{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:57:39Z","timestamp":1760147859565,"version":"build-2065373602"},"reference-count":54,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2023,3,8]],"date-time":"2023-03-08T00:00:00Z","timestamp":1678233600000},"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 present study is concerned with studying the dynamical behavior of two space-dimensional nonlinear time-fractional models governing the unsteady-flow of polytropic-gas (in brief, pGas) that occurred in cosmology and astronomy. For this purpose, two efficient hybrid methods so-called optimal homotopy analysis J-transform method (OHAJTM) and J-variational iteration transform method (J-VITM) have been adopted. The OHAJTM is the hybrid method, where optimal-homotopy analysis method (OHAM) is utilized after implementing the properties of J-transform (JT), and in J-VITM is the J-transform-based variational iteration method. Banach\u2019s fixed point approach is adopted to analyze the convergence of these methods. It is demonstrated that J-VITM is T-stable, and the evaluated dynamics of pGas are described in terms of Mittag\u2013Leffler functions. The proposed evaluation confirms that the implemented methods perform better for the referred model equation of pGas. In addition, for a given iteration, the proposed behavior via OHAJTM performs better in producing more accurate behavior in comparison to J-VITM and the methods introduced recently.<\/jats:p>","DOI":"10.3390\/axioms12030285","type":"journal-article","created":{"date-parts":[[2023,3,9]],"date-time":"2023-03-09T02:01:47Z","timestamp":1678327307000},"page":"285","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Study of Time-Fractional Nonlinear Model Governing Unsteady Flow of Polytropic Gas"],"prefix":"10.3390","volume":"12","author":[{"given":"Brajesh","family":"Singh","sequence":"first","affiliation":[{"name":"School of Physical and Decision Sciences, Department of Mathematics, Babasaheb Bhimrao Ambedkar University Lucknow, Lucknow 226025, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4085-3625","authenticated-orcid":false,"given":"Haci","family":"Baskonus","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Science Education, Faculty of Education, Harran University, Sanliurfa 63100, Turkey"}]},{"given":"Neetu","family":"Singh","sequence":"additional","affiliation":[{"name":"School of Physical and Decision Sciences, Department of Mathematics, Babasaheb Bhimrao Ambedkar University Lucknow, Lucknow 226025, India"},{"name":"Department of Applied Sciences and Humanities, Kendriya Vidyalaya NIT, Banda 210001, India"}]},{"given":"Mukesh","family":"Gupta","sequence":"additional","affiliation":[{"name":"School of Physical and Decision Sciences, Department of Mathematics, Babasaheb Bhimrao Ambedkar University Lucknow, Lucknow 226025, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6453-0308","authenticated-orcid":false,"given":"D.","family":"Prakasha","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Davangere University, Davangere 577007, India"}]}],"member":"1968","published-online":{"date-parts":[[2023,3,8]]},"reference":[{"key":"ref_1","unstructured":"Miller, K.S., and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons."},{"key":"ref_2","unstructured":"Podlubny, I. (1999). Fractional Differential Equations, Academic Press."},{"key":"ref_3","unstructured":"Srivastava, H.M., Trujillo, J.J., and Kilbas, A.A. (2006). Theory and Applications of Fractional Differential Equations, Elsevier."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1016\/j.chaos.2018.07.022","article-title":"Blind in a commutative world: Simple illustrations with functions and chaotic attractors","volume":"114","author":"Atangana","year":"2018","journal-title":"Chaos Solitons Fractals"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"396","DOI":"10.1016\/j.chaos.2017.04.027","article-title":"Fractle-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system","volume":"102","author":"Atangana","year":"2017","journal-title":"Chaos Solitons Fractals"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1006\/jipa.1999.4853","article-title":"Effects of the protozoan parasite ophryocystis elektroscirrha on the fitness of monarch butterflies (danaus plexippus)","volume":"74","author":"Altizer","year":"1999","journal-title":"J. Invertebr. Pathol."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"396","DOI":"10.1016\/j.chaos.2020.110256","article-title":"Fractional order mathematical modelling of covid-19 transmission","volume":"139","author":"Ahmad","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_8","first-page":"1101","article-title":"Numerical simulation of time-fractional black-scholes equation using fractional variational iteration method","volume":"9","author":"Gupta","year":"2019","journal-title":"J. Comput. Math. Sci."},{"key":"ref_9","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_10","doi-asserted-by":"crossref","first-page":"616","DOI":"10.1016\/j.cnsns.2009.04.029","article-title":"Dynamical models of happiness with fractional order","volume":"15","author":"Song","year":"2010","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1650014","DOI":"10.1142\/S0218271816500140","article-title":"Thermodynamic behavior and stability of polytropic gas","volume":"12","author":"Moradpour","year":"2016","journal-title":"Int. J. Mod. Phys. D"},{"key":"ref_12","unstructured":"Liao, S.J. (2003). Beyond Perturbation: Introduction to Homotopy Analysis Method, Chapman and Hall\/CRC."