{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,30]],"date-time":"2026-01-30T22:03:58Z","timestamp":1769810638446,"version":"3.49.0"},"reference-count":28,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2008,8,19]],"date-time":"2008-08-19T00:00:00Z","timestamp":1219104000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2008,11]]},"DOI":"10.1007\/s00211-008-0174-6","type":"journal-article","created":{"date-parts":[[2008,8,18]],"date-time":"2008-08-18T10:12:33Z","timestamp":1219054353000},"page":"81-108","source":"Crossref","is-referenced-by-count":37,"title":["Convergence analysis of sectional methods for solving breakage population balance equations-I: the fixed pivot technique"],"prefix":"10.1007","volume":"111","author":[{"given":"Jitendra","family":"Kumar","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gerald","family":"Warnecke","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2008,8,19]]},"reference":[{"key":"174_CR1","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1007\/BF01013961","volume":"61","author":"J.M. Ball","year":"1990","unstructured":"Ball J.M., Carr J.: The discrete coagulation-fragmentation equations: Existence, uniqueness, and density conservation. J. Stat. Phys. 61, 203\u2013234 (1990)","journal-title":"J. Stat. Phys."},{"key":"174_CR2","doi-asserted-by":"crossref","first-page":"619","DOI":"10.1007\/s003329900061","volume":"8","author":"F.P.D. Costa","year":"1998","unstructured":"Costa F.P.D.: A finite-dimensional dynamical model for gelation in coagulation processes. J. Nonlin. Sci. 8, 619\u2013653 (1998)","journal-title":"J. Nonlin. Sci."},{"key":"174_CR3","doi-asserted-by":"crossref","first-page":"4737","DOI":"10.1088\/0305-4470\/25\/18\/009","volume":"25","author":"P.B. Dubovski\u030c","year":"1992","unstructured":"Dubovski\u030c P.B., Galkin V.A., Stewart I.W.: Exact solutions for the coagulation-fragmentation equations. J. Phys. A Math. Gen. 25, 4737\u20134744 (1992)","journal-title":"J. Phys. A Math. Gen."},{"key":"174_CR4","doi-asserted-by":"crossref","first-page":"571","DOI":"10.1002\/(SICI)1099-1476(19960510)19:7<571::AID-MMA790>3.0.CO;2-Q","volume":"19","author":"P.B. Dubovski\u030c","year":"1996","unstructured":"Dubovski\u030c P.B., Stewart I.W.: Existence, uniqueness and mass conservation for the coagulation-fragmentation equation. Math. Meth. Appl. Sci. 19, 571\u2013591 (1996)","journal-title":"Math. Meth. Appl. Sci."},{"key":"174_CR5","doi-asserted-by":"crossref","first-page":"1433","DOI":"10.1016\/S0098-1354(97)88492-X","volume":"21","author":"R.C. Everson","year":"1997","unstructured":"Everson R.C., Eyre D., Campbell Q.P.: Spline method for solving continuous batch grinding and similarity equations. Comput. Chem. Eng. 21, 1433\u20131440 (1997)","journal-title":"Comput. Chem. Eng."},{"key":"174_CR6","doi-asserted-by":"crossref","first-page":"558","DOI":"10.1007\/s00013-004-1060-9","volume":"83","author":"F. Filbet","year":"2004","unstructured":"Filbet F., Lauren\u00e7ot P.: Mass-conserving solutions and non-conservative approximation to the smoluchowski coagulation equation. Archiv der Mathematik 83, 558\u2013567 (2004)","journal-title":"Archiv der Mathematik"},{"key":"174_CR7","doi-asserted-by":"crossref","first-page":"1204","DOI":"10.1002\/aic.690410516","volume":"41","author":"P.J. Hill","year":"1995","unstructured":"Hill P.J., Ng K.M.: New discretization procedure for the breakage equation. AIChE J. 41, 1204\u20131216 (1995)","journal-title":"AIChE J."