{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,15]],"date-time":"2026-07-15T00:29:06Z","timestamp":1784075346988,"version":"3.55.0"},"reference-count":35,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2023,12,8]],"date-time":"2023-12-08T00:00:00Z","timestamp":1701993600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,12,8]],"date-time":"2023-12-08T00:00:00Z","timestamp":1701993600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2024,2]]},"DOI":"10.1007\/s40314-023-02512-6","type":"journal-article","created":{"date-parts":[[2023,12,8]],"date-time":"2023-12-08T04:23:45Z","timestamp":1702009425000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["An efficient semi-analytical technique to solve multi-dimensional Burgers\u2019 equation"],"prefix":"10.1007","volume":"43","author":[{"given":"Saddam","family":"Hussain","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gourav","family":"Arora","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Rajesh","family":"Kumar","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2023,12,8]]},"reference":[{"issue":"1","key":"2512_CR1","first-page":"95","volume":"170","author":"S Abbasbandy","year":"2005","unstructured":"Abbasbandy S, Darvishi M (2005) A numerical solution of burgers\u2019 equation by time discretization of adomian\u2019s decomposition method. Appl Math Comput 170(1):95\u2013102","journal-title":"Appl Math Comput"},{"issue":"1","key":"2512_CR2","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1016\/j.camwa.2018.04.010","volume":"76","author":"MA Al-Jawary","year":"2018","unstructured":"Al-Jawary MA, Azeez MM, Radhi GH (2018) Analytical and numerical solutions for the nonlinear burgers and advection-diffusion equations by using a semi-analytical iterative method. Comput Math Appl 76(1):155\u2013171","journal-title":"Comput Math Appl"},{"key":"2512_CR3","doi-asserted-by":"publisher","first-page":"101973","DOI":"10.1016\/j.jocs.2023.101973","volume":"67","author":"G Arora","year":"2023","unstructured":"Arora G, Hussain S, Kumar R (2023) Comparison of variational iteration and adomian decomposition methods to solve growth, aggregation and aggregation-breakage equations. J Comput Sci 67:101973","journal-title":"J Comput Sci"},{"key":"2512_CR4","doi-asserted-by":"publisher","first-page":"1339","DOI":"10.1016\/j.ijleo.2016.11.140","volume":"130","author":"H Bakodah","year":"2017","unstructured":"Bakodah H, Al-Zaid N, Mirzazadeh M, Zhou Q (2017) Decomposition method for solving burgers\u2019 equation with dirichlet and neumann boundary conditions. Optik 130:1339\u20131346","journal-title":"Optik"},{"issue":"2","key":"2512_CR5","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1090\/qam\/306736","volume":"30","author":"ER Benton","year":"1972","unstructured":"Benton ER, Platzman GW (1972) A table of solutions of the one-dimensional burgers equation. Q Appl Math 30(2):195\u2013212","journal-title":"Q Appl Math"},{"key":"2512_CR6","first-page":"171","volume-title":"Advances in applied mechanics","author":"JM Burgers","year":"1948","unstructured":"Burgers JM (1948) A mathematical model illustrating the theory of turbulence. Advances in applied mechanics, vol 1. Elsevier, Amsterdam, pp 171\u2013199"},{"issue":"5","key":"2512_CR7","doi-asserted-by":"publisher","first-page":"381","DOI":"10.1016\/S0307-904X(82)80102-9","volume":"6","author":"J Caldwell","year":"1982","unstructured":"Caldwell J, Smith P (1982) Solution of burgers\u2019 equation with a large reynolds number. Appl Math Model 6(5):381\u2013385","journal-title":"Appl Math Model"},{"issue":"3","key":"2512_CR8","doi-asserted-by":"publisher","first-page":"189","DOI":"10.1016\/0307-904X(81)90043-3","volume":"5","author":"J Caldwell","year":"1981","unstructured":"Caldwell J, Wanless P, Cook A (1981) A finite element approach to burgers\u2019 equation. Appl Math Model 5(3):189\u2013193","journal-title":"Appl Math Model"},{"issue":"1","key":"2512_CR9","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1080\/17455030.2016.1205237","volume":"27","author":"Y \u00c7enesiz","year":"2017","unstructured":"\u00c7enesiz Y, Baleanu D, Kurt A, Tasbozan O (2017) New exact solutions of burgers\u2019 type equations with conformable derivative. Waves Random Complex Media 27(1):103\u2013116","journal-title":"Waves Random Complex Media"},{"issue":"15\u201316","key":"2512_CR10","doi-asserted-by":"publisher","first-page":"6805","DOI":"10.1016\/j.apm.2016.02.024","volume":"40","author":"S-H Chang","year":"2016","unstructured":"Chang S-H (2016) Convergence of variational iteration method applied to two-point diffusion problems. Appl Math Model 40(15\u201316):6805\u20136810","journal-title":"Appl