{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T15:15:00Z","timestamp":1774624500479,"version":"3.50.1"},"reference-count":41,"publisher":"Springer Science and Business Media LLC","issue":"3","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"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2026,3]]},"DOI":"10.1007\/s12190-025-02759-8","type":"journal-article","created":{"date-parts":[[2026,2,3]],"date-time":"2026-02-03T14:21:06Z","timestamp":1770128466000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A positivity-preserving Galerkin finite element scheme for a nonlinear reaction-diffusion-advection Mussel-Algal model with Danckwerts boundary conditions"],"prefix":"10.1007","volume":"72","author":[{"given":"Liu","family":"Xing","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wang","family":"Jie","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lang","family":"Yanhua","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yang","family":"Huizi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhou","family":"Zhongxin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2026,2,3]]},"reference":[{"key":"2759_CR1","doi-asserted-by":"crossref","unstructured":"Zou, R., Guo, S.: Dynamics of a diffusive leslie-gower predator-prey model in spatially heterogeneous environment. Discrete Contin. Dyn. Syst.-Ser. B 25(11) (2020)","DOI":"10.3934\/dcdsb.2020093"},{"issue":"4","key":"2759_CR2","doi-asserted-by":"publisher","first-page":"1409","DOI":"10.1016\/j.jde.2015.03.006","volume":"259","author":"S. Guo","year":"2015","unstructured":"Guo, S.: Stability and bifurcation in a reaction\u2013diffusion model with nonlocal delay effect. J. Differ. Eqs. 259(4), 1409\u20131448 (2015)","journal-title":"J. Differ. Equations"},{"issue":"7","key":"2759_CR3","doi-asserted-by":"publisher","first-page":"1141","DOI":"10.1080\/00036811003717954","volume":"89","author":"H. Nie","year":"2010","unstructured":"Nie, H., Wu, J.: Uniqueness and stability for coexistence solutions of the unstirred chemostat model. Applicable Anal. 89(7), 1141\u20131159 (2010)","journal-title":"Applicable Anal."},{"key":"2759_CR4","doi-asserted-by":"crossref","unstructured":"Wang, H., Guo, S., Li, S.: Stationary solutions of advective lotka\u2013volterra models with a weak allee effect and large diffusion. Nonlinear Anal.: Real World Appl. 56(103171) (2020)","DOI":"10.1016\/j.nonrwa.2020.103171"},{"issue":"1","key":"2759_CR5","doi-asserted-by":"publisher","first-page":"347","DOI":"10.3934\/cpaa.2014.13.347","volume":"13","author":"F. Yi","year":"2014","unstructured":"Yi, F., Zhang, H., Cherif, A., Zhang, W.: Spatiotemporal patterns of a homogeneous diffusive system modeling hair growth: global asymptotic behavior and multiple bifurcation analysis. Commun. Pure Appl. Anal. 13(1), 347\u2013369 (2014)","journal-title":"Commun. Pure Appl. Anal."},{"issue":"2","key":"2759_CR6","first-page":"573","volume":"59","author":"M. Ballyk","year":"1999","unstructured":"Ballyk, M., Dung, Le., Jones, D.A., Smith, H.L.: Effects of random motility on microbial growth and competition in a flow reactor. Siam J. Appl. Math. 59(2), 573\u2013596 (1999)","journal-title":"Siam J. Appl. Math."},{"issue":"4","key":"2759_CR7","doi-asserted-by":"publisher","first-page":"281","DOI":"10.1016\/0009-2509(59)80063-4","volume":"10","author":"J. Pearson","year":"1959","unstructured":"Pearson, J.: A note on the \u201cDanckwerts\u201d boundary conditions for continuous flow reactors. Chem. Eng. Sci. 10(4), 281\u2013284 (1959)","journal-title":"Chem. Eng. Sci."},{"key":"2759_CR8","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1016\/j.apnum.2019.08.028","volume":"148","author":"P. Das","year":"2020","unstructured":"Das, P., Rana, S., Vigo-Aguiar, J.: Higher order accurate approximations on equidistributed meshes for boundary layer originated mixed type reaction diffusion systems with multiple scale nature. Appl. Numer. Math. 148, 79\u201397 (2020)","journal-title":"Appl. Numer. Math."},{"issue":"11","key":"2759_CR9","doi-asserted-by":"publisher","first-page":"9419","DOI":"10.1002\/mma.7369","volume":"44","author":"P. Das","year":"2021","unstructured":"Das, P., Rana, S.: Theoretical prospects of fractional order weakly singular volterra integro differential equations and their approximations with convergence analysis. Math. Methods Appl. Sci. 44(11), 9419\u20139440 (2021)","journal-title":"Math. Methods Appl. Sci."