{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,21]],"date-time":"2026-02-21T01:21:01Z","timestamp":1771636861335,"version":"3.50.1"},"reference-count":47,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2025,10,16]],"date-time":"2025-10-16T00:00:00Z","timestamp":1760572800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,10,16]],"date-time":"2025-10-16T00:00:00Z","timestamp":1760572800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100004728","name":"VIT University","doi-asserted-by":"publisher","award":["SG20230081"],"award-info":[{"award-number":["SG20230081"]}],"id":[{"id":"10.13039\/501100004728","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Adv Comput Math"],"published-print":{"date-parts":[[2025,12]]},"DOI":"10.1007\/s10444-025-10260-w","type":"journal-article","created":{"date-parts":[[2025,10,16]],"date-time":"2025-10-16T08:03:41Z","timestamp":1760601821000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A conforming virtual element method for Emden-Fowler model over polygonal meshes"],"prefix":"10.1007","volume":"51","author":[{"given":"Zaffar Mehdi","family":"Dar","sequence":"first","affiliation":[]},{"given":"M.","family":"Arrutselvi","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2228-0969","authenticated-orcid":false,"given":"Chandru","family":"Muthusamy","sequence":"additional","affiliation":[]},{"given":"Sundararajan","family":"Natarajan","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,10,16]]},"reference":[{"key":"10260_CR1","doi-asserted-by":"publisher","first-page":"784","DOI":"10.1016\/j.camwa.2017.05.026","volume":"74","author":"G Acosta","year":"2017","unstructured":"Acosta, G., Bersetche, F., Borthagaray, J.: A short FE implementation for a 2D homogeneous Dirichlet problem of a fractional Laplacian. Comput. Math. Appl. 74, 784\u2013816 (2017). https:\/\/doi.org\/10.1016\/j.camwa.2017.05.026","journal-title":"Comput. Math. Appl."},{"key":"10260_CR2","doi-asserted-by":"publisher","unstructured":"Acosta, G., Borthagaray, J.: A fractional Laplace equation: regularity of solutions and finite element approximations. SIAM J. Numerical Anal. 55 (2015https:\/\/doi.org\/10.1137\/15M1033952","DOI":"10.1137\/15M1033952"},{"key":"10260_CR3","doi-asserted-by":"publisher","unstructured":"Ahmad, B., Alsaedi, A., Brezzi, F., Marini, L.D., Russo, A.: Equivalent projectors for virtual element methods. Comput. Math. Appl. 66, 376\u2013391 (2013https:\/\/doi.org\/10.1016\/j.camwa.2013.05.015","DOI":"10.1016\/j.camwa.2013.05.015"},{"key":"10260_CR4","doi-asserted-by":"crossref","unstructured":"Antonietti, P.F., L. Beirao Da Veiga, Mora, D., Verani, M.: A stream virtual element formulation of the Stokes problem on polygonal meshes. SIAM J. Numer. Anal. 52(1), 386\u2013404 (2014)","DOI":"10.1137\/13091141X"},{"key":"10260_CR5","doi-asserted-by":"crossref","unstructured":"Antonietti, P.F., L. Beirao Da Veiga, Scacchi, S., Verani, M.: A $${C}^1$$ virtual element method for the Cahn-Hilliard equation with polygonal meshes. SIAM J. Numer. Anal. 54(1), 34\u201356 (2016)","DOI":"10.1137\/15M1008117"},{"issue":"9","key":"10260_CR6","doi-asserted-by":"publisher","first-page":"1852","DOI":"10.1080\/00207160.2020.1849637","volume":"98","author":"M Arrutselvi","year":"2021","unstructured":"Arrutselvi, M., Natarajan, E.: Virtual element method for nonlinear convection-diffusion-reaction equation on polygonal meshes. Int. J. Comput. Math. 98(9), 1852\u20131876 (2021)","journal-title":"Int. J. Comput. Math."