{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T07:29:28Z","timestamp":1774596568736,"version":"3.50.1"},"reference-count":55,"publisher":"Institute of Electrical and Electronics Engineers (IEEE)","license":[{"start":{"date-parts":[[2020,1,1]],"date-time":"2020-01-01T00:00:00Z","timestamp":1577836800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["IEEE Access"],"published-print":{"date-parts":[[2020]]},"DOI":"10.1109\/access.2020.2978594","type":"journal-article","created":{"date-parts":[[2020,3,6]],"date-time":"2020-03-06T00:30:28Z","timestamp":1583454628000},"page":"45119-45132","source":"Crossref","is-referenced-by-count":7,"title":["Monotone Finite-Difference Schemes With Second Order Approximation Based on Regularization Approach for the Dirichlet Boundary Problem of the Gamma Equation"],"prefix":"10.1109","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5252-199X","authenticated-orcid":false,"given":"Le Minh","family":"Hieu","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6305-0713","authenticated-orcid":false,"given":"Truong Thi Hieu","family":"Hanh","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2025-8319","authenticated-orcid":false,"given":"Dang Ngoc Hoang","family":"Thanh","sequence":"additional","affiliation":[]}],"member":"263","reference":[{"key":"ref39","doi-asserted-by":"publisher","DOI":"10.1201\/9780203908518"},{"key":"ref38","first-page":"399","article-title":"Second-order accurate finite-difference schemes on nonuniform grids","volume":"38","author":"vabishchevich","year":"1998","journal-title":"Comput Math Math Phys"},{"key":"ref33","doi-asserted-by":"publisher","DOI":"10.1515\/cmam-2016-0046"},{"key":"ref32","first-page":"843","article-title":"Analytical solutions for the Black–Scholes equation","volume":"12","author":"manafian","year":"2017","journal-title":"Applic Appl Math Int J"},{"key":"ref31","first-page":"21","article-title":"Economical monotone difference schemes for multidimensional convection-diffusion problems on non-uniform grids","volume":"44","author":"malafei","year":"2000","journal-title":"Doklady NAS Belarus"},{"key":"ref30","first-page":"3","article-title":"Finite-difference method for the Gamma equation on non-uniform grids","volume":"61","author":"hieu","year":"2019","journal-title":"Vietnam Journal of Science Technology and Engineering"},{"key":"ref37","first-page":"18","article-title":"Monotone difference schemes for linear parabolic equations with mixed boundary conditions","volume":"58","author":"matus","year":"2014","journal-title":"Doklady of the National Academy of Sciences of Belarus"},{"key":"ref36","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2016.04.006"},{"key":"ref35","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2017.09.020"},{"key":"ref34","first-page":"13","article-title":"Maximum principle for finite-difference schemes with non sign-constant input data","volume":"59","author":"matus","year":"2015","journal-title":"Doklady of the National Academy of Sciences of Belarus"},{"key":"ref28","first-page":"149","article-title":"Solution of the first boundary problem in the large for quasi-linear parabolic equations","volume":"7","author":"ladyzhenskaya","year":"1958","journal-title":"Trudi Moskovskogo matematicheskogo obchestva"},{"key":"ref27","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2013.06.082"},{"key":"ref29","article-title":"Lineinye i kvazilineinye uravneniya parabolicheskogo tipa (Linear and Quasilinear Equations of Parabolic Type)","author":"ladyzhenskaya","year":"1967"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1080\/00207160.2012.690034"},{"key":"ref1","first-page":"15","article-title":"Radial basis function methods to solve partial differential equations arising in financial applications—A review","volume":"24","author":"arora","year":"2017","journal-title":"Nonlinear Stud"},{"key":"ref20","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/drr050"},{"key":"ref22","author":"friedman","year":"1964","journal-title":"Partial Differential Equations of Parabolic Type"},{"key":"ref21","author":"frey","year":"2000","journal-title":"Market Illiquidity as a Source of Model Risk in Dynamic Hedging in Model Risk"},{"key":"ref24","article-title":"Difference schemes","author":"godunov","year":"1977"},{"key":"ref23","article-title":"Theory matrices","author":"gantmakher","year":"1967"},{"key":"ref26","doi-asserted-by":"publisher","DOI":"10.1155\/JAM.2005.235"},{"key":"ref25","doi-asserted-by":"crossref","first-page":"3151","DOI":"10.1016\/j.jde.2015.08.036","article-title":"General properties of solutions to inhomogeneous Black–Scholes equations with