{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,13]],"date-time":"2026-10-13T21:47:06Z","timestamp":1791928026475,"version":"4.3.4"},"reference-count":35,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2026,12,1]],"date-time":"2026-12-01T00:00:00Z","timestamp":1796083200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,12,1]],"date-time":"2026-12-01T00:00:00Z","timestamp":1796083200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2026,9,4]],"date-time":"2026-09-04T00:00:00Z","timestamp":1788480000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-nc\/4.0\/"}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["MethodsX"],"published-print":{"date-parts":[[2026,12]]},"DOI":"10.1016\/j.mex.2026.104150","type":"journal-article","created":{"date-parts":[[2026,9,8]],"date-time":"2026-09-08T15:49:30Z","timestamp":1788882570000},"page":"104150","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":0,"special_numbering":"C","title":["A semi-analytical physics-informed neural network framework with hessian analysis for singularly perturbed fractal convection-dominated anomalous diffusion problems"],"prefix":"10.1016","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7884-8640","authenticated-orcid":false,"given":"Navnit","family":"Jha","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mon Bahadur","family":"Rai","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"78","reference":[{"key":"10.1016\/j.mex.2026.104150_bib0001","series-title":"Fitted numerical methods for singular perturbation problems: error estimates in the maximum norm for linear problems in one and two dimensions","author":"Miller","year":"1996"},{"issue":"4","key":"10.1016\/j.mex.2026.104150_bib0002","doi-asserted-by":"crossref","first-page":"501","DOI":"10.1137\/1026104","article-title":"Applications of singular perturbation techniques to control problems","volume":"26","author":"Kokotovi\u0107","year":"1984","journal-title":"SIAM Rev."},{"key":"10.1016\/j.mex.2026.104150_bib0003","volume":"42","author":"de Jager","year":"1996"},{"key":"10.1016\/j.mex.2026.104150_bib0004","series-title":"Singular perturbation theory: mathematical and analytical techniques with applications to engineering","author":"Johnson","year":"2005"},{"key":"10.1016\/j.mex.2026.104150_bib0005","series-title":"Methods and applications of singular perturbations: boundary layers and multiple timescale dynamics","author":"Verhulst","year":"2005"},{"issue":"8","key":"10.1016\/j.mex.2026.104150_bib0006","first-page":"3641","article-title":"A brief survey on numerical methods for solving singularly perturbed problems","volume":"217","author":"Kadalbajoo","year":"2010","journal-title":"Appl. Math. Comput."},{"issue":"5","key":"10.1016\/j.mex.2026.104150_bib0007","doi-asserted-by":"crossref","first-page":"799","DOI":"10.1080\/10236198.2010.513330","article-title":"Novel fitted operator finite difference methods for singularly perturbed elliptic convection\u2013diffusion problems in two dimensions","volume":"18","author":"Munyakazi","year":"2012","journal-title":"J. Differ. Equ. Appl."},{"issue":"2","key":"10.1016\/j.mex.2026.104150_bib0008","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1007\/s42979-024-03566-x","article-title":"Adaptive mesh physics informed neural networks technique for anomalous diffusion equation on a thin circular plate","volume":"6","author":"Jha","year":"2025","journal-title":"SN Comput. Sci."},{"issue":"3","key":"10.1016\/j.mex.2026.104150_bib0009","doi-asserted-by":"crossref","first-page":"1171","DOI":"10.1007\/s41478-024-00785-6","article-title":"The convergence of high accuracy three point compact computational method for structural derivative anomalous diffusion advection model","volume":"33","author":"Jha","year":"2025","journal-title":"J. Anal."},{"key":"10.1016\/j.mex.2026.104150_bib0010","doi-asserted-by":"crossref","DOI":"10.1016\/j.mex.2022.101853","article-title":"Method of approximations for the convection-dominated anomalous diffusion equation in a rectangular plate using high resolution compact discretization","volume":"9","author":"Jha","year":"2022","journal-title":"MethodsX"},{"key":"10.1016\/j.mex.2026.104150_bib0011","series-title":"Singular perturbations and boundary layers","author":"Gie","year":"2018"},{"issue":"1","key":"10.1016\/j.mex.2026.104150_bib0012","article-title":"Solving a system of singular perturbation problems via neural networks","volume":"55","author":"Shwayaa","year":"2020","journal-title":"J. Southwest Jiaotong Univ."