{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:32:41Z","timestamp":1760059961093,"version":"build-2065373602"},"reference-count":49,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,7,21]],"date-time":"2025-07-21T00:00:00Z","timestamp":1753056000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"European Union-NextGenerationEU through the National Recovery and Resilience Plan of the Republic of Bulgaria","award":["BG-RRP-2.013-0001"],"award-info":[{"award-number":["BG-RRP-2.013-0001"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this work we propose a new second-order implicit\u2013explicit difference scheme for a pseudoparabolic equation with a nonlinear flux term. The proposed method is evaluated in comparison with two known numerical approaches. The significance of this study stems from its relevance to physical and mechanical models, including the Benjamin\u2013Bona\u2013Mahony (\u2013Burgers) equations, which arise as specific instances of the considered equation. A stability analysis of the constructed scheme is conducted. Furthermore, the method is generalized to address a two-dimensional case. Several numerical experiments are carried out to assess the accuracy and efficiency of the proposed scheme.<\/jats:p>","DOI":"10.3390\/axioms14070545","type":"journal-article","created":{"date-parts":[[2025,7,21]],"date-time":"2025-07-21T09:33:53Z","timestamp":1753090433000},"page":"545","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Second-Order Implicit\u2013Explicit Difference Scheme for Pseudoparabolic Equations with Nonlinear Flux"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0360-3651","authenticated-orcid":false,"given":"Miglena N.","family":"Koleva","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Natural Sciences and Education, University of Ruse \u201cAngel Kanchev\u201d, 8 Studentska Str., 7017 Ruse, Bulgaria"}]},{"given":"Lubin G.","family":"Vulkov","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Statistics, Faculty of Natural Sciences and Education, University of Ruse \u201cAngel Kanchev\u201d, 8 Studentska Str., 7017 Ruse, Bulgaria"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1016\/j.camwa.2021.10.001","article-title":"Spectral discretizations analysis with time strong stability preserving properties for pseudo-parabolic models","volume":"102","author":"Abreu","year":"2021","journal-title":"Comput. Math. Appl."},{"key":"ref_2","first-page":"47","article-title":"Model equations for long waves in nonlinear dispersive systems","volume":"272","author":"Benjamin","year":"1972","journal-title":"Philas Trans R. Soc. London Ser. A Math. Phys. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Burger, J.M. (1984). A mathematical model illustrating the theory of turbulence. Adv. Appl. Mech., 171\u2013199.","DOI":"10.1016\/S0065-2156(08)70100-5"},{"key":"ref_4","unstructured":"Bona, J. (1981). On solitary waves and their role in the evolution of long waves. Appl. Nonlinear Anal. Phys. Sci., 183\u2013205."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF00250690","article-title":"Certain non-steady flows of second-order fluids","volume":"14","author":"Ting","year":"1963","journal-title":"Arch. Ration. Mech. Anal."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2739","DOI":"10.1090\/S0002-9947-03-03340-3","article-title":"Effect of aggregation on population recovery modelled by a forward-backward pseudoparabolic equation","volume":"356","author":"Padron","year":"2004","journal-title":"Trans. Amer. Math. Soc."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1117","DOI":"10.1080\/03605300600781568","article-title":"Some qualitative properties of solutions to a pseudoparabolic equation modeling solvent uptake in polymeric solids Commun","volume":"31","year":"2006","journal-title":"Partial Differ. Equ."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"614","DOI":"10.1007\/BF01594969","article-title":"On a theory of heat conduction involving two temperatures","volume":"19","author":"Chen","year":"1969","journal-title":"Z. Angew. Math. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1137\/0712028","article-title":"Numerical solution of Sobolev partial differential equations","volume":"12","author":"Ewing","year":"1975","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"112193","DOI":"10.1016\/j.chaos.2022.112193","article-title":"Solvability of pseudoparabolic equation with Caputo fractional derivative","volume":"160","author":"Aitzhanov","year":"2022","journal-title":"Chaos Solitons Fractals"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1137\/0501001","article-title":"Pseudoparabolic partial differential equations","volume":"1","author":"Showalter","year":"1970","journal-title":"SIAM J. Math. Anal."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1414","DOI":"10.1137\/0524082","article-title":"A degenerate pseudoparabolic regularization of a nonlinear forward-backward heat equation arising in the theory of heat and mass exchange in stably stratified turbulent shear flow","volume":"24","author":"Barenblatt","year":"1993","journal-title":"SIAM J. Math. Anal."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"124533","DOI":"10.1016\/j.jmaa.2020.124533","article-title":"Well-posedness of two pseudo-parabolic problems for electrical conduction in heterogeneous media","volume":"493","author":"Amar","year":"2021","journal-title":"J. Math. Anal. Appl."