{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T08:50:52Z","timestamp":1780390252392,"version":"3.54.1"},"reference-count":30,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2025,10,15]],"date-time":"2025-10-15T00:00:00Z","timestamp":1760486400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,10,15]],"date-time":"2025-10-15T00:00:00Z","timestamp":1760486400000},"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":[[2026,3]]},"DOI":"10.1007\/s40314-025-03437-y","type":"journal-article","created":{"date-parts":[[2025,10,15]],"date-time":"2025-10-15T05:22:36Z","timestamp":1760505756000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["An efficient B-spline wavelet algorithm for simulation of viscous Burgers\u2019 and coupled Burgers\u2019 equations"],"prefix":"10.1007","volume":"45","author":[{"given":"Deepak","family":"Kumar","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sapna","family":"Pandit","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,10,15]]},"reference":[{"key":"3437_CR1","doi-asserted-by":"publisher","first-page":"117","DOI":"10.1016\/j.cpc.2015.11.007","volume":"200","author":"HP Bhatt","year":"2016","unstructured":"Bhatt HP, Khaliq A-QM (2016) Fourth-order compact schemes for the numerical simulation of coupled burgers\u2019 equation. Comput Phys Commun 200:117\u2013138","journal-title":"Comput Phys Commun"},{"issue":"3","key":"3437_CR2","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"},{"issue":"4","key":"3437_CR3","doi-asserted-by":"publisher","first-page":"3711","DOI":"10.1103\/PhysRevE.52.3711","volume":"52","author":"SE Esipov","year":"1995","unstructured":"Esipov SE (1995) Coupled burgers equations: a model of polydispersive sedimentation. Phys Rev E 52(4):3711\u20133718","journal-title":"Phys Rev E"},{"issue":"6","key":"3437_CR4","doi-asserted-by":"publisher","first-page":"614","DOI":"10.1109\/8.387178","volume":"43","author":"JC Goswami","year":"1995","unstructured":"Goswami JC, Chan AK, Chui CK (1995) On solving first-kind integral equations using wavelets on a bounded interval. IEEE Trans Antennas Propag 43(6):614\u2013622","journal-title":"IEEE Trans Antennas Propag"},{"issue":"3","key":"3437_CR5","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1002\/cpa.3160030302","volume":"3","author":"E Hopf","year":"1950","unstructured":"Hopf E (1950) The partial differential equation ut+ uux= $$\\mu $$xx. Commun Pure Appl Math 3(3):201\u2013230","journal-title":"Commun Pure Appl Math"},{"issue":"11","key":"3437_CR6","doi-asserted-by":"publisher","first-page":"2413","DOI":"10.1016\/j.cpc.2012.06.009","volume":"183","author":"R Jiwari","year":"2012","unstructured":"Jiwari R (2012) A haar wavelet quasilinearization approach for numerical simulation of burgers\u2019 equation. Comput Phys Commun 183(11):2413\u20132423","journal-title":"Comput Phys Commun"},{"key":"3437_CR7","doi-asserted-by":"publisher","first-page":"106","DOI":"10.1016\/j.matcom.2022.02.024","volume":"198","author":"R Jiwari","year":"2022","unstructured":"Jiwari R (2022) Local radial basis function-finite difference based algorithms for singularly perturbed burgers\u2019 model. Math Comput Simul 198:106\u2013126","journal-title":"Math Comput Simul"},{"issue":"4","key":"3437_CR8","doi-asserted-by":"publisher","first-page":"1142","DOI":"10.1108\/EC-04-2018-0189","volume":"36","author":"R Jiwari","year":"2019","unstructured":"Jiwari R, Kumar S, Mittal RC (2019) Meshfree algorithms based on radial basis functions for numerical simulation and to capture shocks behavior of burgers\u2019 types problems. Eng Comput 36(4):1142\u20131168","journal-title":"Eng Comput"},{"issue":"2","key":"3437_CR9","first-page":"1365","volume":"170","author":"MK Kadalbajoo","year":"2005","unstructured":"Kadalbajoo MK, Sharma KK, Awasthi A (2005) A parameter-uniform implicit difference scheme for solving time-dependent burgers\u2019 equations. Appl Math Comput 170(2):1365\u20131393","journal-title":"Appl Math Comput"},{"issue":"10","key":"3437_CR10","doi-asserted-by":"publisher","first-page":"2863","DOI":"10.1016\/j.jfranklin.2011.09.008","volume":"348","author":"A Korkmaz","year":"2011","unstructured":"Korkmaz A, Da\u01e7 \u0130 (2011) Polynomial based differential