{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,23]],"date-time":"2026-07-23T10:01:47Z","timestamp":1784800907542,"version":"3.55.0"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2026,2,3]],"date-time":"2026-02-03T00:00:00Z","timestamp":1770076800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,2,3]],"date-time":"2026-02-03T00:00:00Z","timestamp":1770076800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"name":"Outstanding Master\u2019s and Doctoral Dissertation Program of Longjiang in the New Era","award":["LJYXL2023-089"],"award-info":[{"award-number":["LJYXL2023-089"]}]},{"name":"Harbin Normal University Science and Technology Innovation Climbing Program","award":["XKB202303"],"award-info":[{"award-number":["XKB202303"]}]},{"name":"the open research fund of Heilongjiang Key Laboratory of Analysis on Machine Learning and Dynamic System","award":["HLJKL2507"],"award-info":[{"award-number":["HLJKL2507"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2026,7]]},"DOI":"10.1007\/s40314-026-03631-6","type":"journal-article","created":{"date-parts":[[2026,2,3]],"date-time":"2026-02-03T05:55:45Z","timestamp":1770098145000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A high-accuracy reproducing kernel approach for the coupled Burgers\u2019 equations"],"prefix":"10.1007","volume":"45","author":[{"given":"Jiabao","family":"Yang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zongwei","family":"Li","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jiaxuan","family":"Chi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Huanmin","family":"Yao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,2,3]]},"reference":[{"key":"3631_CR1","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1016\/j.cnsns.2018.09.007","volume":"69","author":"S Bak","year":"2019","unstructured":"Bak S, Kim P, Kim D (2019) A semi-lagrangian approach for numerical simulation of coupleds\u2019 equations. Communications in Nonlinrear Science And Numerical Simulation 69:31\u201344. https:\/\/doi.org\/10.1016\/j.cnsns.2018.09.007","journal-title":"Communications in Nonlinrear Science And Numerical Simulation"},{"issue":"4","key":"3631_CR2","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\u2019 equations: a model of polydispersive sedimentation. Phys Rev E 52(4):3711. https:\/\/doi.org\/10.1103\/PhysRevE.52.3711","journal-title":"Phys Rev E"},{"issue":"2","key":"3631_CR3","doi-asserted-by":"publisher","first-page":"305","DOI":"10.1088\/0305-4470\/20\/2\/016","volume":"20","author":"G Barker","year":"1987","unstructured":"Barker G, Grimson M (1987) A solitary wave model for sedimentation in colloidal suspensions. J Phys A: Math Gen 20(2):305. https:\/\/doi.org\/10.1088\/0305-4470\/20\/2\/016","journal-title":"J Phys A: Math Gen"},{"issue":"2","key":"3631_CR4","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1209\/0295-5075\/11\/2\/003","volume":"11","author":"W Van Saarloos","year":"1990","unstructured":"Van Saarloos W, Huse D (1990) Layer formation of dilute colloidal suspensions as a result of a concentration-dependent sedimentation velocity: some simple considerations and suggestions for further experiments. Europhys Lett 11(2):107. https:\/\/doi.org\/10.1209\/0295-5075\/11\/2\/003","journal-title":"Europhys Lett"},{"key":"3631_CR5","doi-asserted-by":"publisher","unstructured":"Mohanty, R., Sharma, S.: A new high-accuracy method based on off-step cubic polynomial approximations for the solution of coupled Burgers\u2019 equations and Burgers-Huxley equation. Engineering with Computers 37(4), 3049\u20133066 (2021). https:\/\/doi.org\/10.1007\/s00366-020-00982-4","DOI":"10.1007\/s00366-020-00982-4"},{"key":"3631_CR6","doi-asserted-by":"publisher","unstructured":"Muhannad A.\u00a0Shallal, K.R.R. Khalid K.