{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,12,31]],"date-time":"2023-12-31T01:40:08Z","timestamp":1703986808768},"reference-count":12,"publisher":"Association for Computing Machinery (ACM)","issue":"4","content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["Commun. ACM"],"published-print":{"date-parts":[[1961,4]]},"abstract":"\n Using a generalization of Newton's method, a non-linear parabolic equation of the form\n \n u\n t<\/jats:sub>\n <\/jats:italic>\n -\n \n u\n xx<\/jats:sub>\n <\/jats:italic>\n =\n g<\/jats:italic>\n (\n u<\/jats:italic>\n ), and a non-linear elliptic equation\n \n u\n xx<\/jats:sub>\n <\/jats:italic>\n +\n \n u\n yy<\/jats:sub>\n <\/jats:italic>\n =\n \n e\n u<\/jats:sup>\n <\/jats:italic>\n , are solved numerically. Comparison of these results with results obtained using the Picard iteration procedure show that in many cases the quasilinearization method offers substantial advantages in both time and accuracy.\n <\/jats:p>","DOI":"10.1145\/355578.366508","type":"journal-article","created":{"date-parts":[[2002,7,27]],"date-time":"2002-07-27T11:32:44Z","timestamp":1027769564000},"page":"187-191","update-policy":"http:\/\/dx.doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":26,"title":["Some numerical experiments using Newton's method for nonlinear parabolic and elliptic boundary-value problems"],"prefix":"10.1145","volume":"4","author":[{"given":"Richard","family":"Bellman","sequence":"first","affiliation":[{"name":"The RAND Corp., Santa Monica, CA"}]},{"given":"Mario L.","family":"Juncosa","sequence":"additional","affiliation":[{"name":"The RAND Corp., Santa Monica, CA"}]},{"given":"Robert","family":"Kalaba","sequence":"additional","affiliation":[{"name":"The RAND Corp., Santa Monica, CA"}]}],"member":"320","published-online":{"date-parts":[[1961,4]]},"reference":[{"key":"e_1_2_1_1_2","volume-title":"Numerical methods of obtaining solutions of fixed end point problems in the calculus of variations","author":"HESTENES M. R.","year":"1949","unstructured":"HESTENES , M. R. Numerical methods of obtaining solutions of fixed end point problems in the calculus of variations . The RAND Corporation , Research Memorandum RM-102, 14 August 1949 . HESTENES, M. R. Numerical methods of obtaining solutions of fixed end point problems in the calculus of variations. The RAND Corporation, Research Memorandum RM-102, 14 August 1949."},{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.6028\/jres.050.039"},{"key":"e_1_2_1_3_2","volume-title":"Numerical Solution of Differential Equations J","author":"MILNE W. E.","year":"1953","unstructured":"MILNE , W. E. Numerical Solution of Differential Equations J . Wiley and Sons , New York , 1953 . MILNE, W. E. Numerical Solution of Differential Equations J. Wiley and Sons, New York, 1953."},{"key":"e_1_2_1_4_2","volume-title":"New Method for Approximate Integration of Differential Equations. Recently printed under Classics of the Natural Sciences","author":"CHAPLYGIN S. A.","year":"1950","unstructured":"CHAPLYGIN , S. A. A. New Method for Approximate Integration of Differential Equations. Recently printed under Classics of the Natural Sciences , Moscow , 1950 . CHAPLYGIN, S. A. A. New Method for Approximate Integration of Differential Equations. Recently printed under Classics of the Natural Sciences, Moscow, 1950."},{"key":"e_1_2_1_5_2","first-page":"89","article-title":"Functional analysis and applied mathematics","volume":"3","author":"KANTOROVICH L. V","year":"1948","unstructured":"KANTOROVICH , L. V . Functional analysis and applied mathematics . Uspekhi Matem. Nauk 3 ( 1948 ), 89 - 185 . Also, U. S. Natl. Bur. Stand. translation by C. D. Benster. KANTOROVICH, L. V. Functional analysis and applied mathematics. Uspekhi Matem. Nauk 3 (1948), 89-185. Also, U. S. Natl. Bur. Stand. translation by C. D. Benster.","journal-title":"Uspekhi Matem. Nauk"},{"key":"e_1_2_1_6_2","first-page":"68","article-title":"Certain further applications of Newton's method","volume":"7","author":"KANTOROVICH L. V","year":"1957","unstructured":"KANTOROVICH , L. V , Certain further applications of Newton's method . Vestnik Leningrad. Univ. 7 ( 1957 ), 68 - 103 . --, Certain further applications of Newton's method. Vestnik Leningrad. Univ. 7 (1957), 68-103.","journal-title":"Vestnik Leningrad. Univ."},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF00298003"},{"key":"e_1_2_1_8_2","first-page":"519","article-title":"On nonlinear differential equations, the maximum operation, and monotone convergence","volume":"8","author":"KALABA R","year":"1959","unstructured":"KALABA , R . On nonlinear differential equations, the maximum operation, and monotone convergence . J. Math. Mech. 8 ( 1959 ), 519 - 574 . KALABA, R. On nonlinear differential equations, the maximum operation, and monotone convergence. J. Math. Mech. 8 (1959), 519-574.","journal-title":"J. Math. Mech."},{"key":"e_1_2_1_9_2","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100023197"},{"key":"e_1_2_1_10_2","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100032436"},{"key":"e_1_2_1_11_2","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1954-0059635-7"},{"key":"e_1_2_1_12_2","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.41.10.743"}],"container-title":["Communications of the ACM"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/355578.366508","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,12,31]],"date-time":"2023-12-31T01:25:31Z","timestamp":1703985931000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/355578.366508"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1961,4]]},"references-count":12,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1961,4]]}},"alternative-id":["10.1145\/355578.366508"],"URL":"https:\/\/doi.org\/10.1145\/355578.366508","relation":{},"ISSN":["0001-0782","1557-7317"],"issn-type":[{"value":"0001-0782","type":"print"},{"value":"1557-7317","type":"electronic"}],"subject":[],"published":{"date-parts":[[1961,4]]},"assertion":[{"value":"1961-04-01","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}