quadrature method for numerical solution of nonlinear burgers\u2019 equation. J Frankl Inst 348(10):2863\u20132875","journal-title":"J Frankl Inst"},{"key":"3437_CR11","first-page":"139","volume":"44","author":"KH Kumar","year":"2024","unstructured":"Kumar KH, Jiwari R (2024) Legendre wavelet based solution to Burgers\u2019 and coupled Burgers\u2019 equation. J Comb Syst Sci 44:139\u2013154","journal-title":"J Comb Syst Sci"},{"issue":"3","key":"3437_CR12","doi-asserted-by":"publisher","first-page":"809","DOI":"10.1016\/j.cpc.2013.11.012","volume":"185","author":"M Kumar","year":"2014","unstructured":"Kumar M, Pandit S (2014) A composite numerical scheme for the numerical simulation of coupled burgers\u2019 equation. Comput Phys Commun 185(3):809\u2013817","journal-title":"Comput Phys Commun"},{"issue":"1","key":"3437_CR13","doi-asserted-by":"publisher","first-page":"21","DOI":"10.1016\/j.cam.2003.09.043","volume":"167","author":"S Kutluay","year":"2004","unstructured":"Kutluay S, Esen A, Dag I (2004) Numerical solutions of the burgers\u2019 equation by the least-squares quadratic b-spline finite element method. J Comput Appl Math 167(1):21\u201333","journal-title":"J Comput Appl Math"},{"issue":"3","key":"3437_CR14","doi-asserted-by":"publisher","first-page":"1304","DOI":"10.1016\/j.cnsns.2010.06.028","volume":"16","author":"RC Mittal","year":"2011","unstructured":"Mittal RC, Arora G (2011) Numerical solution of the coupled viscous burgers\u2019 equation. Commun Nonlinear Sci Numer Simul 16(3):1304\u20131313","journal-title":"Commun Nonlinear Sci Numer Simul"},{"issue":"15","key":"3437_CR15","first-page":"7839","volume":"218","author":"RC Mittal","year":"2012","unstructured":"Mittal RC, Jain RK (2012) Numerical solutions of nonlinear burgers\u2019 equation with modified cubic b-splines collocation method. Appl Math Comput 218(15):7839\u20137855","journal-title":"Appl Math Comput"},{"issue":"2","key":"3437_CR16","doi-asserted-by":"publisher","first-page":"88","DOI":"10.1080\/15502287.2011.654175","volume":"13","author":"RC Mittal","year":"2012","unstructured":"Mittal RC, Jiwari R (2012a) Differential quadrature method for numerical solution of coupled viscous burgers\u2019 equations. Int J Comput Methods Eng Sci Mech 13(2):88\u201392","journal-title":"Int J Comput Methods Eng Sci Mech"},{"issue":"7","key":"3437_CR17","doi-asserted-by":"publisher","first-page":"880","DOI":"10.1108\/09615531211255761","volume":"22","author":"RC Mittal","year":"2012","unstructured":"Mittal RC, Jiwari R (2012b) A differential quadrature method for numerical solutions of burgers\u2019-type equations. Int J Numer Methods Heat Fluid Flow 22(7):880\u2013895","journal-title":"Int J Numer Methods Heat Fluid Flow"},{"issue":"3","key":"3437_CR18","doi-asserted-by":"publisher","first-page":"601","DOI":"10.1080\/00207160.2017.1293820","volume":"95","author":"RC Mittal","year":"2017","unstructured":"Mittal RC, Pandit S (2017) Sensitivity analysis of shock wave burgers\u2019 equation via a novel algorithm based on scale-3 haar wavelets. Int J Comput Math 95(3):601\u2013625","journal-title":"Int J Comput Math"},{"issue":"1","key":"3437_CR19","doi-asserted-by":"publisher","first-page":"117","DOI":"10.1108\/02644401311286071","volume":"30","author":"RC Mittal","year":"2012","unstructured":"Mittal RC, Jiwari R, Sharma KK (2012) A numerical scheme based on differential quadrature method to solve time dependent burgers\u2019 equation. Eng Comput 30(1):117\u2013131","journal-title":"Eng Comput"},{"issue":"1","key":"3437_CR20","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1108\/EC-08-2019-0365","volume":"38","author":"T Nazir","year":"2021","unstructured":"Nazir T, Abbas M, Iqbal MK (2021) New cubic b-spline approximation technique for numerical solutions of coupled viscous burgers equations. Eng Comput 38(1):83\u2013106","journal-title":"Eng Comput"},{"issue":"6","key":"3437_CR21","doi-asserted-by":"publisher","first-page":"486","DOI":"10.3390\/math7060486","volume":"7","author":"N Ozdemir","year":"2019","unstructured":"Ozdemir N, Secer A, Bayram M (2019) The gegenbauer wavelets-based computational methods for the coupled system of burgers\u2019 equations with time-fractional derivative. Mathematics 