},{"key":"2759_CR10","doi-asserted-by":"publisher","first-page":"113116","DOI":"10.1016\/j.cam.2020.113116","volume":"404","author":"P. Das","year":"2022","unstructured":"Das, P., Rana, S., Ramos, H.: On the approximate solutions of a class of fractional order nonlinear volterra integro-differential initial value problems and boundary value problems of first kind and their convergence analysis. J. Comput. Appl. Math. 404, 113116 (2022)","journal-title":"J. Comput. Appl. Math."},{"issue":"4","key":"2759_CR11","doi-asserted-by":"publisher","first-page":"e12763","DOI":"10.1111\/sapm.12763","volume":"153","author":"S. Kumar","year":"2024","unstructured":"Kumar, S., R, I., Das, P.: Impact of mixed boundary conditions and nonsmooth data on layer-originated nonpremixed combustion problems: higher-order convergence analysis. Stud. Appl. Math. 153(4), e12763 (2024)","journal-title":"Stud. Appl. Math."},{"key":"2759_CR12","doi-asserted-by":"crossref","unstructured":"Kumar, S., Das, P., et al.: Analysis of an efficient parameter uniform domain decomposition approach for singularly perturbed gierer-meinhardt type nonlinear coupled systems of parabolic problems. Int. J. Numer. Anal. & Modeling. 22(4) (2025)","DOI":"10.4208\/ijnam2025-1020"},{"issue":"3","key":"2759_CR13","doi-asserted-by":"publisher","first-page":"1365","DOI":"10.1007\/s11075-024-01918-5","volume":"99","author":"S. Kumar","year":"2025","unstructured":"Kumar, S., Kumar, S., Das, P.: Second-order a priori and a posteriori error estimations for integral boundary value problems of nonlinear singularly perturbed parameterized form. Numer. Algoritms. 99(3), 1365\u20131392 (2025)","journal-title":"Numer. Algoritms"},{"key":"2759_CR14","doi-asserted-by":"publisher","first-page":"534","DOI":"10.1016\/j.apnum.2024.09.020","volume":"207","author":"S. Kumar","year":"2025","unstructured":"Kumar, S., Das, P.: A uniformly convergent analysis for multiple scale parabolic singularly perturbed convection-diffusion coupled systems: optimal accuracy with less computational time. Appl. Numer. Math. 207, 534\u2013557 (2025)","journal-title":"Appl. Numer. Math."},{"key":"2759_CR15","doi-asserted-by":"crossref","unstructured":"Patawari, A., Das, P.: From traditional to computationally efficient scientific computing algorithms in option pricing: current progresses with future directions. Archiv. Comput. Methods Eng. 1\u201340 (2025)","DOI":"10.1007\/s11831-025-10343-3"},{"issue":"3","key":"2759_CR16","first-page":"1026","volume":"20","author":"A. Patawari","year":"2025","unstructured":"Patawari, A., Kumar, S., Das, P.: A modified iterative pinn algorithm for strongly coupled system of boundary layer originated convection diffusion reaction problems in mhd flows with analysis. Networks Meterogen. Media. 20(3), 1026\u20131060 (2025)","journal-title":"Networks Meterogen. Media"},{"issue":"4","key":"2759_CR17","doi-asserted-by":"publisher","first-page":"459","DOI":"10.4208\/ijnam2025-1020","volume":"22","author":"S.K. Aakansha","year":"2025","unstructured":"Aakansha, S.K., Das, P.: Analysis of an efficient parameter uniform domain decomposition approach for singularly perturbed gierer-meinhardt type nonlinear coupled systems of parabolic problems. Int. J. Numer. Anal. Model. 22(4), 459\u2013482 (2025)","journal-title":"Int. J. Numer. Anal. Model"},{"key":"2759_CR18","doi-asserted-by":"publisher","first-page":"103637","DOI":"10.1016\/j.bulsci.2025.103637","volume":"201","author":"V. Saw","year":"2025","unstructured":"Saw, V., Das, P., Srivastava, H.M.: Investigations of a class of liouville-caputo fractional order pennes bioheat flow partial differential equations through orthogonal polynomials on collocation points. Bull. Des Sci. Math\u00e9matiques. 201, 103637 (2025)","journal-title":"Bull. Des Sci. Math\u00e9matiques"},{"key":"2759_CR19","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1016\/j.camwa.2023.09.008","volume":"150","author":"S. Santra","year":"2023","unstructured":"Santra, S., Mohapatra, J., Das, P., Choudhuri, D.: Higher order approximations for fractional order integro-parabolic partial differential equations on an adaptive mesh with error analysis. Comput. Math. Appl. 150, 87\u2013101 (2023)","journal-title":"Comput. Math. Appl."