\u00a0Ali, Taqi, A.H.: Septic B-spline collocation method for numerical solution of the coupled Burgers\u2019 equations. Arab Journal of Basic and Applied Sciences 26(1), 331\u2013341 (2019) https:\/\/doi.org\/10.1080\/25765299.2019.1628687","DOI":"10.1080\/25765299.2019.1628687"},{"key":"3631_CR7","doi-asserted-by":"publisher","first-page":"391","DOI":"10.1016\/j.apnum.2022.10.014","volume":"184","author":"SA Tahir","year":"2023","unstructured":"Tahir SA, Sari M (2023) A new approach for the coupled advection-diffusion processes including source effects. Applied Numerical Mathematics 184:391\u2013405. https:\/\/doi.org\/10.1016\/j.apnum.2022.10.014","journal-title":"Applied Numerical Mathematics"},{"key":"3631_CR8","doi-asserted-by":"publisher","DOI":"10.1016\/j.padiff.2021.100044","volume":"4","author":"A Babu","year":"2021","unstructured":"Babu A, Han B, Asharaf N (2021) Numerical solution of the viscous $$\\rm B$$urgers\u2019 equation using localized differential quadrature method. Partial Differential Equations in Applied Mathematics 4:100044. https:\/\/doi.org\/10.1016\/j.padiff.2021.100044","journal-title":"Partial Differential Equations in Applied Mathematics"},{"key":"3631_CR9","doi-asserted-by":"publisher","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. https:\/\/doi.org\/10.1016\/j.cpc.2013.11.012","DOI":"10.1016\/j.cpc.2013.11.012"},{"key":"3631_CR10","doi-asserted-by":"publisher","unstructured":"Cui M, Deng Z (1986) On the best operator of interpolation in $${{\\rm W}}^{1}_{2}$$. Mathematic Numerica Sinica 8, 209\u2013216 https:\/\/doi.org\/10.12286\/jssx.1986.2.209","DOI":"10.12286\/jssx.1986.2.209"},{"key":"3631_CR11","unstructured":"Wu B, Cui M, Deng Z (1988) Optimum Hermitean interpolation operator in the $${{\\rm W}}^{2}_{2}$$ space. Chinese Journal of Computational Physics 5(2), 221\u2013231"},{"issue":"2","key":"3631_CR12","first-page":"177","volume":"4","author":"M Cui","year":"1986","unstructured":"Cui M, Zhang M, Deng Z (1986) Two-dimensional reproducing kernel and surface interpolation. J Comput Math 4(2):177\u2013181","journal-title":"J Comput Math"},{"issue":"10","key":"3631_CR13","doi-asserted-by":"publisher","first-page":"1558","DOI":"10.1002\/cnm.1376","volume":"27","author":"H Yao","year":"2011","unstructured":"Yao H, Lin Y (2011) New algorithm for solving a nonlinear hyperbolic telegraph equation with an integral condition. International Journal for Numerical Methods in Biomedical Engineering 27(10):1558\u20131568. https:\/\/doi.org\/10.1002\/cnm.1376","journal-title":"International Journal for Numerical Methods in Biomedical Engineering"},{"key":"3631_CR14","doi-asserted-by":"publisher","first-page":"221","DOI":"10.1016\/j.aml.2017.08.020","volume":"76","author":"J Yang","year":"2018","unstructured":"Yang J, Yao H, Wu B (2018) An efficient numerical method for variable order fractional functional differential equation. Applied Mathematics Letters 76:221\u2013226. https:\/\/doi.org\/10.1016\/j.aml.2017.08.020","journal-title":"Applied Mathematics Letters"},{"issue":"10","key":"3631_CR15","doi-asserted-by":"publisher","first-page":"3841","DOI":"10.1002\/mma.4870","volume":"41","author":"Z Chen","year":"2018","unstructured":"Chen Z, Wu L, Lin Y (2018) Exact solution of a class of fractional integro-differential equations with the weakly singular kernel based on a new fractional reproducing kernel space. Mathematical Methods in the Applied Sciences 41(10):3841\u20133855. https:\/\/doi.org\/10.1002\/mma.4870","journal-title":"Mathematical Methods in the Applied Sciences"},{"key":"3631_CR16","doi-asserted-by":"publisher","first-page":"210","DOI":"10.1016\/j.apnum.2020.06.004","volume":"157","author":"H Du","year":"2020","unstructured":"Du H, Chen Z, Yang T (2020) A stable least residue method in reproducing kernel space for solving a nonlinear fractional integro-differential equation with a weakly singular kernel. Applied Numerical Mathematics 157:210\u2013222. https:\/\/doi.org\/10.1016\/j.apnum.2020.06.004","journal-title":"Applied Numerical Mathematics"},{"issue":"2","key":"3631_CR17","doi-asserted-by":"publisher","first-page":"2171","DOI":"10.1002\/mma.6927","volume":"44","author":"R Zhang","year":"2021","unstructured":"Zhang R, Lin Y (2021) A new algorithm for fractional differential equation based on fractional order reproducing kernel space. Mathematical Methods in the Applied Sciences 44(2):2171\u20132182. https:\/\/doi.org\/10.1002\/mma.6927","journal-title":"Mathematical Methods in the Applied