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2003","DOI":"10.1016\/j.cnsns.2009.09.002","article-title":"An optimal homotopy-analysis approach for strongly nonlinear differential equations","volume":"15","author":"Liao","year":"2010","journal-title":"Commun Nonlinear Sci Numer Simulat."},{"key":"ref_14","unstructured":"Liao, S. (1992). The Proposed Homotopy Analysis Techniques for the Solution of Nonlinear Problems. [Ph.D. Thesis, Shanghai Jiao Tong University]."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1007\/s12043-019-1829-9","article-title":"An efficient technique for a fractional-order system of equations describing the unsteady flow of a polytropic gas","volume":"93","author":"Prakasha","year":"2019","journal-title":"Pramana J. Phys."},{"key":"ref_16","unstructured":"Dalsgard, J.C. (2004). Lecture Notes on Stellar Structure and Evolution, Aarhus University Press."},{"key":"ref_17","first-page":"61","article-title":"Application of he\u2019s variational iteration method for solving the equation governing the unsteady flow of a polytropic gas","volume":"3","author":"Matinfar","year":"2009","journal-title":"J. Math. Ext."},{"key":"ref_18","first-page":"980","article-title":"Homotopy analysis method for solving the equation governing the unsteady flow of a polytropic gas","volume":"9","author":"Matinfar","year":"2010","journal-title":"World Appl. Sci. J."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1088\/0951-7715\/17\/1\/018","article-title":"Dynamics of a strongly nonlocal reaction-diffusion population model","volume":"17","author":"Billingham","year":"2003","journal-title":"Nonlinearity"},{"key":"ref_20","first-page":"753","article-title":"Fractional natural decomposition method for solving fractional system of nonlinear equations of unsteady flow of a polytropic gas","volume":"25","author":"Cherif","year":"2018","journal-title":"Nonlinear Std."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Adel, W., and Srinivasa, K. (2022). A new clique polynomial approach for fractional differential equations. Int. J. Nonlinear Sci. Numer. Simul.","DOI":"10.1515\/ijnsns-2021-0258"},{"key":"ref_22","first-page":"86","article-title":"Time-fractional partial differential equations: A novel technique for analytical and numerical solutions","volume":"29","author":"Yadav","year":"2022","journal-title":"Arab. J. Basic Appl. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Shah, R., Khan, H., Kumam, P., and Arif, M. (2019). An analytical system to solve the system of nonlinear fractional differential equations. Mathematics, 7.","DOI":"10.3390\/math7060505"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"140511","DOI":"10.1098\/rsos.140511","article-title":"Approximate series solution of multi-dimensional, time fractional-order (heat-like) diffusion equations using frdtm","volume":"2","author":"Srivastava","year":"2015","journal-title":"R. Soc. Open Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"290","DOI":"10.1016\/j.cjph.2019.09.005","article-title":"Exact solutions of nonlinear fractional order partial differential equations via singular manifold method","volume":"61","author":"Saleh","year":"2019","journal-title":"Chin. J. Phys."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"100396","DOI":"10.1016\/j.padiff.2022.100396","article-title":"Comments on whether nonlinear fractional partila differential equations have soliton solutions","volume":"5","author":"Weiguo","year":"2022","journal-title":"Partial. Differ. Equ. Appl. Math."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Ali, H.M., Ahmad, H., Askar, S., and Ameen, I.G. (2022). Efficient apporaches for solving system of nonlinear time-fractional partial differential equations. Fractal Fract., 6.","DOI":"10.3390\/fractalfract6010032"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Shakeel, M., Shah, N.A., and Chung, J.D. (2022). Novel analytical technique to find closed form solutions of time fractional partial differential equations. Fractal Fract., 6.","DOI":"10.3390\/fractalfract6010024"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1823","DOI":"10.1007\/s11071-022-07424-4","article-title":"The peridynamic differential opertaor for solving time-fractional partial differential equations","volume":"109","author":"Hosseini","year":"2022","journal-title":"Nonlinear Dyn."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Malagi, N.S., Prakasha, D.G., Veeresha, P., and Prasannakumara, B.C. (2022). Fractional Reaction-Diffusion Model: An Efficient Computational Technique for Nonlinear Time-Fractional Schnackenberg Model, Springer.","DOI":"10.1007\/978-981-19-0179-9_26"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"106620","DOI":"10.1016\/j.cnsns.2022.106620","article-title":"An efficient hybrid numerical method for multi-term time fractional partial differential equations in fluid mechanics with convergence and error analysis","volume":"114","author":"Joujehi","year":"2022","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Alesemi, M., Shahrani, J.S.A., Iqbal, N., Shah, R., and Nonlapon, K. (2023). Analysis and numerical simulation of system of fractional partial differential equations with non-singular kernel operators. Symmetry, 15.","DOI":"10.3390\/sym15010233"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"021001","DOI":"10.1115\/1.4056254","article-title":"Effective optimized decomposition algorithms for solving nonlinear fractional differential equations","volume":"18","author":"Laoubi","year":"2023","journal-title":"J. Comput. Nonlinear Dyn."