{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T22:21:04Z","timestamp":1771539664001,"version":"3.50.1"},"reference-count":32,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2023,7,6]],"date-time":"2023-07-06T00:00:00Z","timestamp":1688601600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"University of Oradea"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In nature, symmetry is all around us. The symmetry framework represents integer partial differential equations and their fractional order in the sense of Caputo derivatives. This article suggests a semi-analytical approach based on Aboodh transform (AT) and the homotopy perturbation scheme (HPS) for achieving the approximate solution of time-fractional porous media and heat transfer equations. The AT converts the fractional problems into simple ones and obtains the recurrence relation without any discretization or assumption. This nonlinear recurrence relation can be decomposed via the use of the HPS to obtain the iterations in terms of series solutions. The initial conditions play an important role in determining the successive iterations and yields towards the exact solution. We provide some numerical applications to analyze the accuracy of this proposed scheme and show that the performance of our scheme has strong agreement with the exact results.<\/jats:p>","DOI":"10.3390\/sym15071374","type":"journal-article","created":{"date-parts":[[2023,7,7]],"date-time":"2023-07-07T01:41:36Z","timestamp":1688694096000},"page":"1374","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Numerical Analysis of Time-Fractional Porous Media and Heat Transfer Equations Using a Semi-Analytical Approach"],"prefix":"10.3390","volume":"15","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"}]},{"given":"Asad","family":"Islam","sequence":"additional","affiliation":[{"name":"Department of Mechanical and Aerospace Engineering, Air University, Islamabad 44000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shazia","family":"Karim","sequence":"additional","affiliation":[{"name":"Department of Basic Sciences, UET Lahore, FSD Campus, Faisalabad 54800, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sorin","family":"Mure\u015fan","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, University of Oradea, 1 University Street, 410087 Oradea, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8845-3095","authenticated-orcid":false,"given":"Loredana Florentina","family":"Iambor","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, University of Oradea, 1 University Street, 410087 Oradea, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,7,6]]},"reference":[{"key":"ref_1","unstructured":"Arrigo, D.J. (2015). Symmetry Analysis of Differential Equations: An Introduction, John Wiley & Sons."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Hussain, A., Bano, S., Khan, I., Baleanu, D., and Sooppy Nisar, K. (2020). Lie symmetry analysis, explicit solutions and conservation laws of a spatially two-dimensional Burgers\u2013Huxley equation. Symmetry, 12.","DOI":"10.3390\/sym12010170"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2096","DOI":"10.1016\/j.camwa.2018.12.009","article-title":"Solitary wave solutions of (3+1)-dimensional extended Zakharov\u2013Kuznetsov equation by Lie symmetry approach","volume":"77","author":"Kumar","year":"2019","journal-title":"Comput. Math. Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"110507","DOI":"10.1016\/j.chaos.2020.110507","article-title":"Lie symmetry analysis for obtaining the abundant exact solutions, optimal system and dynamics of solitons for a higher-dimensional Fokas equation","volume":"142","author":"Kumar","year":"2021","journal-title":"Chaos Solitons Fractals"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1016\/j.matcom.2018.08.008","article-title":"Analysis of (3+1)-dimensional unsteady gas flow using optimal system of Lie symmetries","volume":"156","author":"Rashed","year":"2019","journal-title":"Math. Comput. Simul."},{"key":"ref_6","first-page":"1639","article-title":"Variational Principles and solitary wave solutions of Generalized Nonliner Schrodinger Equation in the Ocean","volume":"7","author":"Liu","year":"2021","journal-title":"J. Appl. Comput. Mech."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"118516","DOI":"10.1016\/j.ces.2023.118516","article-title":"Numerical investigation of the condensation oscillation mechanism of submerged steam jet with high mass flux","volume":"270","author":"Song","year":"2023","journal-title":"Chem. Eng. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"023131","DOI":"10.1063\/5.0138864","article-title":"Resonance and bifurcation of fractional quintic Mathieu\u2013Duffing system","volume":"33","author":"Zhang","year":"2023","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1016\/S0375-9601(01)00522-9","article-title":"Exact solutions for nonlinear partial differential equation: A new approach","volume":"288","author":"Bai","year":"2001","journal-title":"Phys. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2715","DOI":"10.1016\/j.aej.2021.01.009","article-title":"The Schr\u00f6dinger-KdV equation of fractional order with Mittag-Leffler nonsingular kernel","volume":"60","author":"Yavuz","year":"2021","journal-title":"Alex. Eng. J."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"025001","DOI":"10.1088\/1572-9494\/ac4411","article-title":"The Sharma\u2013Tasso\u2013Olver\u2013Burgers equation: Its conservation laws and kink solitons","volume":"74","author":"Hosseini","year":"2022","journal-title":"Commun. Theor. Phys."