Sciences"},{"key":"3631_CR18","doi-asserted-by":"publisher","unstructured":"Su X, Yang J, Yao H (2023) Shifted Legendre reproducing kernel Galerkin method for the quasilinear degenerate parabolic problem. Applied Mathematics Letters 135, 108416 https:\/\/doi.org\/10.1016\/j.aml.2022.108416","DOI":"10.1016\/j.aml.2022.108416"},{"issue":"2","key":"3631_CR19","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1007\/s10092-023-00525-5","volume":"60","author":"R Sun","year":"2023","unstructured":"Sun R, Yang J, Yao H (2023) Numerical solution of the space-time fractional diffusion equation based on fractional reproducing kernel collocation method. Calcolo 60(2):31. https:\/\/doi.org\/10.1007\/s10092-023-00525-5","journal-title":"Calcolo"},{"key":"3631_CR20","doi-asserted-by":"publisher","unstructured":"Mu X, Yang J, Yao H (2023) A binary Caputo-Fabrizio fractional reproducing kernel method for the time-fractional Cattaneo equation. J Appl Math Comput 69(5):3755\u20133791. https:\/\/doi.org\/10.1007\/s12190-023-01902-7","DOI":"10.1007\/s12190-023-01902-7"},{"key":"3631_CR21","doi-asserted-by":"publisher","unstructured":"Chen S-B, Soradi-Zeid S, Dutta H, Mesrizadeh M, Jahanshahi H, Chu Y-M (2021) Reproducing kernel Hilbert space method for nonlinear second order singularly perturbed boundary value problems with time-delay. Chaos Solitons and Fractals 144, 110674 https:\/\/doi.org\/10.1016\/j.chaos.2021.110674","DOI":"10.1016\/j.chaos.2021.110674"},{"key":"3631_CR22","doi-asserted-by":"publisher","unstructured":"Azarnavid B, Parvaneh F, Abbasbandy S (2015) Picard-reproducing kernel Hilbert space method for solving generalized singular nonlinear Lane-Emden type equations. Mathematical Modelling and Analysis 20(6), 754\u2013767 https:\/\/doi.org\/10.3846\/13926292.2015.1111953","DOI":"10.3846\/13926292.2015.1111953"},{"key":"3631_CR23","doi-asserted-by":"publisher","DOI":"10.1155\/2012\/839836","volume":"2012","author":"O Abu Arqub","year":"2012","unstructured":"Abu Arqub O, Al-Smadi M, Momani S et al (2012) Application of reproducing kernel method for solving nonlinear Fredholm-Volterra integrodifferential equations 2012:839836. https:\/\/doi.org\/10.1155\/2012\/839836","journal-title":"Application of reproducing kernel method for solving nonlinear Fredholm-Volterra integrodifferential equations"},{"key":"3631_CR24","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-71041-7","volume-title":"Spectral Methods: Algorithms, Analysis and Applications","author":"J Shen","year":"2011","unstructured":"Shen J, Tang T, Wang L (2011) Spectral Methods: Algorithms, Analysis and Applications, 1st edn. Springer, Heidelberg","edition":"1"},{"issue":"3","key":"3631_CR25","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.65.036708","volume":"65","author":"C Shu","year":"2002","unstructured":"Shu C, Niu X, Chew Y (2002) Taylor-series expansion and least-squares-based lattice Boltzmann method: Two-dimensional formulation and its applications. Phys Rev E 65(3):036708. https:\/\/doi.org\/10.1103\/PhysRevE.65.036708","journal-title":"Phys Rev E"},{"key":"3631_CR26","doi-asserted-by":"publisher","unstructured":"Grigorieff, R.: Stability of multistep-methods on variable grids. Numerische Mathematik 42, 359\u2013377 (1983) https:\/\/doi.org\/10.1007\/BF01389580","DOI":"10.1007\/BF01389580"},{"key":"3631_CR27","doi-asserted-by":"publisher","unstructured":"Emmrich, E.: Stability and error of the variable two-step $$\\rm BDF$$ for semilinear parabolic problems. Journal of Applied Mathematics and Computing 19, 33\u201355 (2005) https:\/\/doi.org\/10.1007\/bf02935787","DOI":"10.1007\/bf02935787"},{"issue":"4","key":"3631_CR28","doi-asserted-by":"publisher","first-page":"633","DOI":"10.1093\/imanum\/20.4.633","volume":"20","author":"A Hill","year":"2000","unstructured":"Hill A, Sli E (2000) Approximation of the global attractor for the incompressible Navier-Stokes equations. Ima Journa of Numerical Analysis 20(4):633\u2013667. https:\/\/doi.org\/10.1093\/imanum\/20.4.633","journal-title":"Ima Journa of Numerical Analysis"},{"key":"3631_CR29","volume-title":"Nonlinear Numerical Analysis in Reproducing