},{"key":"ref_34","first-page":"927","article-title":"Exact solutions and finite time stability of linear conformable fractional systems with pure delay","volume":"134","author":"Elshenhab","year":"2023","journal-title":"Comput. Model. Eng. Sci."},{"key":"ref_35","first-page":"259","article-title":"A fractional order fast repetitive control paradigm ofvienna rectifier for power quality improvement","volume":"135","author":"Wang","year":"2023","journal-title":"Cmes-Comput. Model. Eng. Sci."},{"key":"ref_36","first-page":"1159","article-title":"The fractional investigation of fornberg-whitham equation using an efficient technique","volume":"134","author":"Khan","year":"2023","journal-title":"Cmes-Comput. Model. Eng. Sci."},{"key":"ref_37","first-page":"1013","article-title":"Regarding on the fractional mathematical model of tumor invasion and metastasis","volume":"127","author":"Veeresha","year":"2021","journal-title":"Cmes-Comput. Model. Eng. Sci."},{"key":"ref_38","first-page":"52","article-title":"Adomian decomposition method for solving the equation governing the unsteady flow of a polytropic gas","volume":"4","author":"Mohamed","year":"2009","journal-title":"Appl. Appl. Math."},{"key":"ref_39","unstructured":"Kreyszig, E. (1978). Introductory Functional Analysis with Applications, John Wiley and Sons."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"418971","DOI":"10.1155\/2008\/418971","article-title":"T-Stability of Picard iteration in metric spaces","volume":"2008","author":"Qing","year":"2008","journal-title":"Fixed Point Theory Appl."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"723","DOI":"10.1002\/mma.5375","article-title":"Stability analysis and a numerical scheme for fractional Klein-Gordon equations","volume":"42","author":"Khan","year":"2019","journal-title":"Math. Meth. Appl. Sci."},{"key":"ref_42","first-page":"1223","article-title":"Beyond sumudu transform and natural transform: j-transform properties and applications","volume":"10","author":"Maitama","year":"2020","journal-title":"J. Appl. Anal. Comput."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/S0045-7825(98)00108-X","article-title":"Approximate analytical solution for seepage flow with fractional derivatives in porous media","volume":"167","author":"He","year":"1998","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_44","first-page":"57","article-title":"Variational iteration method\u2014A kind of non-linear analytical technique: Some examples","volume":"167","author":"He","year":"1999","journal-title":"Int. J. -Non-Linear Mech."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1016\/j.cam.2006.07.009","article-title":"Variational iteration method-Some recent results and new interpretations","volume":"207","author":"He","year":"2007","journal-title":"J. Comput. Appl. Math."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"1181","DOI":"10.1016\/j.mcm.2009.12.034","article-title":"A study on the convergence of variational iteration method","volume":"51","author":"Odibat","year":"2010","journal-title":"Math. Comput. Model."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"955","DOI":"10.1016\/j.camwa.2006.12.040","article-title":"Modified variational iteration method for Boussinesq equation","volume":"54","author":"Abassy","year":"2007","journal-title":"Comput. Math. Appl."},{"key":"ref_48","first-page":"1","article-title":"Fractional variational iteration method for solving fractional partial differential equations with proportional delay","volume":"88","author":"Singh","year":"2017","journal-title":"Int. J. Differ. Equ."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"2199","DOI":"10.1016\/j.camwa.2009.03.009","article-title":"The variational iteration method: An efficient scheme for handling fractional partial differential equation in fluid mechanics","volume":"58","author":"Momani","year":"2009","journal-title":"Comput. Math. Appl."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"996","DOI":"10.1002\/num.20567","article-title":"A new method for calculating general Lagrange multiplier in the variational iteration method","volume":"27","author":"Jafari","year":"2011","journal-title":"Numer. Methods Partial. Differ. Equ."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"1944","DOI":"10.22436\/jnsa.009.04.48","article-title":"Solutions of fractional differential equations by sumudu transform and variational iteration method","volume":"9","author":"Goswami","year":"2016","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_52","unstructured":"Finlayson, B.A. (1972). The Method of Weighted Residuals and Variational Principles, Academic Press."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"2298","DOI":"10.1016\/j.aml.2012.06.020","article-title":"A laplace variational iteration strategy for the solution of differential equations","volume":"25","author":"Khuri","year":"2012","journal-title":"Appl. Math. Lett."},{"key":"ref_54","first-page":"2052","article-title":"Modified laplace variational iteration method for solving fourth-order parabolic partial differential equation with variable coefficients","volume":"78","author":"Li","year":"2020","journal-title":"Comput. Math. Appl."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/3\/285\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:51:22Z","timestamp":1760122282000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/3\/285"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,8]]},"references-count":54,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2023,3]]}},"alternative-id":["axioms12030285"],"URL":"https:\/\/doi.org\/10.3390\/axioms12030285","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2023,3,8]]}}}