{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,22]],"date-time":"2025-12-22T18:22:25Z","timestamp":1766427745684,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2022,11,24]],"date-time":"2022-11-24T00:00:00Z","timestamp":1669248000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The main goal of this paper is to introduce a new scheme, known as the Aboodh homotopy integral transform method (AHITM), for the approximate solution of wave problems in multi-dimensional orders. The Aboodh integral transform (AIT) removes the restriction of variables in the recurrence relation, whereas the homotopy perturbation method (HPM) derives the successive iterations using the initial conditions. The convergence analysis is provided to study a wave equation with multiple dimensions. Some computational applications are considered to show the efficiency of this scheme. Graphical representation between the approximate and the exact solution predicts the high rate of convergence of this approach.<\/jats:p>","DOI":"10.3390\/axioms11120665","type":"journal-article","created":{"date-parts":[[2022,11,24]],"date-time":"2022-11-24T03:24:09Z","timestamp":1669260249000},"page":"665","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["A Novel Approach for the Approximate Solution of Wave Problems in Multi-Dimensional Orders with Computational Applications"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9349-4729","authenticated-orcid":false,"given":"Muhammad","family":"Nadeem","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Qujing Normal University, Qujing 655011, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9832-1424","authenticated-orcid":false,"given":"Ali","family":"Akg\u00fcl","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Mathematics, Lebanese American University, Beirut 1102 2801, Lebanon"},{"name":"Art and Science Faculty, Department of Mathematics, Siirt University, 56100 Siirt, Turkey"},{"name":"Department of Mathematics, Mathematics Research Center, Near East University, Near East Boulevard, Mersin 10, 99138 Nicosia, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8304-1574","authenticated-orcid":false,"given":"Liliana","family":"Guran","sequence":"additional","affiliation":[{"name":"Department of Pharmaceutical Sciences, \u201cVasile Goldis\u201d Western University of Arad, L. 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Mech."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2651","DOI":"10.1016\/j.aej.2021.07.028","article-title":"A numerical method for solving a class of systems of nonlinear Pantograph differential equations","volume":"61","author":"Cakmak","year":"2022","journal-title":"Alex. Eng. J."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1016\/j.physleta.2006.02.048","article-title":"Analytical approach to linear fractional partial differential equations arising in fluid mechanics","volume":"355","author":"Momani","year":"2006","journal-title":"Phys. Lett. A"},{"key":"ref_4","unstructured":"Dehghan, M., Manafian, J., and Saadatmandi, A. (2010). The solution of the linear fractional partial differential equations using the homotopy analysis method. Z. Naturforschung-A, 65."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"404","DOI":"10.1016\/j.chaos.2017.06.029","article-title":"The modified extended tanh method with the Riccati equation for solving the space-time fractional EW and MEW equations","volume":"103","author":"Raslan","year":"2017","journal-title":"Chaos Solitons Fractals"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Rezazadeh, H., Ullah, N., Akinyemi, L., Shah, A., Mirhosseini-Alizamin, S.M., Chu, Y.M., and Ahmad, H. (2021). Optical soliton solutions of the generalized non-autonomous nonlinear Schr\u00f6dinger equations by the new Kudryashov\u2019s method. Results Phys., 24.","DOI":"10.1016\/j.rinp.2021.104179"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"944","DOI":"10.1108\/HFF-04-2021-0245","article-title":"Numerical simulation of Chun-Hui He\u2019s iteration method with applications in engineering","volume":"32","author":"Khan","year":"2021","journal-title":"Int. J. Numer. Methods Heat Fluid Flow"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1007\/s12648-013-0407-0","article-title":"Exact solutions of nonlinear partial fractional differential equations using fractional sub-equation method","volume":"88","author":"Gepreel","year":"2014","journal-title":"Indian J. Phys."},{"key":"ref_9","first-page":"102","article-title":"Exp-function Method and Reduction Transformations for Rogue Wave Solutions of the Davey-Stewartson Equations","volume":"7","author":"Zhang","year":"2021","journal-title":"J. Appl. Comput. Mech."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Althobaiti, A., Althobaiti, S., El-Rashidy, K., and Seadawy, A.R. (2021). Exact solutions for the nonlinear extended KdV equation in a stratified shear flow using modified exponential rational method. Results Phys., 29.","DOI":"10.1016\/j.rinp.2021.104723"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"537","DOI":"10.29020\/nybg.ejpam.v11i2.3194","article-title":"Modifications of the multistep optimal homotopy asymptotic method to some nonlinear KdV-equations","volume":"11","author":"Fiza","year":"2018","journal-title":"Eur. J. Pure Appl. Math."