},{"key":"10260_CR7","doi-asserted-by":"publisher","first-page":"813","DOI":"10.1090\/S0002-9947-2014-05887-X","volume":"367","author":"B Baeumer","year":"2014","unstructured":"Baeumer, B., Kov\u2019acs, M., Sankaranarayanan, H.: Higher order Gr\u00fcnwald approximations of fractional derivatives and fractional powers of operators. Trans. Am. Math. Soc. 367, 813\u2013834 (2014)","journal-title":"Trans. Am. Math. Soc."},{"key":"10260_CR8","doi-asserted-by":"publisher","unstructured":"Bandrowski, B., Karczewska, A., Rozmej, P.: Numerical solutions to integral equations equivalent to differential equations with fractional time. Int. J. Appl. Math. Comput. Sci. 20, 261\u2013269 (2010https:\/\/doi.org\/10.2478\/v10006-010-0019-1","DOI":"10.2478\/v10006-010-0019-1"},{"key":"10260_CR9","doi-asserted-by":"publisher","first-page":"18","DOI":"10.1016\/j.cma.2016.07.043","volume":"311","author":"MF Benedetto","year":"2016","unstructured":"Benedetto, M.F., Berrone, S., Borio, A., Pieraccini, S., Scialo, S.: Order preserving SUPG stabilization for the virtual element formulation of advection-diffusion problems. Comput. Methods Appl. Mech. Eng. 311, 18\u201340 (2016)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"10260_CR10","doi-asserted-by":"publisher","first-page":"423","DOI":"10.1016\/j.apnum.2018.09.003","volume":"135","author":"E Caceres","year":"2019","unstructured":"Caceres, E., Gatica, G.N., Sequeira, F.A.: A mixed virtual element method for a pseudostress-based formulation of linear elasticity. Appl. Numer. Math. 135, 423\u2013442 (2019)","journal-title":"Appl. Numer. Math."},{"issue":"3","key":"10260_CR11","doi-asserted-by":"publisher","first-page":"1317","DOI":"10.1093\/imanum\/drw036","volume":"37","author":"A Cangiani","year":"2016","unstructured":"Cangiani, A., Manzini, G., Sutton, O.J.: Conforming and nonconforming virtual element methods for elliptic problems. IMA J. Numer. Anal. 37(3), 1317\u20131354 (2016). https:\/\/doi.org\/10.1093\/imanum\/drw036","journal-title":"IMA J. Numer. Anal."},{"issue":"3","key":"10260_CR12","first-page":"1317","volume":"37","author":"A Cangiani","year":"2017","unstructured":"Cangiani, A., Manzini, G., Sutton, O.J.: Conforming and nonconforming virtual element methods for elliptic problems. IMA J. Numer. Anal. 37(3), 1317\u20131354 (2017)","journal-title":"IMA J. Numer. Anal."},{"issue":"1","key":"10260_CR13","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1214\/11-AIHP447","volume":"49","author":"P Cattiaux","year":"2013","unstructured":"Cattiaux, P., Guillin, A., Zitt, P.A.: Poincar\u00e9 inequalities and hitting times. Annales de l\u2019Institut Henri Poincar\u00e9 - Probabilit\u00e9s et Statistiques 49(1), 95\u2013118 (2013). https:\/\/doi.org\/10.1214\/11-AIHP447","journal-title":"Annales de l\u2019Institut Henri Poincar\u00e9 - Probabilit\u00e9s et Statistiques"},{"key":"10260_CR14","doi-asserted-by":"publisher","unstructured":"Chowdhury, M., Hashim, I.: Solutions of Emden-Fowler equations by homotopy-perturbation method. Nonlinear Anal.: Real World Appl. 10(1), 104\u2013115 (2009https:\/\/doi.org\/10.1016\/j.nonrwa.2007.08.017","DOI":"10.1016\/j.nonrwa.2007.08.017"},{"key":"10260_CR15","doi-asserted-by":"crossref","unstructured":"Dar, Z.M., Arrutselvi, M., Chandru, M., Manzini, G., Natarajan, S.: Analytical and numerical methods for solving fractional-order partial differential equations and the virtual element method. Pre-print, Elsevier (2024). https:\/\/ssrn.com\/abstract=4913214","DOI":"10.2139\/ssrn.4913214"},{"key":"10260_CR16","doi-asserted-by":"publisher","unstructured":"Dar, Z.M., Chandru, M.: A virtual element scheme for the time-fractional parabolic PDEs over distorted polygonal meshes. Alexandria Eng. J. 106, 611\u2013619 (2024https:\/\/doi.org\/10.1016\/j.aej.2024.08.050","DOI":"10.1016\/j.aej.2024.08.050"},{"key":"10260_CR17","doi-asserted-by":"publisher","unstructured":"Dar, Z.M., Muthusamy, C., Ramos, H.: A fractional PDE-based model for nerve impulse transport solved using a conforming virtual element method: Application to prosthetic implants. Axioms 14(6) (2025https:\/\/doi.org\/10.3390\/axioms14060398","DOI":"10.3390\/axioms14060398"},{"key":"10260_CR18","doi-asserted-by":"publisher","unstructured":"Dar, Z.M., Muthusamy, C., Ramos, H.: A virtual element method for a (2+1)-dimensional wave equation with time-fractional dissipation on polygonal meshes. Fractal and Fractional 9(7) (2025https:\/\/doi.org\/10.3390\/fractalfract9070399","DOI":"10.3390\/fractalfract9070399"},{"key":"10260_CR19","doi-asserted-by":"crossref","unstructured":"Diethelm, K.: The Analysis of Fractional Differential Equations. Springer (2010)","DOI":"10.1007\/978-3-642-14574-2"},{"key":"10260_CR20","doi-asserted-by":"publisher","unstructured":"Ervin, V., Roop, J.: Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial Differential Equations 22, 558\u2013576 (2006https:\/\/doi.org\/10.1002\/num.20112","DOI":"10.1002\/num.20112"},{"key":"10260_CR21","doi-asserted-by":"publisher","unstructured":"Esen, A., Ucar, Y., Yagmurlu, N., Tasbozan, O.: A Galerkin finite element method to solve fractional diffusion and fractional diffusion-wave equations. Mathematical Modelling and Analysis 18, 260\u2013273 (2013https:\/\/doi.org\/10.3846\/13926292.2013.783884","DOI":"10.3846\/13926292.2013.783884"},{"key":"10260_CR22","volume-title":"Mittag-Leffler Functions","author":"R Gorenflo","year":"2016","unstructured":"Gorenflo, R., Kilbas, A.A., Mainardi, F., Rogosin, S.V.: Mittag-Leffler Functions. Related Topics and Applications. Springer, Berlin Heidelberg (2016)"},{"key":"10260_CR23","doi-asserted-by":"publisher","unstructured":"Jin, B., Lazarov, R., Zhou, Z.: A Petrov-Galerkin finite element method for fractional convection-diffusion equations. SIAM J. Numer. Anal. 54 (2015https:\/\/doi.org\/10.1137\/140992278","DOI":"10.1137\/140992278"},{"issue":"1","key":"10260_CR24","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1137\/16M1089320","volume":"56","author":"B Jin","year":"2018","unstructured":"Jin, B., Li, B., Zhou, Z.: Numerical analysis of nonlinear subdiffusion equations. SIAM J. Numer. Anal. 56(1), 1\u201323 (2018). https:\/\/doi.org\/10.1137\/16M1089320","journal-title":"SIAM J. Numer. Anal."},{"issue":"6","key":"10260_CR25","doi-asserted-by":"publisher","first-page":"2056","DOI":"10.1002\/num.22399","volume":"35","author":"D Kumar","year":"2019","unstructured":"Kumar, D., Chaudhary, S., Srinivas Kumar, V.: Fractional Crank-Nicolson-Galerkin finite element scheme for the time-fractional nonlinear diffusion equation. Numer. Methods Partial Differential Equations 35(6), 2056\u20132075 (2019). https:\/\/doi.org\/10.1002\/num.22399","journal-title":"Numer. Methods Partial Differential Equations"},{"key":"10260_CR26","doi-asserted-by":"publisher","unstructured":"Lane, H.J.: On the theoretical temperature of the sun, under the hypothesis of a gaseous mass maintaining its volume by its internal heat, and depending on the laws of gases as known to terrestrial experiment. Am. J. Sci. s2-50(148), 57\u201374 (1870https:\/\/doi.org\/10.2475\/ajs.s2-50.148.57","DOI":"10.2475\/ajs.s2-50.148.57"},{"key":"10260_CR27","doi-asserted-by":"publisher","unstructured":"Li, D., Liao, H.l., Sun, W., Wang, J., Zhang, J.: Analysis of L1-Galerkin FEMs for time-fractional nonlinear parabolic problems. Commun. Computat. Phys. 24, 86\u2013103 (2018https:\/\/doi.org\/10.4208\/cicp.OA-2017-0080","DOI":"10.4208\/cicp.OA-2017-0080"},{"key":"10260_CR28","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/drab030","author":"M Li","year":"2022","unstructured":"Li, M., Zhao, J., Huang, C., Shaochun, C.: Conforming and nonconforming VEMs for the fourth-order reaction-subdiffusion equation: a unified framework. IMA J. Numer. Anal. (2022). https:\/\/doi.org\/10.1093\/imanum\/drab030","journal-title":"IMA J. Numer. Anal."