},{"issue":"4","key":"2759_CR20","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10092-025-00671-y","volume":"62","author":"Y.A. Sabawi","year":"2025","unstructured":"Sabawi, Y.A., Santra, S., Das, P., Mohammad, N.A.: h1 stability and convergence analysis of the l1\/l1-2 legendre spectral method for non-local weakly singular integro-pdes. Calcolo 62(4), 1\u201346 (2025)","journal-title":"Calcolo"},{"key":"2759_CR21","doi-asserted-by":"crossref","unstructured":"Anukiruthika, K., Muthukumar, P., Das, P.: Semigroup-based theoretical prospects on optimal control of stochastic second-order gurtin\u2013pipkin integro-differential equations. Optimization 1\u201318 (2025)","DOI":"10.1080\/02331934.2025.2534119"},{"key":"2759_CR22","doi-asserted-by":"publisher","first-page":"264","DOI":"10.1016\/j.camwa.2023.06.018","volume":"144","author":"X. Liu","year":"2023","unstructured":"Liu, X., Yang, Z.W., Zeng, Y.: Global numerical analysis of an improved imex numerical scheme for a reaction diffusion sis model in advective heterogeneous environments. Comput. Math. Appl. 144, 264\u2013273 (2023)","journal-title":"Comput. Math. Appl."},{"key":"2759_CR23","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1016\/j.matcom.2023.06.009","volume":"213","author":"S. Yang","year":"2023","unstructured":"Yang, S., Liu, X., Zhang, M.: Threshold stability of an improved imex numerical method based on conservation law for a nonlinear advection\u2013diffusion lotka\u2013volterra model. Math. Comput. Simul. 213, 127\u2013144 (2023)","journal-title":"Math. Comput. Simul."},{"issue":"2","key":"2759_CR24","doi-asserted-by":"publisher","first-page":"205","DOI":"10.1016\/j.apnum.2003.10.002","volume":"48","author":"A. Galeao","year":"2004","unstructured":"Galeao, A., Almeida, R., Malta, S.: Finite element analysis of convection dominated reaction\u2013diffusion problems. Appl. Numer. Math. 48(2), 205\u2013222 (2004)","journal-title":"Appl. Numer. Math."},{"issue":"1","key":"2759_CR25","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1137\/090775464","volume":"49","author":"G. Wells","year":"2011","unstructured":"Wells, G.: Analysis of an interface stabilized finite element method: the advection-diffusion-reaction equation. SIAM J. Numer. Anal. 49(1), 87\u2013109 (2011)","journal-title":"SIAM J. Numer. Anal."},{"key":"2759_CR26","doi-asserted-by":"publisher","first-page":"576","DOI":"10.1016\/j.apnum.2019.10.021","volume":"150","author":"M. Chowdhury","year":"2020","unstructured":"Chowdhury, M., Kumar, B.: On subgrid multiscale stabilized finite element method for advection-diffusion-reaction equation with variable coefficients. Appl. Numer. Math. 150, 576\u2013586 (2020)","journal-title":"Appl. Numer. Math."},{"key":"2759_CR27","doi-asserted-by":"publisher","first-page":"389","DOI":"10.1016\/j.cma.2018.04.019","volume":"339","author":"G. Barrenechea","year":"2018","unstructured":"Barrenechea, G., Poza, A., Yorston, H.: A stabilised finite element method for the convection\u2013diffusion\u2013reaction equation in mixed form. Comput. Methods Appl. Mech. Eng. 339, 389\u2013415 (2018)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"11","key":"2759_CR28","doi-asserted-by":"publisher","first-page":"2688","DOI":"10.1108\/HFF-02-2018-0077","volume":"28","author":"S. Sheshachala","year":"2018","unstructured":"Sheshachala, S., Codina, R.: Finite element modeling of nonlinear reaction\u2013Diffusion\u2013advection systems of equations. Int. J. Numer. Methods Heat Fluid Flow 28(11), 2688\u20132715 (2018)","journal-title":"Int. J. Numer. Methods Heat Fluid Flow"},{"issue":"8","key":"2759_CR29","doi-asserted-by":"publisher","first-page":"1533","DOI":"10.1142\/S0218202524500283","volume":"34","author":"A. Amiri","year":"2024","unstructured":"Amiri, A., Barrenechea, G., Pryer, T.: A nodally bound-preserving finite element method for reaction\u2013convection\u2013diffusion equations. Math. Model Methods Appl. Sci. 34(8), 1533\u20131565 (2024)","journal-title":"Math. Model Methods Appl. Sci."},{"key":"2759_CR30","doi-asserted-by":"crossref","unstructured":"Kou, J., Chen, H., Wang, X., Sun, S.: A linear, decoupled and positivity-preserving numerical scheme for an epidemic model with advection and diffusion. (2023)","DOI":"10.3934\/cpaa.2021094"},{"key":"2759_CR31","doi-asserted-by":"crossref","unstructured":"Yang, D., Peng, G., Gao, Z.: A positivity-preserving finite volume scheme for nonequilibrium radiation diffusion equations on distorted meshes. Entropy. 24(3) (2022)","DOI":"10.3390\/e24030382"},{"key":"2759_CR32","doi-asserted-by":"publisher","first-page":"773","DOI":"10.1016\/j.jcp.2018.06.052","volume":"372","author":"S. Su","year":"2018","unstructured":"Su, S., Dong, Q., Wu, J.: A decoupled and positivity-preserving discrete duality finite volume scheme for anisotropic diffusion problems on general polygonal meshes. J. Comput. Phys. 372, 773\u2013798 (2018)","journal-title":"J. Comput. Phys."