{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,18]],"date-time":"2026-08-18T05:14:28Z","timestamp":1787030068274,"version":"build-2736575974"},"reference-count":32,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,1,4]],"date-time":"2024-01-04T00:00:00Z","timestamp":1704326400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Science and Technology Program Fund Project of Guizhou Province, China","award":["Qian Ke He Ji Chu-ZK[2022]021"],"award-info":[{"award-number":["Qian Ke He Ji Chu-ZK[2022]021"]}]},{"name":"Science and Technology Program Fund Project of Guizhou Province, China","award":["2022BZXS143"],"award-info":[{"award-number":["2022BZXS143"]}]},{"name":"2023 undergraduate scientific research project of Guizhou University of Finance and Economics, China","award":["Qian Ke He Ji Chu-ZK[2022]021"],"award-info":[{"award-number":["Qian Ke He Ji Chu-ZK[2022]021"]}]},{"name":"2023 undergraduate scientific research project of Guizhou University of Finance and Economics, China","award":["2022BZXS143"],"award-info":[{"award-number":["2022BZXS143"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Population balance equations may be employed to handle a wide variety of particle processes has certainly received unprecedented attention, but the absence of explicit exact solutions necessitates the use of numerical approaches. In this paper, a (2 + 1) dimensional population balance equation with aggregation, nucleation, growth and breakage processes is solved analytically by use of the methods of scaling transformation group, observation and trial function. Symmetries, reduced equations, invariant solutions, exact solutions, existence of solutions, evolution analysis of dynamic behavior for solutions are presented. The exact solutions obtained can be compared with the numerical scheme. The obtained results also show that the method of scaling transformation group can be applied to study integro-partial differential equations.<\/jats:p>","DOI":"10.3390\/sym16010065","type":"journal-article","created":{"date-parts":[[2024,1,4]],"date-time":"2024-01-04T05:23:30Z","timestamp":1704345810000},"page":"65","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Exact Solutions of Population Balance Equation with Aggregation, Nucleation, Growth and Breakage Processes, Using Scaling Group Analysis"],"prefix":"10.3390","volume":"16","author":[{"given":"Fubiao","family":"Lin","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550025, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yang","family":"Yang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550025, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xinxia","family":"Yang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550025, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,1,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Podg\u00f3rska, W. 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