},{"key":"ref_12","first-page":"483","article-title":"Solitary Wave Solution for Fractional-Order General Equal-Width Equation via Semi Analytical Technique","volume":"17","author":"Ali","year":"2023","journal-title":"Appl. Math."},{"key":"ref_13","first-page":"295","article-title":"An analytical method for the solution of two phase Stefan problem in cylindrical geometry","volume":"342","author":"Khalid","year":"2019","journal-title":"Appl. Math. Comput."},{"key":"ref_14","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."},{"key":"ref_15","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_16","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1080\/16583655.2018.1469283","article-title":"Exact solitary wave solution for the fractional and classical GEW-Burgers equations: An application of Kudryashov method","volume":"12","author":"Nuruddeen","year":"2018","journal-title":"J. Taibah Univ. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s13662-022-03709-5","article-title":"Numerical investigation of fractional-order Kersten\u2013Krasil\u2019shchik coupled KdV\u2013mKdV system with Atangana\u2013Baleanu derivative","volume":"2022","author":"Iqbal","year":"2022","journal-title":"Adv. Contin. Discret. Model."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"110200","DOI":"10.1016\/j.chaos.2020.110200","article-title":"Analysis of fractional fishery model with reserve area in the context of time-fractional order derivative","volume":"140","author":"Mansal","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Fang, J., Nadeem, M., Habib, M., Karim, S., and Wahash, H.A. (2022). A New Iterative Method for the Approximate Solution of Klein-Gordon and Sine-Gordon Equations. J. Funct. Spaces, 2022.","DOI":"10.1155\/2022\/5365810"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"3347","DOI":"10.1016\/j.aej.2020.05.003","article-title":"Evaluation of one dimensional fuzzy fractional partial differential equations","volume":"59","author":"Shah","year":"2020","journal-title":"Alex. Eng. J."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1062","DOI":"10.1108\/HFF-08-2022-0499","article-title":"A new strategy for the approximate solution of fourth-order parabolic partial differential equations with fractional derivative","volume":"33","author":"Nadeem","year":"2022","journal-title":"Int. J. Numer. Methods Heat Fluid Flow"},{"key":"ref_22","first-page":"877","article-title":"On the solutions of nonlinear fractional Klein\u2013Gordon equation with modified Riemann\u2013Liouville derivative","volume":"242","author":"Merdan","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1045","DOI":"10.1007\/s10973-019-08007-z","article-title":"Optimal arrangements of a heat sink partially filled with multilayered porous media employing hybrid nanofluid","volume":"137","author":"Arasteh","year":"2019","journal-title":"J. Therm. Anal. Calorim."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"104285","DOI":"10.1016\/j.icheatmasstransfer.2019.104285","article-title":"Two phase natural convection and thermal radiation of Non-Newtonian nanofluid in a porous cavity considering inclined cavity and size of inside cylinders","volume":"108","author":"Barnoon","year":"2019","journal-title":"Int. Commun. Heat Mass Transf."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"129266","DOI":"10.1016\/j.jhydrol.2023.129266","article-title":"Visualization of biochar colloids transport and retention in two-dimensional porous media","volume":"619","author":"Zhao","year":"2023","journal-title":"J. Hydrol."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"102569","DOI":"10.1016\/j.csite.2022.102569","article-title":"Heat and mass transfer analysis of non-Newtonian power-law nanofluid confined within annulus enclosure using Darcy-Brinkman-Forchheimer model","volume":"40","author":"Abderrahmane","year":"2022","journal-title":"Case Stud. Therm. Eng."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"719","DOI":"10.1016\/j.molliq.2019.01.128","article-title":"Heat transfer of magnetic nanoparticles through porous media including exergy analysis","volume":"279","author":"Sheikholeslami","year":"2019","journal-title":"J. Mol. Liq."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1141","DOI":"10.1615\/JPorMedia.2019025699","article-title":"Modeling and numerical simulation of hydromagnetic natural convection Casson fluid flow with nth-order chemical reaction and Newtonian heating in porous medium","volume":"22","author":"Seth","year":"2019","journal-title":"J. Porous Media"},{"key":"ref_29","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."},{"key":"ref_30","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_31","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1016\/j.cam.2006.07.030","article-title":"Application of homotopy-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations","volume":"207","author":"Ganji","year":"2007","journal-title":"J. Comput. Appl. Math."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1409","DOI":"10.2298\/TSCI1305409Y","article-title":"Modified homotopy perturbation method coupled with Laplace transform for fractional heat transfer and porous media equations","volume":"17","author":"Yan","year":"2013","journal-title":"Therm. Sci."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/7\/1374\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:07:23Z","timestamp":1760126843000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/7\/1374"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,6]]},"references-count":32,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2023,7]]}},"alternative-id":["sym15071374"],"URL":"https:\/\/doi.org\/10.3390\/sym15071374","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,7,6]]}}}