{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,9]],"date-time":"2026-03-09T04:27:24Z","timestamp":1773030444268,"version":"3.50.1"},"reference-count":20,"publisher":"Wiley","issue":"3","license":[{"start":{"date-parts":[[2004,2,12]],"date-time":"2004-02-12T00:00:00Z","timestamp":1076544000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Numerical Methods Partial"],"published-print":{"date-parts":[[2004,5]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The boundary value problem \u0394<jats:italic>u<\/jats:italic> + \u03bb<jats:italic>e<\/jats:italic><jats:sup><jats:italic>u<\/jats:italic><\/jats:sup> = 0 where <jats:italic>u<\/jats:italic> = 0 on the boundary is often referred to as \u201cthe Bratu problem.\u201d The Bratu problem with cylindrical radial operators, also known as the cylindrical Bratu\u2010Gelfand problem, is considered here. It is a nonlinear eigenvalue problem with two known bifurcated solutions for \u03bb &lt; \u03bb<jats:sub><jats:italic>c<\/jats:italic><\/jats:sub>, no solutions for \u03bb &gt; \u03bb<jats:sub><jats:italic>c<\/jats:italic><\/jats:sub> and a unique solution when \u03bb = \u03bb<jats:sub><jats:italic>c<\/jats:italic><\/jats:sub>. Numerical solutions to the Bratu\u2010Gelfand problem at the critical value of \u03bb<jats:sub><jats:italic>c<\/jats:italic><\/jats:sub> = 2 are computed using nonstandard finite\u2010difference schemes known as Mickens finite differences. Comparison of numerical results obtained by solving the Bratu\u2010Gelfand problem using a Mickens discretization with results obtained using standard finite differences for \u03bb &lt; 2 are given, which illustrate the superiority of the nonstandard scheme. \u00a9 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 327\u2013337, 2004<\/jats:p>","DOI":"10.1002\/num.10093","type":"journal-article","created":{"date-parts":[[2004,2,13]],"date-time":"2004-02-13T09:00:35Z","timestamp":1076662835000},"page":"327-337","source":"Crossref","is-referenced-by-count":71,"title":["Application of a Mickens finite\u2010difference scheme to the cylindrical Bratu\u2010Gelfand problem"],"prefix":"10.1002","volume":"20","author":[{"given":"Ron","family":"Buckmire","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2004,2,12]]},"reference":[{"key":"e_1_2_1_2_2","first-page":"295","article-title":"Some problems in the theory of quasi\u2010linear equations","volume":"29","author":"Gelfand I. 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