},{"issue":"5","key":"10.1016\/j.mex.2026.104150_bib0013","doi-asserted-by":"crossref","first-page":"1897","DOI":"10.1007\/s00211-025-01491-6","article-title":"Neural networks for singular perturbations","volume":"157","author":"Opschoor","year":"2025","journal-title":"Numer. Math."},{"key":"10.1016\/j.mex.2026.104150_bib0014","first-page":"1","article-title":"Design neural network to solve singular pertubation problems","volume":"3","author":"Tawfiq","year":"2014","journal-title":"J. Appl. Comput. Math."},{"key":"10.1016\/j.mex.2026.104150_bib0015","article-title":"Artificial neural network based predictive modeling of viscosity of an oil based hybrid nanofluid","author":"Furqan","year":"2025","journal-title":"MethodsX"},{"key":"10.1016\/j.mex.2026.104150_bib0016","first-page":"1","article-title":"Semi-analytic pinn methods for singularly perturbed boundary value problems","author":"Gie","year":"2024","journal-title":"Appl. Anal."},{"key":"10.1016\/j.mex.2026.104150_bib0017","series-title":"Fractals in science","author":"Bunde","year":"2013"},{"issue":"3","key":"10.1016\/j.mex.2026.104150_bib0018","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1007\/BF02704566","article-title":"Fractal differential equations and fractal-time dynamical systems","volume":"64","author":"Parvate","year":"2005","journal-title":"Pramana"},{"issue":"06","key":"10.1016\/j.mex.2026.104150_bib0019","doi-asserted-by":"crossref","DOI":"10.1142\/S0218348X21501292","article-title":"Calculation of fractal dimension based on artificial neural network and its application for machined surfaces","volume":"29","author":"Zhou","year":"2021","journal-title":"Fractals"},{"key":"10.1016\/j.mex.2026.104150_bib0020","first-page":"81","article-title":"Solving differential equations of fractional order using an optimization technique based on training artificial neural network","volume":"293","author":"Pakdaman","year":"2017","journal-title":"Appl. Math. Comput."},{"key":"10.1016\/j.mex.2026.104150_bib0021","article-title":"Numerical treatment of fractal boundary value problems for heat conduction in polar bear with spatial variation of thermal conductivity","volume":"2","author":"Jha","year":"2022","journal-title":"Ex. Counterexamples"},{"issue":"11","key":"10.1016\/j.mex.2026.104150_bib0022","doi-asserted-by":"crossref","first-page":"3698","DOI":"10.1007\/s10773-014-2123-8","article-title":"A tutorial review on fractal spacetime and fractional calculus","volume":"53","author":"He","year":"2014","journal-title":"Int. J. Theor. Phys."},{"issue":"1","key":"10.1016\/j.mex.2026.104150_bib0023","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1515\/IJNSNS.2010.11.1.3","article-title":"Investigation on fractional and fractal derivative relaxation-oscillation models","volume":"11","author":"Chen","year":"2010","journal-title":"Int. J. Nonlinear Sci. Numer. Simul."},{"issue":"5","key":"10.1016\/j.mex.2026.104150_bib0024","doi-asserted-by":"crossref","first-page":"1754","DOI":"10.1016\/j.camwa.2009.08.020","article-title":"Anomalous diffusion modeling by fractal and fractional derivatives","volume":"59","author":"Chen","year":"2010","journal-title":"Comput. Math. Appl."},{"issue":"1","key":"10.1016\/j.mex.2026.104150_bib0025","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1186\/s13662-015-0416-8","article-title":"A new discrete economic model involving generalized fractal derivative","volume":"2015","author":"Hu","year":"2015","journal-title":"Adv. Differ. Equ."},{"issue":"06","key":"10.1016\/j.mex.2026.104150_bib0026","doi-asserted-by":"crossref","DOI":"10.1142\/S0218348X23401035","article-title":"Physics-informed neural network for solving hausdorff derivative poisson equations","volume":"31","author":"Wu","year":"2023","journal-title":"Fractals"},{"issue":"3","key":"10.1016\/j.mex.2026.104150_bib0027","doi-asserted-by":"crossref","first-page":"192","DOI":"10.1007\/s11063-024-11644-7","article-title":"GPINN with neural tangent kernel technique for nonlinear two point boundary value problems","volume":"56","author":"Jha","year":"2024","journal-title":"Neural Process. Lett."