Math Model"},{"key":"2512_CR11","unstructured":"Coely A et al. (2001) Backlund and darboux transformations. American Mathematical Society, Providence, Rhode Island, 458\u2013468"},{"issue":"3","key":"2512_CR12","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1090\/qam\/42889","volume":"9","author":"JD Cole","year":"1951","unstructured":"Cole JD (1951) On a quasi-linear parabolic equation occurring in aerodynamics. Q Appl Math 9(3):225\u2013236","journal-title":"Q Appl Math"},{"key":"2512_CR13","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611973242","volume-title":"The method of weighted residuals and variational principles","author":"BA Finlayson","year":"2013","unstructured":"Finlayson BA (2013) The method of weighted residuals and variational principles. SIAM, Philadelphia"},{"issue":"4","key":"2512_CR14","doi-asserted-by":"publisher","first-page":"699","DOI":"10.1016\/S0020-7462(98)00048-1","volume":"34","author":"J-H He","year":"1999","unstructured":"He J-H (1999) Variational iteration method-a kind of non-linear analytical technique: some examples. Int J Non-Linear Mech 34(4):699\u2013708","journal-title":"Int J Non-Linear Mech"},{"issue":"2\u20133","key":"2512_CR15","first-page":"115","volume":"114","author":"J-H He","year":"2000","unstructured":"He J-H (2000) Variational iteration method for autonomous ordinary differential systems. Appl Math Comput 114(2\u20133):115\u2013123","journal-title":"Appl Math Comput"},{"issue":"7\u20138","key":"2512_CR16","doi-asserted-by":"publisher","first-page":"881","DOI":"10.1016\/j.camwa.2006.12.083","volume":"54","author":"J-H He","year":"2007","unstructured":"He J-H, Wu X-H (2007) Variational iteration method: new development and applications. Comput Math Appl 54(7\u20138):881\u2013894","journal-title":"Comput Math Appl"},{"issue":"3","key":"2512_CR17","doi-asserted-by":"publisher","first-page":"179","DOI":"10.1016\/0898-1221(79)90040-3","volume":"5","author":"P Jain","year":"1979","unstructured":"Jain P, Lohar B (1979) Cubic spline technique for coupled non-linear parabolic equations. Comput Math Appl 5(3):179\u2013185","journal-title":"Comput Math Appl"},{"key":"2512_CR18","doi-asserted-by":"crossref","unstructured":"Kaushik S, Hussain S, Kumar R (2023) Laplace transform-based approximation methods for solving pure aggregation and breakage equations. Math Methods Appl Sci 1\u201320","DOI":"10.1002\/mma.9507"},{"issue":"2","key":"2512_CR19","doi-asserted-by":"publisher","first-page":"433","DOI":"10.1016\/0021-9991(81)90162-5","volume":"39","author":"B Lohar","year":"1981","unstructured":"Lohar B, Jain P (1981) Variable mesh cubic spline technique for n-wave solution of burgers\u2019 equation. J Comput Phys 39(2):433\u2013442","journal-title":"J Comput Phys"},{"issue":"4\u20135","key":"2512_CR20","doi-asserted-by":"publisher","first-page":"271","DOI":"10.1016\/j.physleta.2006.02.048","volume":"355","author":"S Momani","year":"2006","unstructured":"Momani S, Odibat Z (2006) Analytical approach to linear fractional partial differential equations arising in fluid mechanics. Phys Lett A 355(4\u20135):271\u2013279","journal-title":"Phys Lett A"},{"issue":"1","key":"2512_CR21","doi-asserted-by":"publisher","first-page":"63","DOI":"10.1090\/qam\/1373838","volume":"54","author":"S Nerney","year":"1996","unstructured":"Nerney S, Schmahl EJ, Musielak Z (1996) Analytic solutions of the vector burgers\u2019 equation. Q Appl Math 54(1):63\u201371","journal-title":"Q Appl Math"},{"issue":"9\u201310","key":"2512_CR22","doi-asserted-by":"publisher","first-page":"1181","DOI":"10.1016\/j.mcm.2009.12.034","volume":"51","author":"ZM Odibat","year":"2010","unstructured":"Odibat ZM (2010) A study on the convergence of variational iteration method. Math Comput Modell 51(9\u201310):1181\u20131192","journal-title":"Math Comput Modell"},{"key":"2512_CR23","unstructured":"Rubin SG, Graves\u00a0Jr RA (1975) A cubic spline approximation for problems in fluid mechanics. Technical report"},{"key":"2512_CR24","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1016\/j.cjph.2019.01.001","volume":"58","author":"M Senol","year":"2019","unstructured":"Senol M, Tasbozan O, Kurt A (2019) Numerical solutions of fractional burgers\u2019 type equations with conformable derivative. Chin J Phys 58:75\u201384","journal-title":"Chin J Phys"},{"issue":"2","key":"2512_CR25","doi-asserted-by":"publisher","first-page":"394","DOI":"10.1016\/j.physa.2005.07.008","volume":"361","author":"A Soliman","year":"2006","unstructured":"Soliman A (2006) The modified extended tanh-function method for solving burgers-type equations. Phys A 361(2):394\u2013404","journal-title":"Phys