},{"issue":"3","key":"2759_CR33","doi-asserted-by":"publisher","first-page":"649","DOI":"10.1007\/s10915-019-00951-0","volume":"80","author":"M. Sulman","year":"2019","unstructured":"Sulman, M., T, N.: A positivity preserving moving mesh finite element method for the keller\u2013segel chemotaxis model. J Sci Comput. 80(3), 649\u2013666 (2019)","journal-title":"J Sci Comput."},{"key":"2759_CR34","doi-asserted-by":"publisher","first-page":"116691","DOI":"10.1016\/j.cam.2025.116691","volume":"470","author":"A. Amiri","year":"2025","unstructured":"Amiri, A., Barrenechea, G.R., Pryer, T.: A nodally bound-preserving finite element method for time-dependent convection\u2013diffusion equations. J. Comput. Appl. Math. 470, 116691 (2025)","journal-title":"J. Comput. Appl. Math."},{"issue":"3","key":"2759_CR35","doi-asserted-by":"publisher","first-page":"117","DOI":"10.1007\/s00033-022-01744-6","volume":"73","author":"J. Wang","year":"2022","unstructured":"Wang, J., Tong, X., Song, Y.: Dynamics and pattern formation in a reaction-diffusion-advection mussel\u2013algae model. Z. f\u00fcr Angew. Math. und Phys. 73(3), 117 (2022)","journal-title":"Z. f\u00fcr Angew. Math. und Phys."},{"key":"2759_CR36","volume-title":"Galerkin Finite Element Methods for Parabolic Problems","author":"V. Thom\u00e9e","year":"2007","unstructured":"Thom\u00e9e, V.: Galerkin finite element methods for parabolic problems. Springer Science & Business Media (2007)"},{"key":"2759_CR37","doi-asserted-by":"crossref","unstructured":"Lang, Y., Liu, X., Li, W., Yang, H.: Threshold stability analysis of an unconditionally positivity-preserving method for a nonlinear reaction-diffusion-advection algae-mussel model under danckwerts boundary conditions. Numerical Algorithms (2025)","DOI":"10.1007\/s11075-025-02108-7"},{"issue":"4","key":"2759_CR38","doi-asserted-by":"publisher","first-page":"1704","DOI":"10.1137\/07068254X","volume":"46","author":"R. Becker","year":"2008","unstructured":"Becker, R., Feng, X., Prohl, A.: Finite element approximations of the ericksen\u2013leslie model for nematic liquid crystal flow. SIAM J. Numer. Anal. 46(4), 1704\u20131731 (2008)","journal-title":"SIAM J. Numer. Anal."},{"issue":"8","key":"2759_CR39","doi-asserted-by":"publisher","first-page":"392","DOI":"10.1007\/s40314-022-02113-9","volume":"41","author":"X. Liu","year":"2022","unstructured":"Liu, X., Yang, Z.: Numerical analysis of a reaction\u2013diffusion susceptible\u2013infected\u2013susceptible epidemic model. Comput. Appl. Math. 41(8), 392 (2022)","journal-title":"Comput. Appl. Math."},{"key":"2759_CR40","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139020411","volume-title":"Matrix Analysis","author":"R. Horn","year":"2012","unstructured":"Horn, R., Johnson, C.: Matrix analysis. Cambridge University Press (2012)"},{"issue":"10","key":"2759_CR41","doi-asserted-by":"publisher","first-page":"1994","DOI":"10.1080\/00207160.2019.1673892","volume":"97","author":"P. Das","year":"2020","unstructured":"Das, P., Rana, S., Ramos, H.: A perturbation-based approach for solving fractional-order volterra\u2013fredholm integro differential equations and its convergence analysis. Int. J. Comput. Math. 97(10), 1994\u20132014 (2020)","journal-title":"Int. J. Comput. Math."}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-025-02759-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-025-02759-8","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-025-02759-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T14:08:14Z","timestamp":1774620494000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-025-02759-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,3]]},"references-count":41,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["2759"],"URL":"https:\/\/doi.org\/10.1007\/s12190-025-02759-8","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"value":"1598-5865","type":"print"},{"value":"1865-2085","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,3]]},"assertion":[{"value":"30 August 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 December 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 December 2025","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"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"110"}}