{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,14]],"date-time":"2026-01-14T00:28:12Z","timestamp":1768350492757,"version":"3.49.0"},"reference-count":57,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2021,5,17]],"date-time":"2021-05-17T00:00:00Z","timestamp":1621209600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, the new iterative transform method and the homotopy perturbation transform method was used to solve fractional-order Equal-Width equations with the help of Caputo-Fabrizio. This method combines the Laplace transform with the new iterative transform method and the homotopy perturbation method. The approximate results are calculated in the series form with easily computable components. The fractional Equal-Width equations play an essential role in describe hydromagnetic waves in cold plasma. Our object is to study the nonlinear behaviour of the plasma system and highlight the critical points. The techniques are very reliable, effective, and efficient, which can solve a wide range of problems arising in engineering and sciences.<\/jats:p>","DOI":"10.3390\/sym13050886","type":"journal-article","created":{"date-parts":[[2021,5,17]],"date-time":"2021-05-17T12:19:57Z","timestamp":1621253997000},"page":"886","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":40,"title":["A New Analysis of Fractional-Order Equal-Width Equations via Novel Techniques"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5112-3711","authenticated-orcid":false,"given":"Muhammad","family":"Naeem","sequence":"first","affiliation":[{"name":"Deanship of Joint First Year, Umm Al-Qura University, Makkah P.O. Box 517, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3270-9124","authenticated-orcid":false,"given":"Ahmed M.","family":"Zidan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Khalid University, Abha 9004, Saudi Arabia"},{"name":"Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7469-5402","authenticated-orcid":false,"given":"Kamsing","family":"Nonlaopon","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3922-8892","authenticated-orcid":false,"given":"Muhammad I.","family":"Syam","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, United Arab Emirates University, Al Ain 15551, United Arab Emirates"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8543-5497","authenticated-orcid":false,"given":"Zeyad","family":"Al-Zhour","sequence":"additional","affiliation":[{"name":"Department of Basic Engineering Sciences, College of Engineering, Imam Abdulrahman Bin Faisal University, Dammam 31441, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4306-8489","authenticated-orcid":false,"given":"Rasool","family":"Shah","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,5,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Baleanu, D., Guvenc, Z.B., and Machado, J.T. (2010). New Trends in Nanotechnology and Fractional Calculus Applications, Springer.","DOI":"10.1007\/978-90-481-3293-5"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Baleanu, D., Machado, J.A., and Luo, A.C. (2011). Fractional Dynamics and Control, Springer Science & Business Media.","DOI":"10.1007\/978-1-4614-0457-6"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1016\/j.jsv.2018.06.022","article-title":"Active vibration suppression of a novel airfoil model with fractional order viscoelastic constitutive relationship","volume":"432","author":"Liu","year":"2018","journal-title":"J. Sound Vib."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2311","DOI":"10.1007\/s11071-015-2482-6","article-title":"A method to stochastic dynamical systems with strong nonlinearity and fractional damping","volume":"83","author":"Xu","year":"2016","journal-title":"Nonlinear Dyn."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"745","DOI":"10.1007\/s11071-013-1002-9","article-title":"Responses of Duffing oscillator with fractional damping and random phase","volume":"74","author":"Xu","year":"2013","journal-title":"Nonlinear Dyn."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Saeed, T., Ibrahim, A., and Marin, M. (2020). A GL model on thermo-elastic interaction in a poroelastic material using finite element method. Symmetry, 3.","DOI":"10.3390\/sym12030488"},{"key":"ref_7","first-page":"529","article-title":"Linear models of dissipation whose Q is almost frequency independent II","volume":"13","author":"Caputo","year":"1967","journal-title":"Geophysics"},{"key":"ref_8","first-page":"333","article-title":"The numerical solution of fractional differential equations: Speed versus accuracy","volume":"26","author":"Ford","year":"2001","journal-title":"Numerai"},{"key":"ref_9","unstructured":"Oldham, K.B., and Spanier, J. (1974). The Fractional Calculus, Academic Press."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00033-019-1095-1","article-title":"New realization method for calculating convective heat transfer near the hypersonic aircraft surface","volume":"70","author":"Ryzhkov","year":"2019","journal-title":"Z. Angew. Math. Phys."},{"key":"ref_11","first-page":"731","article-title":"A new definition of fractional derivative without singular kernel","volume":"2","author":"Caputo","year":"2015","journal-title":"Fract. Differ. Appl."},{"key":"ref_12","first-page":"87","article-title":"Properties of the new fractional derivative without singular kernel","volume":"2","author":"Losada","year":"2015","journal-title":"Fract. Differ. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Yang, X.J., Baleanu, D., and Srivastava, H.M. (2015). Local Fractional Integral Transforms and Their Applications, Academic Press.","DOI":"10.1016\/B978-0-12-804002-7.00004-8"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1016\/j.camwa.2009.08.028","article-title":"On the global existence of solutions to a class of fractional differential equations","volume":"59","author":"Baleanu","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1186\/s13662-019-2137-x","article-title":"Ternary-fractional differential transform schema: Theory and application","volume":"2019","author":"Yousef","year":"2019","journal-title":"Adv. Differ. Equ."