ivan P.","year":"2001","journal-title":"Comput. Methods Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1609-4840","issn-type":"print"},{"issue":"5-6","key":"18","doi-asserted-by":"publisher","first-page":"583","DOI":"10.1080\/01630569408816582","article-title":"The Cayley transform and the solution of an initial value problem for a first order differential equation with an unbounded operator coefficient in Hilbert space","volume":"15","author":"Gavrilyuk, Ivan P.","year":"1994","journal-title":"Numer. Funct. Anal. Optim.","ISSN":"https:\/\/id.crossref.org\/issn\/0163-0563","issn-type":"print"},{"key":"19","doi-asserted-by":"crossref","unstructured":"I.P. Gavrilyuk and V.L. Makarov,  Exponentially convergent parallel discretization methods for the first order differential equations, Doklady of the Ukrainian Academy of Scienses 3, (2002), 1\u20136.","DOI":"10.2478\/cmam-2001-0022"},{"issue":"5","key":"20","doi-asserted-by":"publisher","first-page":"2144","DOI":"10.1137\/040611045","article-title":"Exponentially convergent algorithms for the operator exponential with applications to inhomogeneous problems in Banach spaces","volume":"43","author":"Gavrilyuk, I. P.","year":"2005","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"1","key":"21","doi-asserted-by":"publisher","first-page":"63","DOI":"10.1216\/jiea\/1020282134","article-title":"An explicit boundary integral representation of the solution of the two-dimensional heat equation and its discretization","volume":"12","author":"Gavrilyuk, I. P.","year":"2000","journal-title":"J. Integral Equations Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0897-3962","issn-type":"print"},{"key":"22","unstructured":"I.P. Gavrilyuk and V.L. Makarov, An exponentially convergent algorithm for nonlinear differential equations in Banach spaces, Reports on Numerical Mathematics, Friedrich-Shiller-Universit\u00e4t Jena (http:\/\/www.minet.uni-jena.de\/Math-Net\/reports\/) 02\/05 (2005), 1\u201322."},{"issue":"2","key":"23","doi-asserted-by":"publisher","first-page":"163","DOI":"10.2478\/cmam-2004-0009","article-title":"A new estimate of the sinc method for linear parabolic problems including the initial point","volume":"4","author":"Gavrilyuk, Ivan P.","year":"2004","journal-title":"Comput. Methods Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1609-4840","issn-type":"print"},{"key":"24","series-title":"Oxford Mathematical Monographs","isbn-type":"print","volume-title":"Semigroups of linear operators and applications","author":"Goldstein, Jerome A.","year":"1985","ISBN":"https:\/\/id.crossref.org\/isbn\/0195035402"},{"issue":"3","key":"25","doi-asserted-by":"publisher","first-page":"973","DOI":"10.1137\/S0036142995283412","article-title":"Stability of time-stepping methods for abstract time-dependent parabolic problems","volume":"35","author":"Gonz\u00e1lez, C.","year":"1998","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"26","series-title":"Lecture Notes in Mathematics","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/BFb0089647","volume-title":"Geometric theory of semilinear parabolic equations","volume":"840","author":"Henry, Daniel","year":"1981","ISBN":"https:\/\/id.crossref.org\/isbn\/3540105573"},{"key":"27","first-page":"499","article-title":"Fractional powers of operators acting in Banach spaces","volume":"129","author":"Krasnosel\u2032ski\u012d, M. A.","year":"1959","journal-title":"Dokl. Akad. Nauk SSSR","ISSN":"https:\/\/id.crossref.org\/issn\/0002-3264","issn-type":"print"},{"issue":"41-42","key":"28","doi-asserted-by":"publisher","first-page":"4641","DOI":"10.1016\/S0045-7825(03)00442-0","article-title":"A parallel method for the numerical solution of integro-differential equation with positive memory","volume":"192","author":"Kwon, Kiwoon","year":"2003","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"key":"29","unstructured":"M. L\u00f3pez-Fern\u00e1ndez, C. Palencia and A. Sch\u00e4dle, A spectral order method for inverting sectorial Laplace transforms , ZIB-Report 05-26, April 2005."