},{"issue":"1","key":"10.1016\/j.mex.2026.104150_bib0028","doi-asserted-by":"crossref","first-page":"505","DOI":"10.1007\/s00366-022-01607-8","article-title":"A new iterative technique for solving fractal-fractional differential equations based on artificial neural network in the new generalized Caputo sense","volume":"39","author":"Shloof","year":"2023","journal-title":"Eng.Comput"},{"issue":"1","key":"10.1016\/j.mex.2026.104150_bib0029","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1016\/j.ijsrc.2022.07.007","article-title":"Application of hausdorff fractal derivative to the determination of the vertical sediment concentration distribution","volume":"38","author":"Sun","year":"2023","journal-title":"Int. J. Sediment Res."},{"key":"10.1016\/j.mex.2026.104150_bib0030","doi-asserted-by":"crossref","first-page":"292","DOI":"10.1016\/j.advwatres.2012.11.005","article-title":"A fractal richards\u2019 equation to capture the non-boltzmann scaling of water transport in unsaturated media","volume":"52","author":"Sun","year":"2013","journal-title":"Adv. Water Resour."},{"key":"10.1016\/j.mex.2026.104150_bib0031","doi-asserted-by":"crossref","first-page":"346","DOI":"10.1016\/j.chaos.2017.03.060","article-title":"Fractional and fractal derivative models for transient anomalous diffusion: model comparison","volume":"102","author":"Sun","year":"2017","journal-title":"Chaos Solit. Fractals"},{"key":"10.1016\/j.mex.2026.104150_bib0032","doi-asserted-by":"crossref","first-page":"636","DOI":"10.1016\/j.jpowsour.2015.12.037","article-title":"A fractal time thermal model for predicting the surface temperature of air-cooled cylindrical li-ion cells based on experimental measurements","volume":"306","author":"Reyes-Marambio","year":"2016","journal-title":"J. Power Sour."},{"issue":"4","key":"10.1016\/j.mex.2026.104150_bib0033","doi-asserted-by":"crossref","first-page":"923","DOI":"10.1016\/j.chaos.2005.08.199","article-title":"Time-space fabric underlying anomalous diffusion","volume":"28","author":"Chen","year":"2006","journal-title":"Chaos Solit. Fractals"},{"issue":"244","key":"10.1016\/j.mex.2026.104150_bib0035","first-page":"1","article-title":"A high-resolution convergent radial basis functions compact-fdd for boundary layer problems on a scattered mesh network appearing in viscous elastic fluid","volume":"8","author":"Jha","year":"2022","journal-title":"Int. J. Appl. Comput. Math."},{"issue":"108748","key":"10.1016\/j.mex.2026.104150_bib0034","first-page":"1","article-title":"Order-preserving fuzzy transform for singular boundary value problems of polytropic gas flow and sewage diffusion","volume":"475","author":"Jha","year":"2024","journal-title":"Fuzzy Sets Syst."}],"container-title":["MethodsX"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S2215016126003663?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S2215016126003663?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,10,13]],"date-time":"2026-10-13T21:15:44Z","timestamp":1791926144000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S2215016126003663"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,12]]},"references-count":35,"alternative-id":["S2215016126003663"],"URL":"https:\/\/doi.org\/10.1016\/j.mex.2026.104150","relation":{},"ISSN":["2215-0161"],"issn-type":[{"value":"2215-0161","type":"print"}],"subject":[],"published":{"date-parts":[[2026,12]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"A semi-analytical physics-informed neural network framework with hessian analysis for singularly perturbed fractal convection-dominated anomalous diffusion problems","name":"articletitle","label":"Article Title"},{"value":"MethodsX","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.mex.2026.104150","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 The Authors. Published by Elsevier B.V.","name":"copyright","label":"Copyright"}],"article-number":"104150"}}