{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,12]],"date-time":"2025-11-12T03:10:45Z","timestamp":1762917045766},"reference-count":31,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2006,6,1]],"date-time":"2006-06-01T00:00:00Z","timestamp":1149120000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2006,6]]},"DOI":"10.1007\/s11075-006-9034-6","type":"journal-article","created":{"date-parts":[[2006,8,16]],"date-time":"2006-08-16T12:26:44Z","timestamp":1155731204000},"page":"137-164","source":"Crossref","is-referenced-by-count":37,"title":["Efficient spectral-Galerkin algorithms for direct solution for second-order differential equations using Jacobi polynomials"],"prefix":"10.1007","volume":"42","author":[{"given":"E. H.","family":"Doha","sequence":"first","affiliation":[]},{"given":"A. H.","family":"Bhrawy","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2006,8,17]]},"reference":[{"key":"9034_CR1","doi-asserted-by":"crossref","first-page":"306","DOI":"10.1006\/jcph.1998.6155","volume":"149","author":"F. Auteri","year":"1999","unstructured":"Auteri, F., Quartapelle, L.: Galerkin spectral method for the vorticity and stream function equations. J. Comput. Phys. 149, 306\u2013332 (1999)","journal-title":"J. Comput. Phys."},{"key":"9034_CR2","doi-asserted-by":"crossref","first-page":"454","DOI":"10.1006\/jcph.2000.6504","volume":"161","author":"F. Auteri","year":"2000","unstructured":"Auteri, F., Quartapelle, L.: Galerkin\u2013Legendre spectral method for the 3D Helmholtz equation. J. Comput. Phys. 161, 454\u2013483 (2000)","journal-title":"J. Comput. Phys."},{"key":"9034_CR3","doi-asserted-by":"crossref","first-page":"427","DOI":"10.1016\/S0021-9991(02)00064-5","volume":"185","author":"F. Auteri","year":"2003","unstructured":"Auteri, F., Parolini, N., Quartapelle, L.: Essential imposition of Neumann Galerkin\u2013Legendre elliptic solvers. J. Comput. Phys. 185, 427\u2013444 (2003)","journal-title":"J. Comput. Phys."},{"key":"9034_CR4","first-page":"95","volume":"18","author":"G. Ben-Yu","year":"2000","unstructured":"Ben-Yu, G.: Jacobi spectral approximations to differential equations on the half line. J. Comput. Math. 18, 95\u2013112 (2000)","journal-title":"J. Comput. Math."},{"key":"9034_CR5","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/S0898-1221(03)90083-6","volume":"46","author":"G. Ben-Yu","year":"2003","unstructured":"Ben-Yu, G.: Jacobi spectral method for differential equations with rough asymptotic behaviors at infinity. Comput. Math. Appl. 46, 95\u2013104 (2003)","journal-title":"Comput. Math. Appl."},{"key":"9034_CR6","volume-title":"Chebyshev and Fourier Spectral Methods","author":"J.P. Boyd","year":"2001","unstructured":"Boyd, J.P.: Chebyshev and Fourier Spectral Methods, 2nd ed. Dover, Mineola (2001)","edition":"2"},{"key":"9034_CR7","doi-asserted-by":"crossref","first-page":"627","DOI":"10.1137\/0707049","volume":"7","author":"B.L. Buzbee","year":"1970","unstructured":"Buzbee, B.L., Golub, G.H., Neilson, C.W.: On direct methods for solving Poisson's equations. SIAM J. Numer. Anal. 7, 627\u2013656 (1970)","journal-title":"SIAM J. Numer. Anal."},{"key":"9034_CR8","volume-title":"Spectral Methods in Fluid Dynamics","author":"C. Canuto","year":"1989","unstructured":"Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, New York (1988)"},{"key":"9034_CR9","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1006\/jcph.1993.1021","volume":"104","author":"H. Dang-Vu","year":"1993","unstructured":"Dang-Vu, H., Delcarte, C.: An accurate solution of the Poisson equation by the Chebyshev collocation method. J. Comput. Phys. 104, 211\u2013220 (1993)","journal-title":"J. Comput. Phys."