{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,18]],"date-time":"2026-06-18T13:45:27Z","timestamp":1781790327152,"version":"3.54.5"},"reference-count":14,"publisher":"Wiley","license":[{"start":{"date-parts":[[2020,4,1]],"date-time":"2020-04-01T00:00:00Z","timestamp":1585699200000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100007345","name":"King Mongkut's University of Technology North Bangkok","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100007345","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2020,4,1]]},"abstract":"<jats:p>The study of pollution movement is an important basis for solving water quality problems, which is of vital importance in almost every country. This research proposes the motion of flowing pollution by using a mathematical model in one-dimensional advection-dispersion equation which includes terms of decay and enlargement process. We are assuming an added pollutant sources along the river in two cases: uniformly and exponentially increasing terms. The unsteady state analytical solutions are obtained by using the Laplace transformation, and the finite difference technique is utilized for numerical solutions. Solutions are compared by relative error values. The result appears acceptable between the analytical and numerical solutions. Varying the value of the rate of pollutant addition along the river (<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mi>q<\/mml:mi><\/mml:math>) and the arbitrary constant of exponential pollution source term (<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mi>\u03bb<\/mml:mi><\/mml:math>) is displayed to explain the behavior of the incremental concentration. It is shown that the concentration increases as<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mi>q<\/mml:mi><\/mml:math>and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mi>\u03bb<\/mml:mi><\/mml:math>increase, and the exponentially increasing pollution source is a suitable model for the behavior of incremental pollution along the river. The results are presented and discussed graphically. This work can be applied to other physical situations described by advection-dispersion phenomena which are affected by the increase of those source concentrations.<\/jats:p>","DOI":"10.1155\/2020\/9504835","type":"journal-article","created":{"date-parts":[[2020,4,1]],"date-time":"2020-04-01T23:31:14Z","timestamp":1585783874000},"page":"1-9","source":"Crossref","is-referenced-by-count":9,"title":["Analytical and Numerical Solutions of Pollution Concentration with Uniformly and Exponentially Increasing Forms of Sources"],"prefix":"10.1155","volume":"2020","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7053-2665","authenticated-orcid":true,"given":"N.","family":"Manitcharoen","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, King Mongkut\u2019s Institute of Technology Ladkrabang, Bangkok 10520, Thailand"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2550-7484","authenticated-orcid":true,"given":"B.","family":"Pimpunchat","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, King Mongkut\u2019s Institute of Technology Ladkrabang, Bangkok 10520, Thailand"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","reference":[{"key":"1","first-page":"2431","volume-title":"Modelling river pollution and removal by Aeration","year":"2007"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2008.03.026"},{"key":"3"},{"key":"6"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1080\/02508060903115134"},{"key":"8","year":"2000","edition":"4"},{"key":"9","volume-title":"Analytical solutions of the one-dimensional convective-dispersive solute transport equation","volume":"1661","year":"1982"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.1016\/j.jhydrol.2009.11.008"},{"key":"11","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijheatmasstransfer.2012.03.073"},{"issue":"3","key":"12","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1061\/JSEDAI.0000495","volume":"90","year":"1964","journal-title":"Journal of the Sanitary Engineering Division"},{"key":"13","year":"1997"},{"key":"14","year":"1974"},{"issue":"2","key":"15","first-page":"193","volume":"5","year":"2017","journal-title":"New Trends in Mathematical Science"},{"key":"16","year":"1969"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2020\/9504835.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2020\/9504835.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2020\/9504835.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,3,7]],"date-time":"2021-03-07T01:53:01Z","timestamp":1615081981000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.hindawi.com\/journals\/jam\/2020\/9504835\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,4,1]]},"references-count":14,"alternative-id":["9504835","9504835"],"URL":"https:\/\/doi.org\/10.1155\/2020\/9504835","relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"value":"1110-757X","type":"print"},{"value":"1687-0042","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,4,1]]}}}