Kernel Space","author":"M Cui","year":"2009","unstructured":"Cui M, Lin Y (2009) Nonlinear Numerical Analysis in Reproducing Kernel Space, 1st edn. Nova Science Publishers Inc, USA","edition":"1"},{"issue":"2","key":"3631_CR30","doi-asserted-by":"publisher","first-page":"415","DOI":"10.1016\/S0096-3003(03)00721-5","volume":"154","author":"G Wang","year":"2004","unstructured":"Wang G, Sun J (2004) A cramer rule for solution of the general restricted matrix equation. Appl Math Comput 154(2):415\u2013422. https:\/\/doi.org\/10.1016\/S0096-3003(03)00721-5","journal-title":"Appl Math Comput"},{"key":"3631_CR31","doi-asserted-by":"publisher","first-page":"1399","DOI":"10.1109\/ICTTA.2006.1684586","volume":"1","author":"SB Sadkhan","year":"2006","unstructured":"Sadkhan SB, Ruma KH (2006) Evaluation of polynomial reconstruction problem using Lagrange interpolation method 1:1399\u20131403. https:\/\/doi.org\/10.1109\/ICTTA.2006.1684586. (IEEE)","journal-title":"Evaluation of polynomial reconstruction problem using Lagrange interpolation method"},{"issue":"5","key":"3631_CR32","doi-asserted-by":"publisher","first-page":"6025","DOI":"10.1002\/mma.8887","volume":"46","author":"Y U\u00e7ar","year":"2023","unstructured":"U\u00e7ar Y, Ya\u011fmurlu NM, Yi\u011fit MK (2023) Numerical solution of the coupled Burgers\u2019 equation by trigonometric B-spline collocation method. Mathematical Methods in the Applied Sciences 46(5):6025\u20136041. https:\/\/doi.org\/10.1002\/mma.8887","journal-title":"Mathematical Methods in the Applied Sciences"},{"key":"3631_CR33","doi-asserted-by":"publisher","unstructured":"Ba\u015fhan, A.: A numerical treatment of the coupled viscous Burgers\u2019 equation in the presence of very large Reynolds number. Physica A: Statistical Mechanics and its Applications 545, 123755 (2020) https:\/\/doi.org\/10.1016\/j.physa.2019.123755","DOI":"10.1016\/j.physa.2019.123755"},{"key":"3631_CR34","doi-asserted-by":"publisher","unstructured":"Kutluay S, Ucar Y (2013) Numerical solutions of the coupled Burgers\u2019 equation by the Galerkin quadratic B-spline finite element method. Mathematical Methods in the Applied Sciences 36(17):2403\u20132415. https:\/\/doi.org\/10.1002\/mma.2767","DOI":"10.1002\/mma.2767"},{"key":"3631_CR35","doi-asserted-by":"publisher","unstructured":"Sreelakshmi, A., Shyaman, V.P., Awasthi, A.: An adaptive finite point scheme for the two-dimensional coupled Burgers\u2019 equation. Numerical Algorithms 99, 1871\u20131924 (2025) https:\/\/doi.org\/10.1007\/s11075-024-01936-3","DOI":"10.1007\/s11075-024-01936-3"},{"issue":"5","key":"3631_CR36","doi-asserted-by":"publisher","first-page":"1275","DOI":"10.13140\/RG.2.1.2072.2086","volume":"32","author":"RC Mittal","year":"2015","unstructured":"Mittal RC, Tripathi A (2015) Numerical solutions of two-dimensional Burgers\u2019 equations using modified Bi-cubic B-spline finite elements. Engineering Computations 32(5):1275\u20131306. https:\/\/doi.org\/10.13140\/RG.2.1.2072.2086","journal-title":"Engineering Computations"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-026-03631-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-026-03631-6","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-026-03631-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,23]],"date-time":"2026-07-23T09:06:36Z","timestamp":1784797596000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-026-03631-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,3]]},"references-count":36,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2026,7]]}},"alternative-id":["3631"],"URL":"https:\/\/doi.org\/10.1007\/s40314-026-03631-6","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,3]]},"assertion":[{"value":"9 May 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 November 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 January 2026","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 February 2026","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no conflict of interest.","order":1,"name":"Ethics","label":"Conflict of Interest","group":{"name":"EthicsHeading","label":"Declarations"}}],"article-number":"249"}}