},{"key":"ref_16","first-page":"101","article-title":"Application of Shehu transform to Atangana-Baleanu derivatives","volume":"20","author":"Bokhari","year":"2019","journal-title":"Int. J. Math. Comput. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2131","DOI":"10.2298\/TSCI1904131H","article-title":"Two-scale mathematics and fractional calculus for thermodynamics","volume":"21","author":"He","year":"2019","journal-title":"Therm. Sci."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2351","DOI":"10.2298\/TSCI1904351W","article-title":"A fractal derivative model for snow\u2019s thermal insulation property","volume":"23","author":"Wang","year":"2019","journal-title":"Therm. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1305","DOI":"10.1143\/JPSJ.26.1305","article-title":"Weak non-linear hydromagnetic waves in a cold collision-free plasma","volume":"26","author":"Kakutani","year":"1969","journal-title":"J. Phys. Soc. Japan."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"753","DOI":"10.2298\/TSCI151224222Y","article-title":"A new fractional derivative without singular kernel: Application to the modelling of the steady heat flow","volume":"20","author":"Yang","year":"2016","journal-title":"Therm. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1161","DOI":"10.2298\/TSCI161216326Y","article-title":"Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problems","volume":"21","author":"Yang","year":"2017","journal-title":"Therm. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1515\/nleng-2013-0022","article-title":"A new fractional model of nonlinear shock wave equation arising in flow of gases","volume":"3","author":"Singh","year":"2014","journal-title":"Nonlinear Eng."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"276","DOI":"10.1016\/j.physa.2017.04.054","article-title":"A new fractional operator of variable order: Application in the description of anomalous diffusion","volume":"481","author":"Yang","year":"2017","journal-title":"Physica"},{"key":"ref_24","first-page":"165","article-title":"Electrostatic field in terms of geometric curvature in membrane MEMS devices","volume":"8","author":"Paolo","year":"2017","journal-title":"Comm. Appl. Ind. Math."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1016\/j.conbuildmat.2018.04.094","article-title":"Shear property, high-temperature rheological performance and low-temperature flexibility of asphalt mastics modified with bio-oil","volume":"174","author":"Yong","year":"2018","journal-title":"Constr. Build. Mater."},{"key":"ref_26","first-page":"467","article-title":"Generalized differential transform method: Application to differential equations of fractional order","volume":"197","author":"Odibat","year":"2008","journal-title":"Appl. Math. Comput."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1473","DOI":"10.1016\/j.chaos.2006.09.004","article-title":"Solution of fractional differential equations by using differential transform method","volume":"34","author":"Arikoglu","year":"2007","journal-title":"Chaos Solitons Fractals"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"5545","DOI":"10.1016\/j.apm.2014.04.018","article-title":"Homotopy perturbation method for two dimensional time-fractional wave equation","volume":"38","author":"Zhang","year":"2014","journal-title":"Appl. Math. Model."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1515\/nleng-2016-0008","article-title":"Analytical method for space-fractional telegraph equation by homotopy perturbation transform method","volume":"5","author":"Prakash","year":"2016","journal-title":"Nonlinear Eng."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"684","DOI":"10.2478\/s13540-012-0046-8","article-title":"Solution of fractional partial differential equations using iterative method","volume":"15","author":"Dhaigude","year":"2012","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"2091","DOI":"10.1016\/j.camwa.2009.03.043","article-title":"Application of He\u2019s variational iteration method and Adomian\u2019s decomposition method to the fractional KdV-Burgers-Kuramoto equation","volume":"58","author":"Safari","year":"2009","journal-title":"Comput. Math. Appl."},{"key":"ref_32","unstructured":"Liao, S.J. (1992). The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems. [Ph.D. Thesis, Shanghai Jiao Tong University]."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/S1007-5704(97)90047-2","article-title":"Homotopy analysis method: A new analytical technique for nonlinear problems","volume":"2","author":"Liao","year":"1997","journal-title":"Comm. Nonlinear Sci. Numer. Simulat."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1016\/j.apnum.2005.02.008","article-title":"Finite difference approximations for two-sided space-fractional partial differential equations","volume":"56","author":"Meerschaert","year":"2006","journal-title":"Appl. Numer. Math."},{"key":"ref_35","first-page":"398","article-title":"Analytical solution of the Bagley Torvik equation by Adomian decomposition method","volume":"168","author":"Ray","year":"2005","journal-title":"Appl. Math. Comput."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"3285","DOI":"10.1016\/j.cam.2011.01.011","article-title":"High-order finite element methods for time-fractional partial differential equations","volume":"235","author":"Jiang","year":"2011","journal-title":"J. Comput. Appl. Math."},{"key":"ref_37","first-page":"499","article-title":"On the homotopy analysis method for nonlinear problems","volume":"147","author":"Liao","year":"2004","journal-title":"Appl. Math. Comput."