{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,26]],"date-time":"2026-03-26T12:40:45Z","timestamp":1774528845501,"version":"3.50.1"},"reference-count":45,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2023,2,26]],"date-time":"2023-02-26T00:00:00Z","timestamp":1677369600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,2,26]],"date-time":"2023-02-26T00:00:00Z","timestamp":1677369600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Engineering with Computers"],"published-print":{"date-parts":[[2024,2]]},"DOI":"10.1007\/s00366-023-01794-y","type":"journal-article","created":{"date-parts":[[2023,2,26]],"date-time":"2023-02-26T11:02:30Z","timestamp":1677409350000},"page":"313-322","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["A meshless collocation method based on Pascal polynomial approximation and implicit closest point method for solving reaction\u2013diffusion systems on surfaces"],"prefix":"10.1007","volume":"40","author":[{"given":"Hasan","family":"Zamani-Gharaghoshi","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2573-9755","authenticated-orcid":false,"given":"Mehdi","family":"Dehghan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mostafa","family":"Abbaszadeh","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,2,26]]},"reference":[{"issue":"3","key":"1794_CR1","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1006\/bulm.1998.0093","volume":"61","author":"RA Barrio","year":"1999","unstructured":"Barrio RA, Varea C, Arag\u00f3n JL, Maini PK (1999) A two-dimensional numerical study of spatial pattern formation in interacting Turing systems. Bull Math Biol 61(3):483\u2013505","journal-title":"Bull Math Biol"},{"key":"1794_CR2","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1016\/j.enganabound.2016.06.004","volume":"70","author":"CW Chang","year":"2016","unstructured":"Chang CW (2016) A new meshless method for solving steady-state nonlinear heat conduction problems in arbitrary plane domain. Eng Anal Bound Elem 70:56\u201371","journal-title":"Eng Anal Bound Elem"},{"key":"1794_CR3","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1016\/j.enganabound.2022.03.019","volume":"140","author":"W Chang","year":"2022","unstructured":"Chang W, Zhang J, Wang Y, Wang J (2022) The local meshless method based on Pascal polynomial basis functions for solving fourth-order PDEs. Eng Anal Bound Elem 140:159\u2013166","journal-title":"Eng Anal Bound Elem"},{"key":"1794_CR4","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1007\/s002850000067","volume":"42","author":"MAJ Chaplain","year":"2001","unstructured":"Chaplain MAJ, Ganesh M, Graham IG (2001) Spatio-temporal pattern formation on spherical surfaces: numerical simulation and application to solid tumour growth. J Math Biol 42:387\u2013423","journal-title":"J Math Biol"},{"issue":"1","key":"1794_CR5","doi-asserted-by":"crossref","first-page":"A134","DOI":"10.1137\/130929497","volume":"37","author":"Y Chen","year":"2015","unstructured":"Chen Y, Macdonald CB (2015) The closest point method and multigrid solvers for elliptic equations on surfaces. SIAM J Sci Comput 37(1):A134\u2013A155","journal-title":"SIAM J Sci Comput"},{"issue":"1","key":"1794_CR6","doi-asserted-by":"crossref","first-page":"266","DOI":"10.1137\/16M1080410","volume":"40","author":"KC Cheung","year":"2018","unstructured":"Cheung KC, Ling L (2018) A kernel-based embedding method and convergence analysis for surfaces PDEs. SIAM J Sci Comput 40(1):266\u2013287","journal-title":"SIAM J Sci Comput"},{"issue":"4","key":"1794_CR7","doi-asserted-by":"crossref","first-page":"868","DOI":"10.1103\/PhysRevLett.85.868","volume":"85","author":"VA Davydov","year":"2000","unstructured":"Davydov VA, Manz N, Steinbock O, Zykov VS, M\u00fcller SC (2000) Excitation fronts on a periodically modulated curved surface. Phys Rev Lett 85(4):868\u2013871","journal-title":"Phys Rev Lett"},{"issue":"1","key":"1794_CR8","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1108\/HFF-10-2019-0731","volume":"31","author":"M Dehghan","year":"2021","unstructured":"Dehghan M, Mohammadi V (2021) The boundary knot method for solving two-dimensional (advection) reaction-diffusion equation. Int J Numer Methods Heat Fluid Flow 31(1):106\u2013133","journal-title":"Int J Numer Methods Heat Fluid Flow"},{"issue":"5","key":"1794_CR9","doi-asserted-by":"crossref","first-page":"6955","DOI":"10.1007\/s40314-018-0716-1","volume":"37","author":"M