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Nuruddeen, R.I., Aboodh, K.S., and Ali, K.K. (2018). Analytical investigation of soliton solutions to three quantum Zakharov-Kuznetsov equations. Commun. Theor. Phys., 70.","DOI":"10.1088\/0253-6102\/70\/4\/405"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"589","DOI":"10.1108\/HFF-02-2021-0136","article-title":"New variational theory for coupled nonlinear fractal Schr\u00f6dinger system","volume":"32","author":"Wang","year":"2021","journal-title":"Int. J. Numer. Methods Heat Fluid Flow"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Alaroud, M., Al-Smadi, M., Rozita Ahmad, R., and Salma Din, U.K. (2019). An analytical numerical method for solving fuzzy fractional Volterra integro-differential equations. Symmetry, 11.","DOI":"10.3390\/sym11020205"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1080\/00207160.2015.1100299","article-title":"Higher order numeric solutions of the Lane\u2013Emden-type equations derived from the multi-stage modified Adomian decomposition method","volume":"94","author":"Duan","year":"2017","journal-title":"Int. J. Comput. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2633","DOI":"10.1016\/j.nonrwa.2008.07.002","article-title":"Convergence of the homotopy perturbation method for partial differential equations","volume":"10","author":"Biazar","year":"2009","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1515\/zna-2009-3-402","article-title":"Homotopy perturbation method for solving partial differential equations","volume":"64","author":"Noor","year":"2009","journal-title":"Z. Naturforschung"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"He, J.H., El-Dib, Y.O., and Mady, A.A. (2021). Homotopy perturbation method for the fractal toda oscillator. Fractal Fract., 5.","DOI":"10.3390\/fractalfract5030093"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Jornet, M. (2021). Exact solution to a multidimensional wave equation with delay. Appl. Math. Comput., 409.","DOI":"10.1016\/j.amc.2021.126421"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1016\/j.matcom.2020.10.017","article-title":"Exact solutions of the generalized multidimensional mathematical physics models via sub-equation method","volume":"182","author":"Akinyemi","year":"2021","journal-title":"Math. Comput. Simul."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"926","DOI":"10.1016\/j.camwa.2006.12.038","article-title":"The variational iteration method: A reliable analytic tool for solving linear and nonlinear wave equations","volume":"54","author":"Wazwaz","year":"2007","journal-title":"Comput. Math. Appl."},{"key":"ref_22","first-page":"341","article-title":"Numerical solution of two-dimensional nonlinear differential equation by homotopy perturbation method","volume":"189","author":"Ghasemi","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"ref_23","first-page":"113","article-title":"Reduced differential transform method for solving linear and nonlinear wave equations","volume":"34","author":"Keskin","year":"2010","journal-title":"Iran. J. Sci. Technol. Trans.-Sci."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Ullah, H., Islam, S., Dennis, L., Abdelhameed, T., Khan, I., and Fiza, M. (2015). Approximate solution of two-dimensional nonlinear wave equation by optimal homotopy asymptotic method. Math. Probl. Eng., 2015.","DOI":"10.1155\/2015\/380104"},{"key":"ref_25","first-page":"166","article-title":"Analytic and numerical solutions for linear and nonlinear multidimensional wave equations","volume":"27","author":"Adwan","year":"2020","journal-title":"Arab. J. Basic Appl. Sci."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2859","DOI":"10.1016\/j.aej.2019.12.022","article-title":"Analytical approach for time fractional wave equations in the sense of Yang-Abdel-Aty-Cattani via the homotopy perturbation transform method","volume":"59","author":"Jleli","year":"2020","journal-title":"Alex. Eng. J."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1002\/nme.1620180103","article-title":"Dispersion analysis of finite element semidiscretizations of the two-dimensional wave equation","volume":"18","author":"Mullen","year":"1982","journal-title":"Int. J. Numer. Methods Eng."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Ojo, G.O., and Mahmudov, N.I. (2021). Aboodh transform iterative method for spatial diffusion of a biological population with fractional-order. Mathematics, 9.","DOI":"10.3390\/math9020155"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"520","DOI":"10.32622\/ijrat.712019107","article-title":"A comparative study of Mohand and Aboodh transforms","volume":"7","author":"Aggarwal","year":"2019","journal-title":"Int. J. Res. Advent Technol."},{"key":"ref_30","first-page":"317","article-title":"Solution of Abel\u2019s integral equation by Aboodh transform method","volume":"6","author":"Aggarwal","year":"2019","journal-title":"J. Emerg. Technol. Innov. Res."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/12\/665\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:25:45Z","timestamp":1760145945000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/12\/665"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,24]]},"references-count":30,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["axioms11120665"],"URL":"https:\/\/doi.org\/10.3390\/axioms11120665","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2022,11,24]]}}}