{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,18]],"date-time":"2026-03-18T00:46:09Z","timestamp":1773794769870,"version":"3.50.1"},"reference-count":19,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2025,10,30]],"date-time":"2025-10-30T00:00:00Z","timestamp":1761782400000},"content-version":"vor","delay-in-days":302,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2025,1,1]],"date-time":"2025-01-01T00:00:00Z","timestamp":1735689600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/doi.wiley.com\/10.1002\/tdm_license_1.1"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2025,1]]},"abstract":"<jats:p>In this paper, we derive both a two\u2010rank solution and an atomic solution for a Bateman\u2013Burgers type partial differential equation. Specifically, we study an equation in which the second mixed derivative of the unknown function with respect to space and time, together with the second derivative in space, equals the second derivative in time. Using functional analysis techniques, particularly the Hahn\u2013Banach theorem and tensor product theory, we construct explicit two\u2010rank and atomic solutions. These results illustrate how functional analytic tools provide alternative strategies for analyzing nonlinear equations where traditional separation of variables does not apply.<\/jats:p>","DOI":"10.1155\/jama\/2323749","type":"journal-article","created":{"date-parts":[[2025,10,31]],"date-time":"2025-10-31T04:51:28Z","timestamp":1761886288000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Two\u2010Rank and Atomic Solutions of a Bateman\u2013Burgers Type Equation via Functional Analytic Methods"],"prefix":"10.1155","volume":"2025","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6403-3448","authenticated-orcid":false,"given":"Waseem Ghazi","family":"Alshanti","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1099-3600","authenticated-orcid":false,"given":"Ma\u2019mon Abu","family":"Hammad","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0684-5867","authenticated-orcid":false,"given":"Roshdi","family":"Khalil","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2025,10,30]]},"reference":[{"key":"e_1_2_8_1_2","doi-asserted-by":"publisher","DOI":"10.1175\/1520-0493(1915)43<163:srrotm>2.0.co;2"},{"key":"e_1_2_8_2_2","doi-asserted-by":"publisher","DOI":"10.1090\/qam\/42889"},{"key":"e_1_2_8_3_2","volume-title":"Supersonic Flow and Shock Waves","author":"Courant R.","year":"1948"},{"key":"e_1_2_8_4_2","volume-title":"Perturbation Methods","author":"Nayfeh A. H.","year":"1973"},{"key":"e_1_2_8_5_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-58239-4"},{"key":"e_1_2_8_6_2","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511791253"},{"key":"e_1_2_8_7_2","first-page":"1","article-title":"Tuning the Fractional-Order PID-Controller for Blood Glucose Level of Diabetic Patients","volume":"13","author":"Batiha I.","year":"2021","journal-title":"International Journal of Advances in Soft Computing & Its Applications"},{"key":"e_1_2_8_8_2","doi-asserted-by":"publisher","DOI":"10.15849\/IJASCA.220720.07"},{"key":"e_1_2_8_9_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4684-0274-2"},{"key":"e_1_2_8_10_2","first-page":"711","article-title":"Point Solution Concept for Analysis of Reaction-Diffusion in Porous Catalysts","volume":"56","author":"He J.-H.","year":"2025","journal-title":"Journal of Computational Applied Mechanics"},{"key":"e_1_2_8_11_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2006.03.020"},{"key":"e_1_2_8_12_2","volume-title":"Introductory Functional Analysis With Applications","author":"Kreyszig E.","year":"1991"},{"key":"e_1_2_8_13_2","first-page":"725","article-title":"Atomic Solution of Certain Inverse Problems","volume":"3","author":"Khalil R. R.","year":"2010","journal-title":"European Journal of Pure and Applied Mathematics"},{"key":"e_1_2_8_14_2","doi-asserted-by":"publisher","DOI":"10.1080\/01630568608816217"},{"key":"e_1_2_8_15_2","article-title":"Simultaneous Approximation in Operator and Tensor Product Spaces","volume":"4","author":"Abu-Sirhan E.","year":"2019","journal-title":"Journal of Applied Functional Analysis"},{"key":"e_1_2_8_16_2","first-page":"77","article-title":"Isometries of Lp\u2217\u2297\u2322Lp","volume":"16","author":"Khalil R.","year":"1985","journal-title":"Tamkang Journal of Mathematics"},{"key":"e_1_2_8_17_2","doi-asserted-by":"publisher","DOI":"10.1090\/gsm\/101"},{"key":"e_1_2_8_18_2","article-title":"Operators of Exponential Type and the Abstract Cauchy Problem","volume":"2020","author":"Khalil R.","year":"2020","journal-title":"Journal of Semigroup Theory and Applications"},{"key":"e_1_2_8_19_2","doi-asserted-by":"publisher","DOI":"10.58997\/ejde.2024.28"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/jama\/2323749","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/full-xml\/10.1155\/jama\/2323749","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/jama\/2323749","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,17]],"date-time":"2026-03-17T20:55:54Z","timestamp":1773780954000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/jama\/2323749"}},"subtitle":[],"editor":[{"given":"Huaiqin","family":"Wu","sequence":"additional","affiliation":[]}],"short-title":[],"issued":{"date-parts":[[2025,1]]},"references-count":19,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2025,1]]}},"alternative-id":["10.1155\/jama\/2323749"],"URL":"https:\/\/doi.org\/10.1155\/jama\/2323749","archive":["Portico"],"relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"value":"1110-757X","type":"print"},{"value":"1687-0042","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,1]]},"assertion":[{"value":"2025-05-25","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2025-09-23","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2025-10-30","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"2323749"}}