{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,17]],"date-time":"2026-03-17T23:35:32Z","timestamp":1773790532111,"version":"3.50.1"},"reference-count":37,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2024,1,30]],"date-time":"2024-01-30T00:00:00Z","timestamp":1706572800000},"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>In this paper, we apply two different methods, namely, the G\u2032G-expansion method and the G\u2032G2-expansion method to investigate the nonlinear time fractional Harry Dym equation in the Caputo sense and the symmetric regularized long wave equation in the conformable sense. The mentioned nonlinear partial differential equations (NPDEs) arise in diverse physical applications such as ion sound waves in plasma and waves on shallow water surfaces. There exist multiple wave solutions to many NPDEs and researchers are interested in analytical approaches to obtain these multiple wave solutions. The multi-exp-function method (MEFM) formulates a solution algorithm for calculating multiple wave solutions to NPDEs and at the end of paper, we apply the MEFM for calculating multiple wave solutions to the (2 + 1)-dimensional equation.<\/jats:p>","DOI":"10.3390\/axioms13020092","type":"journal-article","created":{"date-parts":[[2024,1,30]],"date-time":"2024-01-30T12:06:58Z","timestamp":1706616418000},"page":"92","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Soliton Solution of the Nonlinear Time Fractional Equations: Comprehensive Methods to Solve Physical Models"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4096-1469","authenticated-orcid":false,"given":"Donal","family":"O\u2019Regan","sequence":"first","affiliation":[{"name":"School of Mathematical and Statistical Sciences, University of Galway, University Road, H91 TK33 Galway, Ireland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3108-6524","authenticated-orcid":false,"given":"Safoura Rezaei","family":"Aderyani","sequence":"additional","affiliation":[{"name":"School of Mathematics, Iran University of Science and Technology, Narmak, Tehran 13114-16846, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6770-6951","authenticated-orcid":false,"given":"Reza","family":"Saadati","sequence":"additional","affiliation":[{"name":"School of Mathematics, Iran University of Science and Technology, Narmak, Tehran 13114-16846, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4996-8373","authenticated-orcid":false,"given":"Mustafa","family":"Inc","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Science Faculty, Firat University, Elazig 23119, Turkey"},{"name":"Department of Medical Research, China Medical University, Taichung 40402, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,1,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"4318192","DOI":"10.1155\/2022\/4318192","article-title":"The exact solutions of the conformable time-fractional modified nonlinear Schr\u00f6dinger equation by the Trial equation method and modified Trial equation method","volume":"2022","author":"Aderyani","year":"2022","journal-title":"Adv. Math. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"218","DOI":"10.1007\/s11082-022-03605-y","article-title":"The exact solutions of conformable time-fractional modified nonlinear Schr\u00f6dinger equation by first integral method and functional variable method","volume":"54","author":"Aderyani","year":"2022","journal-title":"Opt. Quantum Electron."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Aderyani, S.R., Saadati, R., O\u2019Regan, D., and Alshammari, F.S. (2022). Describing Water Wave Propagation Using the G\u2032G\u2013Expansion Method. Mathematics, 11.","DOI":"10.3390\/math11010191"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1007\/s11082-022-03527-9","article-title":"New optical solitons based on the perturbed Chen-Lee-Liu model through Jacobi elliptic function method","volume":"54","author":"Tarla","year":"2022","journal-title":"Opt. Quantum Electron."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Yasmin, H., Aljahdaly, N.H., Saeed, A.M., and Shah, R. (2023). Probing families of optical soliton solutions in fractional perturbed Radhakrishnan\u2013Kundu\u2013Lakshmanan model with improved versions of extended direct algebraic method. Fractal Fract., 7.","DOI":"10.3390\/fractalfract7070512"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"10807","DOI":"10.3934\/math.2022604","article-title":"The exact solutions of conformable time-fractional modified nonlinear Schr\u00f6dinger equation by Direct algebraic method and Sine-Gordon expansion method","volume":"7","author":"Aderyani","year":"2022","journal-title":"AIMS Math."