{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,28]],"date-time":"2025-11-28T17:25:39Z","timestamp":1764350739425,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,8,8]],"date-time":"2022-08-08T00:00:00Z","timestamp":1659916800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research at King Khalid University","award":["R.G.P2.\/41\/43"],"award-info":[{"award-number":["R.G.P2.\/41\/43"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This study examines approximate long wave and the modified Boussinesq equations, as well as their complexities with the Atangana\u2013Baleanu fractional derivative operator in the Caputo sense. The analytical solution of the aforementioned model is discussed using the Elzaki transform and the Adomian decomposition method. These problems are indispensable for defining the characteristics of surface water waves by applying a particular relationship of dispersion. We used Elzaki transformation on time-fractional approximate long wave and modified Boussinesq equations, followed by inverse Elzaki transformation, to achieve the results of the equations. To validate the methodology, we concentrated on two systems and compared them to the actual solutions. The numerical and graphical results demonstrate that the proposed method is computationally precise and straightforward for investigating and resolving fractionally coupled nonlinear phenomena that occur in scientific and technological.<\/jats:p>","DOI":"10.3390\/sym14081632","type":"journal-article","created":{"date-parts":[[2022,8,10]],"date-time":"2022-08-10T09:47:06Z","timestamp":1660124826000},"page":"1632","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Fractional Numerical Simulation of Coupled Approximate Long Wave and Modified Boussinesq Equations Involving Mittag-Leffler Kernel"],"prefix":"10.3390","volume":"14","author":[{"given":"Aisha Abdullah","family":"Alderremy","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, King Khalid University, Abha 61413, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,8]]},"reference":[{"key":"ref_1","unstructured":"Podlubny, I. (1999). Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Khan, H., Kumam, P., Baleanu, D., and Arif, M. (2019). An efficient analytical technique, for the solution of fractional-order telegraph equations. Mathematics, 7.","DOI":"10.3390\/math7050426"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1016\/S0377-0427(00)00292-2","article-title":"Multiple (multiindex) Mittag-Leffler functions and relations to generalized fractional calculus","volume":"118","author":"Kiryakova","year":"2000","journal-title":"J. Comput. Appl. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2941","DOI":"10.1016\/j.aej.2020.03.029","article-title":"The analytical investigation of time-fractional multi-dimensional Navier-Stokes equation","volume":"59","author":"Shah","year":"2020","journal-title":"Alex. Eng. J."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Khan, H., Farooq, U., Baleanu, D., Kumam, P., and Arif, M. (2019). Analytical solutions of (2+ time fractional order) dimensional physical models, using modified decomposition method. Appl. Sci., 10.","DOI":"10.3390\/app10010122"},{"key":"ref_6","first-page":"1083","article-title":"Decomposition method for solving fractional Riccati differential equations","volume":"182","author":"Momani","year":"2006","journal-title":"Appl. Math. Comput."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Shymanskyi, V., and Sokolovskyy, Y. (2020, January 23\u201326). Variational Formulation of the Stress-Strain Problem in Capillary-Porous Materials with Fractal Structure. Proceedings of the 2020 IEEE 15th International Conference on Computer Sciences and Information Technologies (CSIT), Zbarazh, Ukraine.","DOI":"10.1109\/CSIT49958.2020.9321996"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"114","DOI":"10.2174\/18750362021140100114","article-title":"Finite Element Calculation of the Linear Elasticity Problem for Biomaterials with Fractal Structure","volume":"14","author":"Shymanskyi","year":"2021","journal-title":"Open Bioinform. J."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1615\/JAutomatInfScien.v44.i1.50","article-title":"Numerical solution of Burgers equation by Petrov-Galerkin method with adaptive weighting functions","volume":"44","author":"Siryk","year":"2012","journal-title":"J. Autom. Inf. Sci."},{"key":"ref_10","unstructured":"Roos, H.-G., Stynes, M., and Tobiska, L. (2008). Robust Numerical Methods for Singularly Perturbed Differential Equations, Springer."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"6936","DOI":"10.3934\/math.2022385","article-title":"Analytical investigation of fractional-order Newell-Whitehead-Segel equations via a novel transform","volume":"7","author":"Areshi","year":"2022","journal-title":"AIMS Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"805","DOI":"10.1007\/s10559-014-9671-z","article-title":"Construction of Weight Functions of the Petrov-Galerkin Method for Convection-Diffusion-Reaction Equations in the Three-Dimensional Case","volume":"50","author":"Salnikov","year":"2014","journal-title":"Cybern. Syst. Anal."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1007\/s00791-018-0290-5","article-title":"Finite elements for scalar convection-dominated equations and incompressible flow problems: A never ending story?","volume":"19","author":"John","year":"2018","journal-title":"Comput. Vis. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1098\/rspa.1967.0119","article-title":"Variational methods and applications to water waves","volume":"299","author":"Whitham","year":"1967","journal-title":"Proc. R. Soc. Lond. Ser. A"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1007\/BF00418048","article-title":"Approximate equations for long water waves","volume":"31","author":"Broer","year":"1975","journal-title":"Appl. Sci. Res."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"396","DOI":"10.1143\/PTP.54.396","article-title":"A higher-order water-wave equation and the method for solving it","volume":"54","author":"Kaup","year":"1975","journal-title":"Prog. Theor. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1352","DOI":"10.1002\/mma.3151","article-title":"A novel method for travelling wave solutions of fractional Whitham-Broer-Kaup, fractional modified Boussinesq and fractional approximate long wave equations in shallow water","volume":"38","year":"2015","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Shah, N., Alyousef, H., El-Tantawy, S., and Chung, J. (2022). Analytical Investigation of Fractional-Order Korteweg\u2013De-Vries-Type Equations under Atangana\u2013Baleanu\u2013Caputo Operator: Modeling Nonlinear Waves in a Plasma and Fluid. Symmetry, 14.","DOI":"10.3390\/sym14040739"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1017\/S0022112095001170","article-title":"Modified Boussinesq equations and associated parabolic models for water wave propagation","volume":"288","author":"Chen","year":"1995","journal-title":"J. Fluid Mech."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1016\/j.joes.2017.07.001","article-title":"New exact solutions for the time fractional coupled Boussinesq-Burger equation and approximate long water wave equation in shallow water","volume":"2","author":"Khater","year":"2017","journal-title":"J. Ocean Eng. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18576\/sjm\/040201","article-title":"Elzaki decomposition method and its applications in solving linear and nonlinear Schrodinger equations","volume":"4","author":"Nuruddeen","year":"2017","journal-title":"Sohag J. Math."},{"key":"ref_22","first-page":"3484482","article-title":"Numerical analysis of fractional-order parabolic equations via Elzaki transform","volume":"2021","author":"Naeem","year":"2021","journal-title":"J. Funct. Spaces"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"3248376","DOI":"10.1155\/2021\/3248376","article-title":"Analytical investigation of Noyes-Field model for time-fractional Belousov-Zhabotinsky reaction","volume":"2021","author":"Alaoui","year":"2021","journal-title":"Complexity"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1002\/mma.5846","article-title":"Some analytical and numerical investigation of a family of fractional-order Helmholtz equations in two space dimensions","volume":"43","author":"Srivastava","year":"2020","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_25","first-page":"603","article-title":"Modification of Sumudu transform Elzaki transform and Adomian decomposition method","volume":"9","author":"Elzaki","year":"2015","journal-title":"Appl. Math. Sci."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Shah, R., Kumam, P., and Arif, M. (2019). An analytical technique to solve the system of nonlinear fractional partial differential equations. Mathematics, 7.","DOI":"10.3390\/math7060505"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"8876149","DOI":"10.1155\/2022\/8876149","article-title":"A comparative analysis of the fractional-order coupled Korteweg-De Vries equations with the Mittag-Leffler law","volume":"2022","author":"Aljahdaly","year":"2022","journal-title":"J. Math."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"763","DOI":"10.2298\/TSCI160111018A","article-title":"New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model","volume":"20","author":"Atangana","year":"2016","journal-title":"Therm. Sci."},{"key":"ref_29","first-page":"57","article-title":"The new integral transform Elzaki transform","volume":"7","author":"Elzaki","year":"2011","journal-title":"Glob. J. Pure Appl. Math."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"655","DOI":"10.1016\/j.rinp.2018.07.004","article-title":"New transform iterative method for solving some Klein-Gordon equations","volume":"10","author":"Alderremy","year":"2018","journal-title":"Results Phys."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"261","DOI":"10.12732\/ijpam.v87i2.6","article-title":"The time shifting theorem and the convolution for Elzaki transform","volume":"87","author":"Kim","year":"2013","journal-title":"Int. J. Pure Appl. Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/8\/1632\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:05:45Z","timestamp":1760141145000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/8\/1632"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,8]]},"references-count":31,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2022,8]]}},"alternative-id":["sym14081632"],"URL":"https:\/\/doi.org\/10.3390\/sym14081632","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,8,8]]}}}