},{"key":"174_CR8","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-09017-6","volume-title":"Numerical solution of time-dependent advection\u2013diffusion-reaction equations","author":"W. Hundsdorfer","year":"2003","unstructured":"Hundsdorfer W., Verwer J.G.: Numerical solution of time-dependent advection\u2013diffusion-reaction equations, 1st edn. Springer, New York (2003)","edition":"1"},{"key":"174_CR9","doi-asserted-by":"crossref","first-page":"116","DOI":"10.1016\/S0032-5910(02)00110-9","volume":"127","author":"M. Kostoglou","year":"2002","unstructured":"Kostoglou M., Karabelas A.J.: An assessment of low-order methods for solving the breakage equation. Powder Technol. 127, 116\u2013127 (2002)","journal-title":"Powder Technol."},{"issue":"144","key":"174_CR10","doi-asserted-by":"crossref","first-page":"280","DOI":"10.1016\/j.powtec.2004.04.020","volume":"143","author":"M. Kostoglou","year":"2004","unstructured":"Kostoglou M., Karabelas A.J.: Optimal low order methods of moments for solving the fragmentation equation. Powder Technol. 143(144), 280\u2013290 (2004)","journal-title":"Powder Technol."},{"key":"174_CR11","doi-asserted-by":"crossref","first-page":"571","DOI":"10.1016\/j.jcis.2004.10.029","volume":"284","author":"M. Kostoglou","year":"2005","unstructured":"Kostoglou M., Karabelas A.J.: On the self similar solution of fragmentation equation: Numerical evaluation with implication for the inverse problem. J. Colloid Interface Sci. 284, 571\u2013581 (2005)","journal-title":"J. Colloid Interface Sci."},{"key":"174_CR12","first-page":"205","volume":"179","author":"J. Kumar","year":"2007","unstructured":"Kumar J., Peglow M., Warnecke G., Heinrich S.: An efficient numerical technique for solving population balance equation involving aggregation, breakage, growth and nucleation. Powder Technol. 179, 205\u2013228 (2007)","journal-title":"Powder Technol."},{"key":"174_CR13","doi-asserted-by":"crossref","first-page":"3327","DOI":"10.1016\/j.ces.2005.12.014","volume":"61","author":"J. Kumar","year":"2006","unstructured":"Kumar J., Peglow M., Warnecke G., Heinrich S., M\u00f6rl L.: Improved accuracy and convergence of discretized population balance for aggregation: the cell average technique. Chem. Eng. Sci. 61, 3327\u20133342 (2006)","journal-title":"Chem. Eng. Sci."},{"key":"174_CR14","doi-asserted-by":"crossref","unstructured":"Kumar, J., Warnecke, G.: Convergence analysis of sectional methods for solving breakage population balance equation-II. The cell average technique. Accepted for publication in Numerische Mathematik (2008)","DOI":"10.1007\/s00211-008-0173-7"},{"key":"174_CR15","doi-asserted-by":"crossref","unstructured":"Kumar, J., Warnecke, G., Peglow, M., Heinrich, S.: Comparison of numerical methods for solving population balance equations incorporating aggregation and breakage. Powder Technol. (in press, 2008)","DOI":"10.1016\/j.powtec.2007.05.028"},{"key":"174_CR16","doi-asserted-by":"crossref","first-page":"1311","DOI":"10.1016\/0009-2509(96)88489-2","volume":"51","author":"S. Kumar","year":"1996","unstructured":"Kumar S., Ramkrishna D.: On the solution of population balance equations by discretization-I. A fixed pivot technique. Chem. Eng. Sci. 51, 1311\u20131332 (1996)","journal-title":"Chem. Eng. Sci."},{"key":"174_CR17","doi-asserted-by":"crossref","first-page":"1333","DOI":"10.1016\/0009-2509(95)00355-X","volume":"51","author":"S. Kumar","year":"1996","unstructured":"Kumar S., Ramkrishna D.: On the solution of population balance equations by discretization-II. A moving pivot technique. Chem. Eng. Sci. 51, 1333\u20131342 (1996)","journal-title":"Chem. Eng. Sci."},{"key":"174_CR18","doi-asserted-by":"crossref","first-page":"703","DOI":"10.1002\/mma.496","volume":"27","author":"W. Lamb","year":"2004","unstructured":"Lamb W.: Existence and uniqueness results for the continuous coagulation and fragmentation equation. Math. Meth. Appl. Sci. 27, 703\u2013721 (2004)","journal-title":"Math. Meth. Appl. Sci."