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Yasmin, H., Aljahdaly, N.H., Saeed, A.M., and Shah, R. (2023). Investigating Families of Soliton Solutions for the Complex Structured Coupled Fractional Biswas\u2013Arshed Model in Birefringent Fibers Using a Novel Analytical Technique. Fractal Fract., 7.","DOI":"10.3390\/fractalfract7070491"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Yasmin, H., Aljahdaly, N.H., Saeed, A.M., and Shah, R. (2023). Investigating Symmetric Soliton Solutions for the Fractional Coupled Konno\u2013Onno System Using Improved Versions of a Novel Analytical Technique. Mathematics, 11.","DOI":"10.3390\/math11122686"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Hirota, R. (2004). The Direct Method in Soliton Theory, Cambridge University Press. No. 155.","DOI":"10.1017\/CBO9780511543043"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1007\/s10013-015-0157-8","article-title":"Soliton solution of good Boussinesq equation","volume":"44","author":"Nguyen","year":"2016","journal-title":"Vietnam J. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"4245","DOI":"10.1016\/j.na.2008.09.010","article-title":"A second Wronskian formulation of the Boussinesq equation","volume":"70","author":"Ma","year":"2009","journal-title":"Nonlinear Anal. Theory Methods Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"449","DOI":"10.1007\/s10013-015-0145-z","article-title":"Wronskian formulation and Ansatz method for bad Boussinesq equation","volume":"44","author":"Nguyen","year":"2016","journal-title":"Vietnam J. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1016\/j.joes.2021.08.013","article-title":"On the modified (G\u2032G2)-expansion method for finding some analytical solutions of the traveling waves","volume":"7","author":"Behera","year":"2022","journal-title":"J. Ocean. Eng. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"4360735","DOI":"10.1155\/2022\/4360735","article-title":"Analytical Approach for the Approximate Solution of Harry Dym Equation with Caputo Fractional Derivative","volume":"2022","author":"Nadeem","year":"2022","journal-title":"Math. Probl. Eng."},{"key":"ref_15","first-page":"165","article-title":"Homotopy perturbation Sumudu transform method for nonlinear equations","volume":"4","author":"Singh","year":"2011","journal-title":"Adv. Theor. Appl. Mech."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1546","DOI":"10.4236\/am.2017.811113","article-title":"A mathematical approach based on the homotopy analysis method: Application to solve the nonlinear Harry-Dym (HD) equation","volume":"8","author":"Ghiasi","year":"2017","journal-title":"Appl. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1088\/0253-6102\/55\/2\/03","article-title":"Exact Solutions of the Harry-Dym Equation","volume":"55","author":"Mokhtari","year":"2011","journal-title":"Commun. Theor. Phys."},{"key":"ref_18","first-page":"2579","article-title":"A Solution of the Harry-Dym Equation Using Lattice-Boltzmannn and a Solitary Wave Methods","volume":"11","author":"Fonseca","year":"2017","journal-title":"Appl. Math. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"553","DOI":"10.12732\/ijpam.v95i4.8","article-title":"A new approach to solve the fractional Harry Dym equation using the FRDTM","volume":"95","author":"Rawashdeh","year":"2014","journal-title":"Int. J. Pure Appl. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"409","DOI":"10.18576\/amis\/100204","article-title":"An analytical approach to time-fractional Harry Dym equation","volume":"10","author":"Iyiola","year":"2016","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"481","DOI":"10.47372\/uajnas.2020.n2.a15","article-title":"Exact solutions of the Harry Dym Equation using Lie group method","volume":"24","author":"Assabaai","year":"2020","journal-title":"Univ. Aden J. Nat. Appl. Sci."},{"key":"ref_22","first-page":"561","article-title":"An Efficient Approach for Fractional Harry Dym Equation by Using Homotopy Analysis Method","volume":"5","author":"Shunmugarajan","year":"2016","journal-title":"Int. J. Eng. Res. Technol."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1007\/s40096-015-0137-x","article-title":"An approximate solution for a fractional model of generalized Harry Dym equation","volume":"8","author":"Alquran","year":"2014","journal-title":"Math. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"100031","DOI":"10.1016\/j.padiff.2021.100031","article-title":"Distinct solutions of nonlinear space-time fractional evolution equations appearing in mathematical physics via a new technique","volume":"3","author":"Islam","year":"2021","journal-title":"Partial Differ. Equ. Appl. Math."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Pavlidou, E., Papadopoulou, S.K., Seroglou, K., and Giaginis, C. (2023). Revised harris\u2013benedict equation: New human resting metabolic rate equation. Metabolites, 13.","DOI":"10.3390\/metabo13020189"},{"key":"ref_26","first-page":"a09","article-title":"Body length estimation of Neogene macrophagous lamniform sharks (Carcharodon and Otodus) derived from associated fossil dentitions","volume":"24","author":"Perez","year":"2021","journal-title":"Palaeontol. Electron."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"134046","DOI":"10.1016\/j.physd.2023.134046","article-title":"On asymptotic stability of multi-solitons for the focusing modified Korteweg\u2013de Vries equation","volume":"459","author":"Liu","year":"2024","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"106072","DOI":"10.1016\/j.cnsns.2021.106072","article-title":"A generalized (1 + 2)-dimensional Bogoyavlenskii-Kadomtsev-Petviashvili (BKP) equation: Multiple exp-function algorithm; conservation laws; similarity solutions","volume":"106","author":"Moretlo","year":"2022","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Polyanin, A.D., and Zaitsev, V.F. (2016). Handbook of Nonlinear Partial Differential Equations, Chapman and Hall\/CRC.","DOI":"10.1201\/b11412"},{"key":"ref_30","first-page":"11871","article-title":"Solving the (3 + 1)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm","volume":"218","author":"Ma","year":"2012","journal-title":"Appl. Math. Comput."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"070201","DOI":"10.1088\/1674-1056\/26\/7\/070201","article-title":"Multiple exp-function method for soliton solutions of nonlinear evolution equations","volume":"26","author":"Yildirim","year":"2017","journal-title":"Chin. Phys. B"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1248","DOI":"10.1016\/j.camwa.2016.02.005","article-title":"The generalized (1 + 1)-dimensional and (2 + 1)-dimensional Ito equations: Multiple exp-function algorithm and multiple wave solutions","volume":"71","author":"Adem","year":"2016","journal-title":"Comput. Math. Appl."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/j.aml.2018.01.010","article-title":"Multiple soliton solutions for the new (2 + 1)-dimensional Korteweg\u2013de Vries equation by multiple exp-function method","volume":"80","author":"Liu","year":"2018","journal-title":"Appl. Math. Lett."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"775","DOI":"10.1515\/zna-2015-0151","article-title":"The multiple exp-function method and the linear superposition principle for solving the (2 + 1)-dimensional Calogero\u2013Bogoyavlenskii\u2013Schiff equation","volume":"70","author":"Zayed","year":"2015","journal-title":"Z. Naturforsch. A"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"418793","DOI":"10.1155\/2014\/418793","article-title":"Multiple soliton solutions for a new generalization of the associated camassa-holm equation by exp-function method","volume":"2014","author":"Long","year":"2014","journal-title":"Math. Probl. Eng."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"2350213","DOI":"10.1142\/S0217979223502132","article-title":"Multiple exp-function method to solve the nonlinear space\u2013time fractional partial differential symmetric regularized long wave (SRLW) equation and the (1 + 1)-dimensional Benjamin\u2013Ono equation","volume":"37","author":"Aderyani","year":"2022","journal-title":"Int. J. Mod. Phys. B"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"148","DOI":"10.1016\/j.chaos.2015.01.017","article-title":"Modified homogeneous balance method: Applications and new solutions","volume":"73","author":"Nguyen","year":"2015","journal-title":"Chaos Solitons Fractals"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/2\/92\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T13:51:39Z","timestamp":1760104299000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/2\/92"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,1,30]]},"references-count":37,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2024,2]]}},"alternative-id":["axioms13020092"],"URL":"https:\/\/doi.org\/10.3390\/axioms13020092","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,1,30]]}}}