},{"key":"9034_CR10","volume-title":"Methods of Numerical Integration","author":"P.J. Davis","year":"1975","unstructured":"Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration. Academic, New York (1975)"},{"key":"9034_CR11","first-page":"243","volume":"10","author":"E.H. Doha","year":"1990","unstructured":"Doha, E.H.: An accurate double Chebyshev spectral approximation for Poisson's equation. Ann. Univ. Sci. Budapest Sect. Comp. 10, 243\u2013276 (1990)","journal-title":"Ann. Univ. Sci. Budapest Sect. Comp."},{"key":"9034_CR12","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/0898-1221(90)90139-B","volume":"19","author":"E.H. Doha","year":"1990","unstructured":"Doha, E.H.: An accurate solution of parabolic equations by expansion in ultraspherical polynomials. J. Comput. Math. Appl. 19, 75\u201388 (1990)","journal-title":"J. Comput. Math. Appl."},{"key":"9034_CR13","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/0898-1221(91)90089-M","volume":"21","author":"E.H. Doha","year":"1991","unstructured":"Doha, E.H.: The coefficients of differentiated expansions and derivatives of ultraspherical polynomials. J. Comput. Math. Appl. 21, 115\u2013122 (1991)","journal-title":"J. Comput. Math. Appl."},{"key":"9034_CR14","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1016\/S0377-0427(97)00228-8","volume":"89","author":"E.H. Doha","year":"1998","unstructured":"Doha, E.H.: The ultraspherical coefficients of the moments of a general-order derivative of an infinitely differentiable function. J. Comput. Appl. Math. 89, 53\u201372 (1998)","journal-title":"J. Comput. Appl. Math."},{"key":"9034_CR15","doi-asserted-by":"crossref","first-page":"3467","DOI":"10.1088\/0305-4470\/35\/15\/308","volume":"35","author":"E.H. Doha","year":"2002","unstructured":"Doha, E.H.: On the coefficients of differentiated expansions and derivatives of Jacobi polynomials. J. Phys. A: Math. Gen. 35, 3467\u20133478 (2002)","journal-title":"J. Phys. A: Math. Gen."},{"key":"9034_CR16","doi-asserted-by":"crossref","first-page":"657","DOI":"10.1088\/0305-4470\/37\/3\/010","volume":"37","author":"E.H. Doha","year":"2004","unstructured":"Doha, E.H.: On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials. J. Phys. A: Math. Gen. 37, 657\u2013675 (2004)","journal-title":"J. Phys. A: Math. Gen."},{"key":"9034_CR17","doi-asserted-by":"crossref","first-page":"548","DOI":"10.1137\/S1064827500378933","volume":"24","author":"E.H. Doha","year":"2002","unstructured":"Doha, E.H., Abd-Elhameed, W.M.: Efficient spectral-Galerkin algorithms for direct solution of second-order equations using ultraspherical polynomials. SIAM J. Sci. Comput. 24, 548\u2013571 (2002)","journal-title":"SIAM J. Sci. Comput."},{"key":"9034_CR18","doi-asserted-by":"crossref","unstructured":"Gottlieb, D., Orszag, S.A.: Numerical Analysis of Spectral Methods: Theory and Applications, SIAM, Philadelphia, Pennsylvania (1977)","DOI":"10.1137\/1.9781611970425"},{"key":"9034_CR19","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1016\/0021-9991(79)90097-4","volume":"30","author":"D.B. Haidvogel","year":"1979","unstructured":"Haidvogel, D.B., Zang, T.: The accurate solution of Poisson's equation by expansion in Chebyshev polynomials. J. Comput. Phys. 30, 167\u2013180 (1979)","journal-title":"J. Comput. Phys."},{"key":"9034_CR20","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1090\/S0025-5718-1989-0972370-0","volume":"53","author":"W. Heinrichs","year":"1989","unstructured":"Heinrichs, W.: Improved condition number of spectral methods. Math. Comp. 53, 103\u2013119 (1989)","journal-title":"Math. Comp."