},{"key":"174_CR19","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1006\/jdeq.2000.3809","volume":"167","author":"P. Lauren\u00e7ot","year":"2000","unstructured":"Lauren\u00e7ot P.: On a class of continuous coagulation\u2013fragmentation equations. J. Diff. Eq. 167, 245\u2013274 (2000)","journal-title":"J. Diff. Eq."},{"key":"174_CR20","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1006\/icar.1999.6239","volume":"143","author":"M.H. Lee","year":"2000","unstructured":"Lee M.H.: On the validity of the coagulation equation and the nature of runaway growth. Icarus 143, 74\u201386 (2000)","journal-title":"Icarus"},{"key":"174_CR21","doi-asserted-by":"crossref","first-page":"1173","DOI":"10.1137\/S0036141095291713","volume":"28","author":"D.J. McLaughlin","year":"1997","unstructured":"McLaughlin D.J., Lamb W., McBride A.C.: An existence and uniqueness result for a coagulation and multiple-fragmentation equation. SIAM J. Math. Anal. 28, 1173\u20131190 (1997)","journal-title":"SIAM J. Math. Anal."},{"key":"174_CR22","doi-asserted-by":"crossref","first-page":"1067","DOI":"10.1002\/(SICI)1099-1476(19980725)21:11<1067::AID-MMA985>3.0.CO;2-X","volume":"21","author":"D.J. McLaughlin","year":"1998","unstructured":"McLaughlin D.J., Lamb W., McBride A.C.: Existence and uniqueness results for the non-autonomous coagulation and multiple-fragmentation equation. Math. Meth. Appl. Sci. 21, 1067\u20131084 (1998)","journal-title":"Math. Meth. Appl. Sci."},{"key":"174_CR23","doi-asserted-by":"crossref","first-page":"246","DOI":"10.1016\/S0032-5910(99)00281-8","volume":"110","author":"B.K. Mishra","year":"2000","unstructured":"Mishra B.K.: Monte Carlo simulation of particle breakage process during grinding. Powder Technol. 110, 246\u2013252 (2000)","journal-title":"Powder Technol."},{"key":"174_CR24","unstructured":"Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical recipes in C++, 2nd edn. Cambridge University Press, The Edinburgh Building, Cambridge (2002)"},{"key":"174_CR25","doi-asserted-by":"crossref","first-page":"916","DOI":"10.1002\/aic.690450422","volume":"45","author":"M. Vanni","year":"1999","unstructured":"Vanni M.: Discretization procedure for the breakage equation. AIChE J. 45, 916\u2013919 (1999)","journal-title":"AIChE J."},{"key":"174_CR26","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1006\/jcis.1999.6571","volume":"221","author":"M. Vanni","year":"2002","unstructured":"Vanni M.: Approximate population balance equations for aggregation-breakage processes. J. Collid Interface Sci. 221, 143\u2013160 (2002)","journal-title":"J. Collid Interface Sci."},{"key":"174_CR27","doi-asserted-by":"crossref","first-page":"2821","DOI":"10.1088\/0305-4470\/24\/12\/020","volume":"24","author":"R.M. Ziff","year":"1991","unstructured":"Ziff R.M.: New solution to the fragmentation equation. J. Phys. A Math. Gen. 24, 2821\u20132828 (1991)","journal-title":"J. Phys. A Math. Gen."},{"key":"174_CR28","doi-asserted-by":"crossref","first-page":"3027","DOI":"10.1088\/0305-4470\/18\/15\/026","volume":"18","author":"R.M. Ziff","year":"1985","unstructured":"Ziff R.M., McGrady E.D.: The kinetics of cluster fragmentation and depolymerization. J. Phys. A Math. Gen. 18, 3027\u20133037 (1985)","journal-title":"J. Phys. A Math. Gen."}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-008-0174-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00211-008-0174-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-008-0174-6","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,23]],"date-time":"2019-05-23T16:22:41Z","timestamp":1558628561000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00211-008-0174-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,8,19]]},"references-count":28,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2008,11]]}},"alternative-id":["174"],"URL":"https:\/\/doi.org\/10.1007\/s00211-008-0174-6","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,8,19]]}}}