},{"key":"10260_CR29","doi-asserted-by":"publisher","DOI":"10.4208\/jcm.2209-m2021-0038","author":"M Li","year":"2023","unstructured":"Li, M., Zhao, J., Wang, Z., Shaochun, C.: Conservative conforming and nonconforming VEMs for fourth order nonlinear Schr\u00f6dinger equations with trapped term. J. Computat. Math. (2023). https:\/\/doi.org\/10.4208\/jcm.2209-m2021-0038","journal-title":"J. Computat. Math."},{"key":"10260_CR30","doi-asserted-by":"publisher","unstructured":"Mall, S., Chakraverty, S.: Numerical solution of nonlinear singular initial value problems of Emden-Fowler type using Chebyshev neural network method. Neurocomputing 149, 975\u2013982 (2015https:\/\/doi.org\/10.1016\/j.neucom.2014.07.036","DOI":"10.1016\/j.neucom.2014.07.036"},{"key":"10260_CR31","doi-asserted-by":"publisher","unstructured":"Mall, S., Chakraverty, S.: A novel Chebyshev neural network approach for solving singular arbitrary order Lane-Emden equation arising in astrophysics. Network: Computat. Neural Syst. 31(1-4), 142\u2013165 (2020https:\/\/doi.org\/10.1080\/0954898X.2020.1807636. PMID: 33148086","DOI":"10.1080\/0954898X.2020.1807636"},{"issue":"4","key":"10260_CR32","doi-asserted-by":"publisher","first-page":"1258","DOI":"10.1002\/num.22257","volume":"34","author":"L Mascotto","year":"2018","unstructured":"Mascotto, L.: Ill-conditioning in the virtual element method: stabilizations and bases. Numer. Methods Partial Differerential Equations 34(4), 1258\u20131281 (2018)","journal-title":"Numer. Methods Partial Differerential Equations"},{"key":"10260_CR33","doi-asserted-by":"publisher","unstructured":"Oldham, K.B., Spanier, J.E.: Chapter 3: fractional derivatives and integrals: definitions and equivalences. In: The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Mathematics in Science and Engineering, vol. 111, pp. 45\u201360. Academic Press (1974https:\/\/doi.org\/10.1016\/S0076-5392(09)60225-3","DOI":"10.1016\/S0076-5392(09)60225-3"},{"key":"10260_CR34","unstructured":"Podlubny, I. (ed.): Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of their Applications, vol. 198. Elsevier (1999)"},{"key":"10260_CR35","doi-asserted-by":"publisher","first-page":"4038","DOI":"10.1016\/j.jcp.2009.02.011","volume":"228","author":"E Sousa","year":"2009","unstructured":"Sousa, E.: Finite difference approximations for a fractional advection diffusion problem. J. Computat. Phys. 228, 4038\u20134054 (2009)","journal-title":"J. Computat. Phys."},{"issue":"2","key":"10260_CR36","doi-asserted-by":"publisher","first-page":"1057","DOI":"10.1137\/16M1082329","volume":"55","author":"M Stynes","year":"2017","unstructured":"Stynes, M., O\u2019Riordan, E., Gracia, J.L.: Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. SIAM J. Numer. Anal. 55(2), 1057\u20131079 (2017)","journal-title":"SIAM J. Numer. Anal."},{"key":"10260_CR37","doi-asserted-by":"publisher","unstructured":"Swati, Singh, K., Verma, A.K., Singh, M.: Higher order Emden-Fowler type equations via uniform Haar wavelet resolution technique. J. Computat. Appl. Math. 376, 112836 (2020https:\/\/doi.org\/10.1016\/j.cam.2020.112836","DOI":"10.1016\/j.cam.2020.112836"},{"key":"10260_CR38","doi-asserted-by":"publisher","unstructured":"Syam, M.: Analytical solution of the fractional initial Emden-Fowler equation using the fractional residual power series method. Int. J. Appl. Computat. Math. 4 (2018https:\/\/doi.org\/10.1007\/s40819-018-0538-2","DOI":"10.1007\/s40819-018-0538-2"},{"key":"10260_CR39","doi-asserted-by":"publisher","unstructured":"Thom\u00e9e, V.: Galerkin Finite Element Method for Parabolic Problems, vol. 1054. Springer (2006https:\/\/doi.org\/10.1007\/BFb0071790","DOI":"10.1007\/BFb0071790"},{"issue":"5","key":"10260_CR40","doi-asserted-by":"publisher","first-page":"882","DOI":"10.1016\/j.camwa.2016.04.029","volume":"74","author":"G Vacca","year":"2017","unstructured":"Vacca, G.: Virtual element methods for hyperbolic problems on polygonal meshes. Comput. Math. Appl. 74(5), 882\u2013898 (2017)","journal-title":"Comput. Math. Appl."},{"key":"10260_CR41","doi-asserted-by":"crossref","unstructured":"Vacca, G., L. Beirao Da Veiga: Virtual element methods for parabolic problems on polygonal meshes. Numer. Meth. Part D E 31(6), 2110\u20132134 (2015)","DOI":"10.1002\/num.21982"},{"key":"10260_CR42","doi-asserted-by":"publisher","unstructured":"Beir\u00e3o\u00a0da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L.D., Russo, A.: Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23 (2012https:\/\/doi.org\/10.1142\/S0218202512500492","DOI":"10.1142\/S0218202512500492"},{"key":"10260_CR43","doi-asserted-by":"crossref","unstructured":"The hitchhiker\u2019s guide to the virtual element method: Beir\u00e3o da veiga, l., brezzi, f., marini, l.d., russo, a. Math. Models Methods Appl. Sci. 24, 1541\u20131573 (2014)","DOI":"10.1142\/S021820251440003X"},{"key":"10260_CR44","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-21449-3","volume-title":"Linear and Nonlinear Integral Equations: Methods and Applications","author":"AM Wazwaz","year":"2011","unstructured":"Wazwaz, A.M.: Linear and Nonlinear Integral Equations: Methods and Applications, 1st edn. Springer Publishing Company, Incorporated (2011)","edition":"1"},{"key":"10260_CR45","doi-asserted-by":"publisher","unstructured":"Zhang, Y., Feng, M.: A mixed virtual element method for the time-fractional fourth-order subdiffusion equation. Numer. Algorithms 90, 1\u201321 (2022https:\/\/doi.org\/10.1007\/s11075-021-01244-0","DOI":"10.1007\/s11075-021-01244-0"},{"key":"10260_CR46","doi-asserted-by":"publisher","unstructured":"Zhang, Y., Feng, M.: The virtual element method for the time fractional convection diffusion reaction equation with non-smooth data. Comput. Math. Appl. 110, 1\u201318 (2022https:\/\/doi.org\/10.1016\/j.camwa.2022.01.033","DOI":"10.1016\/j.camwa.2022.01.033"},{"key":"10260_CR47","doi-asserted-by":"crossref","unstructured":"Zhang, Y., Feng, M.: A local projection stabilization virtual element method for the time-fractional burgers equation with high Reynolds numbers. Appl. Math. Comput. 436,(2023)","DOI":"10.1016\/j.amc.2022.127509"}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-025-10260-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10444-025-10260-w","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-025-10260-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T12:03:38Z","timestamp":1767960218000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10444-025-10260-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,10,16]]},"references-count":47,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2025,12]]}},"alternative-id":["10260"],"URL":"https:\/\/doi.org\/10.1007\/s10444-025-10260-w","relation":{},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"value":"1019-7168","type":"print"},{"value":"1572-9044","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,10,16]]},"assertion":[{"value":"6 January 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 September 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 October 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"47"}}