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Leveque, R.J. (1992). Numerical Methods for Conservation Laws, Birkhauser.","DOI":"10.1007\/978-3-0348-8629-1"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1016\/j.camwa.2024.04.015","article-title":"Mathematical properties and numerical approximation of pseudoparabolic systems","volume":"165","author":"Abren","year":"2024","journal-title":"Comput. Math. Appl."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"988","DOI":"10.1080\/01630560903405412","article-title":"The stability of finite-difference schemes for a pseudoparabolic equation with nonlocal conditions","volume":"30","author":"Sapagovas","year":"2009","journal-title":"Numer. Funct. Anal. Optim."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Sapagovas, M., \u0160tikonas, A., and \u0160tikoniene, O. (2023). ADI method for pseudoparabolic equation with nonlocal boundary conditions. Mathematics, 11.","DOI":"10.3390\/math11061303"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/j.matcom.2016.10.006","article-title":"Computing numerical solutions of the pseudoparabolic Buckley-Leveret equation with dynamic capillary pressure","volume":"137","author":"Abreu","year":"2017","journal-title":"Math. Comput. Simul."},{"key":"ref_19","first-page":"1","article-title":"An efficient second order linearized Banjamin-Bona-Mahony equation with artificial boundary conditions","volume":"22\/23","author":"Zheng","year":"2022","journal-title":"Preprint BUW-IMACM"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2037","DOI":"10.1007\/s11075-024-01864-2","article-title":"A fast second-order absorbing boundary condition for the linearized Benjamin-Bona-Mahony equation","volume":"98","author":"Zheng","year":"2025","journal-title":"Numer. Algor."},{"key":"ref_21","first-page":"1835","article-title":"Three-dimensional nonlinear evolutionary pseudoparabolic equations in mathematical physics","volume":"43","author":"Korpusov","year":"2003","journal-title":"Zh. Vychis. Mat. Mat. Fisica"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2625","DOI":"10.1016\/j.nonrwa.2011.03.010","article-title":"Existence results for nonlinear pseudoparabolic problems","volume":"12","author":"Seam","year":"2011","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_23","first-page":"35","article-title":"Existence of weak solutions for a pseudo-parabolic system coupling chemical reactions, diffusion and momentum equations","volume":"1703","author":"Vromans","year":"2017","journal-title":"Tech. Univ. Eindh. CASA-Rep."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"883","DOI":"10.1016\/j.na.2003.07.011","article-title":"Solvability of nonlinear pseudoparabolic equation with a nonlocal boundary condition","volume":"55","author":"Bouziani","year":"2003","journal-title":"Nonlinear Anal. Theory Methods Appl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"585","DOI":"10.3934\/cpaa.2021190","article-title":"On a system of non-linear pseudoparabolic equations with Robin-Dirichlet boundary conditions","volume":"21","author":"Ngoc","year":"2022","journal-title":"Commun. Pure Appl. Anal."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"174","DOI":"10.1016\/j.camwa.2024.05.032","article-title":"Two-grid methods for nonlinear pseudo-parabolic integro-differential equations by finite element method","volume":"168","author":"Wang","year":"2024","journal-title":"Comput. Math. Appl."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"125182","DOI":"10.1016\/j.jmaa.2021.125182","article-title":"A high-order linearized difference scheme preserving dissipation property for the 2D Benjamin-Bona-Mahony-Burgers equation","volume":"500","author":"Cheng","year":"2021","journal-title":"J. Math. Anal. Appl."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Jhangeer, A., Ibraheem, F., Jamal, T., Rahimzai, A.A., and Khan, I. (2024). Investigating pseudo parabolic dynamics through phase portraits, sensitivity, chaos and soliton behavior. Sci. Rep., 14.","DOI":"10.1038\/s41598-024-64985-7"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1759","DOI":"10.1016\/j.cnsns.2009.08.003","article-title":"The first integral method for modified Benjamin-Bona-Mahony equation","volume":"15","author":"Abbasbandy","year":"2010","journal-title":"Commun. Nonlin. Sci. Numer. Simul."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1016\/0022-247X(89)90195-9","article-title":"On the decay of solutions of the generalized Benjamin-Bona-Mahony equation","volume":"141","author":"Albert","year":"1989","journal-title":"J. Math. Anal. Appl."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"116377","DOI":"10.1016\/j.cam.2024.116377","article-title":"Numerical simulation of the generalized modified Benjamin\u2013Bona\u2013Mahony equation using SBP-SAT in time","volume":"459","author":"Kjelldahl","year":"2025","journal-title":"J. Comput. Appl. Math."},{"key":"ref_32","first-page":"1818","article-title":"A solution by the finite difference method of a nonlinear regularized equation of shallow water","volume":"19","author":"Chernikov","year":"1983","journal-title":"Differ. Uravn."