},{"key":"9034_CR21","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1007\/BF01395776","volume":"56","author":"W. Heinrichs","year":"1989","unstructured":"Heinrichs, W.: Spectral methods with sparse matrices. Numer. Math. 56, 25\u201341 (1989)","journal-title":"Numer. Math."},{"key":"9034_CR22","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1016\/0045-7825(90)90031-G","volume":"80","author":"W. Heinrichs","year":"1990","unstructured":"Heinrichs, W.: Algebraic spectral multigrid methods. Comput. Methods Appl. Mech. Eng. 80, 281\u2013286 (1990)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"9034_CR23","volume-title":"Applied Analysis","author":"C. Lanczos","year":"1957","unstructured":"Lanczos, C.: Applied Analysis. Pitman, London (1957)"},{"key":"9034_CR24","volume-title":"The Special Functions and Their Approximations, vol. 1","author":"Y. Luke","year":"1969","unstructured":"Luke, Y.: The Special Functions and Their Approximations, vol. 1. Academic, New York (1969)"},{"key":"9034_CR25","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1016\/0021-9991(80)90005-4","volume":"37","author":"S.A. Orszag","year":"1980","unstructured":"Orszag, S.A.: Spectral methods for problems in complex geometrics. J. Comput. Phys. 37, 70\u201392 (1980)","journal-title":"J. Comput. Phys."},{"key":"9034_CR26","volume-title":"A First Course in Numerical Analysis","author":"A. Ralston","year":"1965","unstructured":"Ralston, A.: A First Course in Numerical Analysis. McGraw-Hill, New York (1965)"},{"key":"9034_CR27","doi-asserted-by":"crossref","first-page":"1489","DOI":"10.1137\/0915089","volume":"15","author":"J. Shen","year":"1994","unstructured":"Shen, J.: Efficient spectral-Galerkin method. I. Direct solvers of second- and fourth-order equations using Legendre polynomials. SIAM J. Sci. Comput. 15, 1489\u20131505 (1994)","journal-title":"SIAM J. Sci. Comput."},{"key":"9034_CR28","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1137\/0916006","volume":"16","author":"J. Shen","year":"1995","unstructured":"Shen, J.: Efficient spectral-Galerkin method. II. Direct solvers of second- and fourth-order equations using Chebyshev polynomials. SIAM J. Sci. Comput. 16, 74\u201387 (1995)","journal-title":"SIAM J. Sci. Comput."},{"key":"9034_CR29","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0377-0427(97)00104-0","volume":"85","author":"H.I. Siyyam","year":"1997","unstructured":"Siyyam, H.I., Syam, M.I.: An accurate solution of Poisson equation by the Chebyshev\u2013Tau method. J. Comput. Appl. Math. 85, 1\u201310 (1997)","journal-title":"J. Comput. Appl. Math."},{"key":"9034_CR30","unstructured":"Szeg\u00f6, G.: Orthogonal Polynomials. Am. Math. Soc. Colloq. Pub. 23 (1985)"},{"issue":"2","key":"9034_CR31","first-page":"xiii","volume":"23","author":"G.N. Watson","year":"1925","unstructured":"Watson, G.N.: A note on generalized hypergeometric series. Proc. Lond. Math. Soc. 23(2), xiii\u2013xv (1925) (Records for 8 Nov. 1923)","journal-title":"Proc. Lond. Math. Soc."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-006-9034-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11075-006-9034-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-006-9034-6","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,1]],"date-time":"2019-06-01T05:08:07Z","timestamp":1559365687000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11075-006-9034-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,6]]},"references-count":31,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2006,6]]}},"alternative-id":["9034"],"URL":"https:\/\/doi.org\/10.1007\/s11075-006-9034-6","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2006,6]]}}}