},{"issue":"2-3","key":"30","doi-asserted-by":"publisher","first-page":"289","DOI":"10.1016\/j.apnum.2004.06.015","article-title":"On the numerical inversion of the Laplace transform of certain holomorphic mappings","volume":"51","author":"L\u00f3pez-Fern\u00e1ndez, M.","year":"2004","journal-title":"Appl. Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0168-9274","issn-type":"print"},{"key":"31","doi-asserted-by":"crossref","unstructured":"M. L\u00f3pez-Fern\u00e1ndez, C. Palencia and A. Sch\u00e4dle, Fast Runge-Kutta approximation of inhomogeneous parabolic differential equations, Preprint 2005.","DOI":"10.1007\/s00211-005-0624-3"},{"key":"32","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971637","volume-title":"Sinc methods for quadrature and differential equations","author":"Lund, John","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/089871298X"},{"key":"33","series-title":"Applied Mathematical Sciences","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-5561-1","volume-title":"Semigroups of linear operators and applications to partial differential equations","volume":"44","author":"Pazy, A.","year":"1983","ISBN":"https:\/\/id.crossref.org\/isbn\/0387908455"},{"issue":"229","key":"34","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1090\/S0025-5718-99-01098-4","article-title":"A parallel method for time-discretization of parabolic problems based on contour integral representation and quadrature","volume":"69","author":"Sheen, Dongwoo","year":"2000","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"2","key":"35","doi-asserted-by":"publisher","first-page":"269","DOI":"10.1093\/imanum\/23.2.269","article-title":"A parallel method for time discretization of parabolic equations based on Laplace transformation and quadrature","volume":"23","author":"Sheen, Dongwoo","year":"2003","journal-title":"IMA J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0272-4979","issn-type":"print"},{"key":"36","first-page":"766","article-title":"Application of semigroup theory to the study of differential equations in Banach spaces","volume":"122","author":"Solomjak, M. Z.","year":"1958","journal-title":"Dokl. Akad. Nauk SSSR","ISSN":"https:\/\/id.crossref.org\/issn\/0002-3264","issn-type":"print"},{"key":"37","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-2706-9","volume-title":"Numerical methods based on sinc and analytic functions","volume":"20","author":"Stenger, Frank","year":"1993","ISBN":"https:\/\/id.crossref.org\/isbn\/0387940081"},{"key":"38","series-title":"American Mathematical Society Colloquium Publications, Vol. 23","volume-title":"Orthogonal polynomials","author":"Szeg\u00f6, Gabor","year":"1959"},{"key":"39","unstructured":"G. Szeg\u00f6, Orthogonal Polynomials (with an Introduction and a Complement by J.L. Geronimus), State Publishing House of Physical and Mathematical Literature, Moscow, 1962."},{"issue":"1","key":"40","first-page":"85","article-title":"A high order parallel method for time discretization of parabolic type equations based on Laplace transformation and quadrature","volume":"2","author":"Thom\u00e9e, Vidar","year":"2005","journal-title":"Int. J. Numer. Anal. Model.","ISSN":"https:\/\/id.crossref.org\/issn\/1705-5105","issn-type":"print"},{"key":"41","unstructured":"V. Vasylyk, Uniform exponentially convergent method for the first order evolution equation with unbounded operator coefficient, Journal of Numerical and Applied Mathematics (ISSN 0868-6912), 1, (2003), 99-104 (in Russian)."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2007-76-260\/S0025-5718-07-01987-4\/S0025-5718-07-01987-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2007-76-260\/S0025-5718-07-01987-4\/S0025-5718-07-01987-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:03:17Z","timestamp":1776783797000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2007-76-260\/S0025-5718-07-01987-4\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,4,19]]},"references-count":41,"journal-issue":{"issue":"260","published-print":{"date-parts":[[2007,10]]}},"alternative-id":["S0025-5718-07-01987-4"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-07-01987-4","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":[[2007,4,19]]}}}