7(6):486","journal-title":"Mathematics"},{"issue":"2\u20133","key":"3437_CR22","first-page":"417","volume":"139","author":"T \u00d6zi\u015f","year":"2003","unstructured":"\u00d6zi\u015f T, Aksan EN, \u00d6zde\u015f A (2003) A finite element approach for solution of burgers\u2019 equation. Appl Math Comput 139(2\u20133):417\u2013428","journal-title":"Appl Math Comput"},{"issue":"8","key":"3437_CR23","doi-asserted-by":"publisher","first-page":"1814","DOI":"10.1108\/HFF-05-2016-0188","volume":"27","author":"S Pandit","year":"2017","unstructured":"Pandit S, Kumar M, Mohapatra RN, Alshomrani AS (2017) Shock waves analysis of planar and non planar nonlinear burgers\u2019 equation using scale-2 haar wavelets. Int J Numer Methods Heat Fluid Flow 27(8):1814\u20131850","journal-title":"Int J Numer Methods Heat Fluid Flow"},{"issue":"3","key":"3437_CR24","doi-asserted-by":"publisher","first-page":"217","DOI":"10.1006\/acha.1994.1009","volume":"1","author":"E Quak","year":"1994","unstructured":"Quak E, Weyrich N (1994) Decomposition and reconstruction algorithms for spline wavelets on a bounded interval. Appl Comput Harmon Anal 1(3):217\u2013231","journal-title":"Appl Comput Harmon Anal"},{"issue":"4","key":"3437_CR25","doi-asserted-by":"publisher","first-page":"412","DOI":"10.2478\/cmam-2009-0026","volume":"9","author":"A Rashid","year":"2009","unstructured":"Rashid A, Ismail AIBM (2009) A fourier pseudospectral method for solving coupled viscous burgers equations. Comput Methods Appl Math 9(4):412\u2013420","journal-title":"Comput Methods Appl Math"},{"issue":"4","key":"3437_CR26","first-page":"283","volume":"7","author":"VK Srivastava","year":"2013","unstructured":"Srivastava VK, Awasthi MK, Tamsir M (2013) A fully implicit finite-difference solution to one dimensional coupled nonlinear burgers\u2019 equations. Int J Math Sci 7(4):283\u2013288","journal-title":"Int J Math Sci"},{"issue":"2","key":"3437_CR27","first-page":"560","volume":"191","author":"AHAE Tabatabaei","year":"2007","unstructured":"Tabatabaei AHAE, Shakour E, Dehghan M (2007) Some implicit methods for the numerical solution of burgers\u2019 equation. Appl Math Comput 191(2):560\u2013570","journal-title":"Appl Math Comput"},{"issue":"9","key":"3437_CR28","first-page":"4473","volume":"217","author":"M Xu","year":"2011","unstructured":"Xu M, Wang R-H, Zhang J-H, Fang Q (2011) A novel numerical scheme for solving burgers\u2019 equation. Appl Math Comput 217(9):4473\u20134482","journal-title":"Appl Math Comput"},{"issue":"5","key":"3437_CR29","doi-asserted-by":"publisher","first-page":"1652","DOI":"10.1002\/num.22158","volume":"33","author":"OP Yadav","year":"2017","unstructured":"Yadav OP, Jiwari R (2017) Finite element analysis and approximation of burgers\u2019-fisher equation. Numer Methods Partial Differ Equ 33(5):1652\u20131677","journal-title":"Numer Methods Partial Differ Equ"},{"issue":"6","key":"3437_CR30","doi-asserted-by":"publisher","first-page":"1853","DOI":"10.1063\/1.869305","volume":"9","author":"DS Zhang","year":"1997","unstructured":"Zhang DS, Wei GW, Kouri DJ, Hoffman DK (1997) Burgers\u2019 equation with high reynolds number. Phys Fluids 9(6):1853\u20131855","journal-title":"Phys Fluids"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03437-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-025-03437-y","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03437-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,31]],"date-time":"2026-01-31T04:46:08Z","timestamp":1769834768000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-025-03437-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,10,15]]},"references-count":30,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["3437"],"URL":"https:\/\/doi.org\/10.1007\/s40314-025-03437-y","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,10,15]]},"assertion":[{"value":"22 February 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 August 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 September 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 October 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"No conflict of interest is disclosed by the authors.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"55"}}