A"},{"issue":"2","key":"2512_CR26","first-page":"115","volume":"1","author":"VK Srivastava","year":"2014","unstructured":"Srivastava VK, Mishra N, Kumar S, Singh BK, Awasthi MK (2014) Reduced differential transform method for solving (1+ n)-dimensional burgers\u2019 equation. Egypt J Basic Appl Sci 1(2):115\u2013119","journal-title":"Egypt J Basic Appl Sci"},{"issue":"2","key":"2512_CR27","doi-asserted-by":"publisher","first-page":"419","DOI":"10.1143\/PTP.53.419","volume":"53","author":"M Wadati","year":"1975","unstructured":"Wadati M, Sanuki H, Konno K (1975) Relationships among inverse method, b\u00e4cklund transformation and an infinite number of conservation laws. Progress Theor Phys 53(2):419\u2013436","journal-title":"Progress Theor Phys"},{"issue":"3","key":"2512_CR28","first-page":"713","volume":"154","author":"A-M Wazwaz","year":"2004","unstructured":"Wazwaz A-M (2004) The tanh method for traveling wave solutions of nonlinear equations. Appl Math Comput 154(3):713\u2013723","journal-title":"Appl Math Comput"},{"issue":"5\u20136","key":"2512_CR29","doi-asserted-by":"publisher","first-page":"499","DOI":"10.1016\/j.mcm.2003.12.010","volume":"40","author":"A-M Wazwaz","year":"2004","unstructured":"Wazwaz A-M (2004) A sine-cosine method for handlingnonlinear wave equations. Math Comput Modell 40(5\u20136):499\u2013508","journal-title":"Math Comput Modell"},{"issue":"1","key":"2512_CR30","first-page":"485","volume":"188","author":"A-M Wazwaz","year":"2007","unstructured":"Wazwaz A-M (2007) The variational iteration method for solving two forms of blasius equation on a half-infinite domain. Appl Math Comput 188(1):485\u2013491","journal-title":"Appl Math Comput"},{"issue":"4","key":"2512_CR31","doi-asserted-by":"publisher","first-page":"1136","DOI":"10.1016\/j.chaos.2006.10.009","volume":"37","author":"A-M Wazwaz","year":"2008","unstructured":"Wazwaz A-M (2008) A study on linear and nonlinear schrodinger equations by the variational iteration method. Chaos Solitons Fractals 37(4):1136\u20131142","journal-title":"Chaos Solitons Fractals"},{"key":"2512_CR32","doi-asserted-by":"publisher","first-page":"479","DOI":"10.1007\/978-3-642-00251-9_12","volume-title":"Partial differential equations and solitary waves theory","author":"A-M Wazwaz","year":"2009","unstructured":"Wazwaz A-M (2009) Solitary waves theory. Partial differential equations and solitary waves theory. Springer, Heidelberg, Berlin, pp 479\u2013502"},{"issue":"5\u20136","key":"2512_CR33","doi-asserted-by":"publisher","first-page":"355","DOI":"10.1016\/S0375-9601(01)00376-0","volume":"285","author":"Z Yan","year":"2001","unstructured":"Yan Z, Zhang H (2001) New explicit solitary wave solutions and periodic wave solutions for whitham-broer-kaup equation in shallow water. Phys Lett A 285(5\u20136):355\u2013362","journal-title":"Phys Lett A"},{"issue":"5","key":"2512_CR34","doi-asserted-by":"publisher","first-page":"2171","DOI":"10.1002\/mma.5982","volume":"43","author":"D Zeidan","year":"2020","unstructured":"Zeidan D, Chau CK, Lu T-T, Zheng W-Q (2020) Mathematical studies of the solution of burgers\u2019 equations by adomian decomposition method. Math Methods Appl Sci 43(5):2171\u20132188","journal-title":"Math Methods Appl Sci"},{"issue":"3","key":"2512_CR35","doi-asserted-by":"publisher","first-page":"840","DOI":"10.1016\/j.camwa.2010.05.031","volume":"60","author":"H Zhu","year":"2010","unstructured":"Zhu H, Shu H, Ding M (2010) Numerical solutions of two-dimensional burgers\u2019 equations by discrete adomian decomposition method. Comput Math Appl 60(3):840\u2013848","journal-title":"Comput Math Appl"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-023-02512-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-023-02512-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-023-02512-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,11]],"date-time":"2024-02-11T11:20:10Z","timestamp":1707650410000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-023-02512-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,12,8]]},"references-count":35,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2024,2]]}},"alternative-id":["2512"],"URL":"https:\/\/doi.org\/10.1007\/s40314-023-02512-6","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,12,8]]},"assertion":[{"value":"8 February 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 October 2023","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 October 2023","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 December 2023","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"This work does not have any conflicts of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"11"}}