},{"key":"ref_33","first-page":"725","article-title":"Existence, uniqueness and analyticity of space-periodic solutions to the regularised long-wave equation","volume":"19","author":"Chertovskih","year":"2014","journal-title":"Adv. Differ. Equ."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1737","DOI":"10.1016\/j.na.2011.06.032","article-title":"On the blow-up of solutions of the Benjamin-Bona-Mahony-Burgers and Rosenau-Burgers equations","volume":"75","author":"Korpusov","year":"2012","journal-title":"Nonlinear Anal. Theory Methods Appl."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"127","DOI":"10.5269\/bspm.v35i1.28804","article-title":"Numerical study of the Benjamin-Bona-Mahony-Burgers equation","volume":"35","author":"Zarebnia","year":"2017","journal-title":"Bol. Soc. Paran. Mat."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"537","DOI":"10.55730\/1300-0098.3377","article-title":"Numerical solution for Benjamin-Bona-Mahony-Burgers equation with Strang time-splitting technique","volume":"47","author":"Karta","year":"2023","journal-title":"Turk. J. Math."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.apnum.2020.03.015","article-title":"Solution of Benjamin-Bona-Mahony-Burgers equation using collocation method with quintic Hermite splines","volume":"154","author":"Arora","year":"2020","journal-title":"Appl. Numer. Math."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1016\/j.joes.2021.07.001","article-title":"Kukreja. Numerical treatment of Benjamin-Bona-Mahony-Burgers equation with fourth-order improvised B-spline collocation method","volume":"7","author":"Shallu","year":"2022","journal-title":"J. Ocean Eng. Sci."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1002\/num.20256","article-title":"Finite difference discretization of the Benjamin-Bona-Mahony-Burgers equation","volume":"24","author":"Omrani","year":"2008","journal-title":"Numer. Methods Partial Differ. Equ."},{"key":"ref_40","first-page":"237","article-title":"Numerical solutions of Benjamin-Bona-Mahony-Burgers equation via nonstandard finite difference scheme","volume":"6","author":"Sweilam","year":"2018","journal-title":"Electron. J. Math. Anal. Appl."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1108\/EC-08-2024-0706","article-title":"An efficient spectral method for two-dimensional Benjamin\u2013Bona\u2013Mahony\u2013Burgers equation","volume":"42","author":"Jiao","year":"2025","journal-title":"Eng. Comput."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1790","DOI":"10.1002\/num.22504","article-title":"The numerical analysis of two linearized difference schemes for the Benjamin\u2013Bona\u2013Mahony\u2013Burgers equation","volume":"36","author":"Zhang","year":"2020","journal-title":"Numer. Methods Partial Differ. Equ."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"212","DOI":"10.1016\/j.camwa.2014.05.019","article-title":"The numerical solution of nonlinear high dimensional generalized Benjamin-Bona-Mahony-Burgers equation via the meshless method of radial basis functions","volume":"68","author":"Dehghan","year":"2014","journal-title":"Comput. Math. Appl."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1016\/j.cam.2015.03.012","article-title":"The use of interpolating element-free Galerkin technique for solving 2D generalized Benjamin-Bona-Mahony-Burgers and regularized long-wave equations on non-rectangular domains with error estimate","volume":"286","author":"Dehghan","year":"2015","journal-title":"J. Comput. Appl. Math."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"360","DOI":"10.1002\/num.22531","article-title":"Numerical and theoretical discussions for solving nonlinear generalized Benjamin\u2013Bona\u2013Mahony\u2013Burgers equation based on the Legendre spectral element method","volume":"37","author":"Dehghan","year":"2021","journal-title":"Numer. Methods Partial. Differ. Equ."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"3042","DOI":"10.1016\/j.camwa.2017.07.046","article-title":"A new algorithm based on Lucas polynomials for approximate solution of 1D and 2D nonlinear generalized Benjamin\u2013Bona\u2013Mahony\u2013Burgers equation","volume":"74","year":"2017","journal-title":"Comput. Math. Appl."},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Samarskii, A.A. (2001). The Theory of Difference Schemes, Marcel Dekker, Inc.. (In Russian).","DOI":"10.1201\/9780203908518"},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Hunsdorfer, W., and Verwer, J. (2003). Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Springer. [1st ed.].","DOI":"10.1007\/978-3-662-09017-6"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1002\/cpa.3160130205","article-title":"Systems of conservation laws","volume":"13","author":"Lax","year":"1960","journal-title":"Comp. Pure Appl. Math."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/7\/545\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:13:06Z","timestamp":1760033586000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/7\/545"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,21]]},"references-count":49,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2025,7]]}},"alternative-id":["axioms14070545"],"URL":"https:\/\/doi.org\/10.3390\/axioms14070545","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,7,21]]}}}