discontinuous maturity payoffs","volume":"260","author":"hyong-chol","year":"2016","journal-title":"J Differe Equ"},{"key":"ref50","doi-asserted-by":"publisher","DOI":"10.1109\/CCE.2018.8465764"},{"key":"ref51","article-title":"Second order monotone difference schemes with approximation on non-uniform grids for two-dimensional quasilinear parabolic convection-diffusion equations","author":"hieu","year":"0","journal-title":"Vestnik St. Petersburg University: Mathematics"},{"key":"ref55","doi-asserted-by":"publisher","DOI":"10.1007\/s10614-017-9730-4"},{"key":"ref54","first-page":"707","article-title":"Standard Galerkin formulation with high order langrange finite elements for option markets pricing","volume":"195","author":"markolefas","year":"2008","journal-title":"Appl Math Comput"},{"key":"ref53","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-15431-2"},{"key":"ref52","article-title":"Solving nonlinear and high-dimensional partial differential equations via deep learning","author":"al-aradi","year":"2018","journal-title":"arXiv 1811 08782"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2007.01.035"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2008.02.010"},{"key":"ref40","first-page":"269","article-title":"Difference schemes on non-uniform grids for two-dimensional parabolic equations","volume":"34","author":"samarskii","year":"1998","journal-title":"Differents Uravn"},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2013.12.008"},{"key":"ref13","doi-asserted-by":"publisher","DOI":"10.4236\/jmf.2018.82024"},{"key":"ref14","doi-asserted-by":"publisher","DOI":"10.1007\/s007800050046"},{"key":"ref15","doi-asserted-by":"publisher","DOI":"10.1086\/260062"},{"key":"ref16","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2010.02.014"},{"key":"ref17","doi-asserted-by":"publisher","DOI":"10.1080\/00207160.2011.558574"},{"key":"ref18","doi-asserted-by":"publisher","DOI":"10.1137\/050627241"},{"key":"ref19","doi-asserted-by":"publisher","DOI":"10.1002\/num.20547"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2018.04.045"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1080\/1350486X.2013.850902"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1016\/j.mcm.2006.03.009"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2008.02.005"},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.1007\/s00366-019-00760-x"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1016\/j.mcm.2006.04.006"},{"key":"ref49","doi-asserted-by":"publisher","DOI":"10.5194\/isprsarchives-XL-5-W6-73-2015"},{"key":"ref9","first-page":"66","article-title":"Comparative study analytic and numerical methods for solving non-linear Black–Scholes equation with European call option","volume":"3","author":"g\u00fclka\u00e7","year":"2015","journal-title":"International Journal of Innovation in Science and Mathematics"},{"key":"ref46","doi-asserted-by":"publisher","DOI":"10.12693\/APhysPolA.132.1050"},{"key":"ref45","doi-asserted-by":"publisher","DOI":"10.3329\/ganit.v33i0.17664"},{"key":"ref48","doi-asserted-by":"publisher","DOI":"10.31449\/inf.v43i2.2179"},{"key":"ref47","first-page":"12","article-title":"Monotone difference scheme for parabolic and elliptic equations with lower order derivatives on nonuniform grids","volume":"2","author":"zhuk","year":"1998","journal-title":"Vestsi NAN Bel Ser Fiz -Mat"},{"key":"ref42","doi-asserted-by":"publisher","DOI":"10.22436\/jmcs.020.04.02"},{"key":"ref41","first-page":"265","article-title":"Difference schemes of increased order of accuracy on non-uniform grids","volume":"32","author":"samarskii","year":"1996","journal-title":"Differents Uravn"},{"key":"ref44","article-title":"Numerical methods for solution of convection–diffusion problems","author":"samarskii","year":"1999"},{"key":"ref43","first-page":"95","article-title":"High-order accurate monotone difference schemes on nonuniform grids for convection–diffusion problems","volume":"9","author":"samarskii","year":"1997","journal-title":"Mat Model"}],"container-title":["IEEE Access"],"original-title":[],"link":[{"URL":"http:\/\/xplorestaging.ieee.org\/ielx7\/6287639\/8948470\/09025045.pdf?arnumber=9025045","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,12,17]],"date-time":"2021-12-17T19:52:07Z","timestamp":1639770727000},"score":1,"resource":{"primary":{"URL":"https:\/\/ieeexplore.ieee.org\/document\/9025045\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020]]},"references-count":55,"URL":"https:\/\/doi.org\/10.1109\/access.2020.2978594","relation":{},"ISSN":["2169-3536"],"issn-type":[{"value":"2169-3536","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020]]}}}