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"93","DOI":"10.2989\/16073606.2013.780336","article-title":"On convergence of homotopy analysis method and its application to fractional integro-differential equations","volume":"36","author":"Abbasbandy","year":"2013","journal-title":"Quaest. Math."},{"key":"ref_39","first-page":"103","article-title":"A fractional model of convective radial fins with temperature-dependent thermal conductivity","volume":"69","author":"Kumar","year":"2017","journal-title":"Rom. Rep. Phys."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1016\/j.cam.2017.03.011","article-title":"A modified numerical scheme and convergence analysis for fractional model of Lienards equation","volume":"339","author":"Kumar","year":"2018","journal-title":"J. Comput. Appl. Math."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1250","DOI":"10.1016\/j.physleta.2007.09.039","article-title":"Analysis of a time fractional wave-like equation with the homotopy analysis method","volume":"372","author":"Hang","year":"2008","journal-title":"Phys. Lett. A"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"935","DOI":"10.1515\/zna-2010-1106","article-title":"The solution of the linear fractional partial differential equations using the homotopy analysis method","volume":"65","author":"Dehghan","year":"2010","journal-title":"Z. Naturforsch. A"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"839","DOI":"10.1515\/math-2015-0078","article-title":"Duplication in a model of rock fracture with fractional derivative without singular kernel","volume":"13","author":"Goufo","year":"2015","journal-title":"Open Math."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1016\/j.jksus.2010.06.023","article-title":"Solving a multi-order fractional differential equation using homotopy analysis method","volume":"23","author":"Jafari","year":"2011","journal-title":"J. King Saud Univ. Sci."},{"key":"ref_45","first-page":"621","article-title":"Multi-order fractional differential equations and their numerical solution","volume":"154","author":"Diethelm","year":"2004","journal-title":"Appl. Math. Comput."},{"key":"ref_46","first-page":"73","article-title":"Homotopy perturbation method: A new nonlinear analytical technique","volume":"135","author":"He","year":"2003","journal-title":"Appl. Math. Comput."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1007\/s42452-018-0016-9","article-title":"Solving time-fractional Navier\u2013Stokes equations using homotopy perturbation Elzaki transform","volume":"1","author":"Jena","year":"2018","journal-title":"SN Appl. Sci."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"1","DOI":"10.9734\/BJMCS\/2016\/29922","article-title":"A comparative study for solving nonlinear fractional heat-like equations via Elzaki transform","volume":"19","author":"Mahgoub","year":"2016","journal-title":"Br. J. Math. Comp. Sci."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1515\/zna-2010-0305","article-title":"An approximate analytical solution of the fractional diffusion equation with absorbent term and external force by homotopy perturbation method","volume":"65","author":"Das","year":"2010","journal-title":"Z. Naturforsch."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1515\/nleng-2018-0136","article-title":"Comparative study of homotopy perturbation transformation with homotopy perturbation Elzaki transform method for solving nonlinear fractional PDE","volume":"9","author":"Singh","year":"2019","journal-title":"Nonlinear Eng."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"753","DOI":"10.1016\/j.jmaa.2005.05.009","article-title":"An iterative method for solving nonlinear functional equations","volume":"316","author":"Jafari","year":"2006","journal-title":"J. Math. Anal. Appl."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"1799","DOI":"10.1016\/j.aml.2011.04.037","article-title":"Application of the Laplace decomposition method for solving linear and nonlinear fractional diffusion-wave equations","volume":"24","author":"Jafari","year":"2011","journal-title":"Appl. Math. Lett."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"838","DOI":"10.1016\/j.camwa.2012.11.014","article-title":"A new approach for solving a system of fractional partial differential equations","volume":"66","author":"Jafari","year":"2013","journal-title":"Comput. Math. Appl."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"465160","DOI":"10.1155\/2013\/465160","article-title":"Numerical solutions of fractional Fokker-Planck equations using iterative Laplace transform method","volume":"2013","author":"Yan","year":"2013","journal-title":"Abstr. Appl. Anal."},{"key":"ref_55","first-page":"30","article-title":"A new iterative technique for a fractional model of nonlinear Zakharov-Kuznetsov equations via Sumudu transform","volume":"334","author":"Prakash","year":"2018","journal-title":"Appl. Math. Comput."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"1213","DOI":"10.9734\/BJMCS\/2014\/8534","article-title":"New iterative method for solving the Fornberg-Whitham equation and comparison with homotopy perturbation transform method","volume":"4","author":"Ramadan","year":"2014","journal-title":"Br. J. Math. Comp. Sci."},{"key":"ref_57","first-page":"1","article-title":"Laplace homotopy analysis method for solving linear partial differential equations using a fractional derivative with and without kernel singular","volume":"2016","author":"Baleanu","year":"2016","journal-title":"Adv. Differ. Equ."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/5\/886\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:02:43Z","timestamp":1760162563000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/5\/886"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,5,17]]},"references-count":57,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2021,5]]}},"alternative-id":["sym13050886"],"URL":"https:\/\/doi.org\/10.3390\/sym13050886","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,5,17]]}}}