Dehghan","year":"2018","unstructured":"Dehghan M, Narimani N (2018) Approximation of continuous surface differential operators with the generalized moving least-squares (GMLS) method for solving reaction-diffusion equation. Comput Appl Math 37(5):6955\u20136971","journal-title":"Comput Appl Math"},{"key":"1794_CR10","doi-asserted-by":"crossref","DOI":"10.1016\/j.cam.2022.114128","volume":"408","author":"M Dehghan","year":"2022","unstructured":"Dehghan M, Shirilord A (2022) A fast computational algorithm for computing outer pseudo-inverses with numerical experiments. J Comput Appl Math 408:114128","journal-title":"J Comput Appl Math"},{"issue":"7","key":"1794_CR11","doi-asserted-by":"crossref","first-page":"2506","DOI":"10.1108\/HFF-09-2021-0598","volume":"32","author":"M Dehghan","year":"2021","unstructured":"Dehghan M, Hooshyarfarzin B, Abbaszadeh M (2021) Proper orthogonal decomposition Pascal polynomial-based method for solving Sobolev equation. Int J Numer Methods Heat Fluid Flow 32(7):2506\u20132542","journal-title":"Int J Numer Methods Heat Fluid Flow"},{"issue":"3","key":"1794_CR12","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1007\/s10915-013-9688-x","volume":"56","author":"EJ Fuselier","year":"2013","unstructured":"Fuselier EJ, Wright GB (2013) A high-order kernel method for diffusion and reaction-diffusion equations on surfaces. J Sci Comput 56(3):535\u2013565","journal-title":"J Sci Comput"},{"issue":"4","key":"1794_CR13","doi-asserted-by":"crossref","first-page":"3913","DOI":"10.1103\/PhysRevE.56.3913","volume":"56","author":"J Gomatam","year":"1997","unstructured":"Gomatam J, Amdjadi F (1997) Reaction-diffusion equations on a sphere: meandering of spiral waves. Phys Rev E 56(4):3913","journal-title":"Phys Rev E"},{"issue":"1887","key":"1794_CR14","first-page":"151","volume":"433","author":"P Grindrod","year":"1991","unstructured":"Grindrod P, Lewis MA, Murray JD (1991) A geometrical approach to wave-type solutions of excitable reaction-diffusion systems. Proc R Soc A 433(1887):151\u2013164","journal-title":"Proc R Soc A"},{"issue":"5","key":"1794_CR15","doi-asserted-by":"crossref","first-page":"2129","DOI":"10.1137\/16M1095457","volume":"39","author":"E Lehto","year":"2017","unstructured":"Lehto E, Shankar V, Wright GB (2017) A radial basis function (RBF) compact finite difference (FD) scheme for reaction-diffusion equations on surfaces. SIAM J Sci Comput 39(5):2129\u20132151","journal-title":"SIAM J Sci Comput"},{"key":"1794_CR16","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1016\/j.enganabound.2015.09.003","volume":"62","author":"CS Liu","year":"2016","unstructured":"Liu CS, Kuo CL (2016) A multiple-scale Pascal polynomial triangle solving elliptic equations and inverse Cauchy problems. Eng Anal Bound Elem 62:35\u201343","journal-title":"Eng Anal Bound Elem"},{"key":"1794_CR17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jcp.2016.02.017","volume":"312","author":"CS Liu","year":"2016","unstructured":"Liu CS, Young DL (2016) A multiple-scale Pascal polynomial for 2D Stokes and inverse Cauchy-Stokes problems. J Comput Phys 312:1\u201313","journal-title":"J Comput Phys"},{"key":"1794_CR18","first-page":"430","volume":"331","author":"G Liu","year":"2018","unstructured":"Liu G, Ma W, Ma H, Zhu L (2018) A multiple-scale higher order polynomial collocation method for 2D and 3D elliptic partial differential equations with variable coefficients. Appl Math Comput 331:430\u2013444","journal-title":"Appl Math Comput"},{"issue":"2","key":"1794_CR19","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1007\/s10915-008-9196-6","volume":"35","author":"CB Macdonald","year":"2008","unstructured":"Macdonald CB, Ruuth SJ (2008) Level set equations on surfaces via the closest point method. J Sci Comput 35(2):219\u2013240","journal-title":"J Sci Comput"},{"issue":"6","key":"1794_CR20","doi-asserted-by":"crossref","first-page":"4330","DOI":"10.1137\/080740003","volume":"31","author":"CB Macdonald","year":"2010","unstructured":"Macdonald CB, Ruuth SJ (2010) The implicit closest point method for the numerical solution of partial differential equations on surfaces. SIAM J Sci Comput 31(6):4330\u20134350","journal-title":"SIAM J Sci Comput"},{"issue":"22","key":"1794_CR21","doi-asserted-by":"crossref","first-page":"7944","DOI":"10.1016\/j.jcp.2011.06.021","volume":"230","author":"CB Macdonald","year":"2011","unstructured":"Macdonald CB, Brandman J, Ruuth SJ (2011) Solving eigenvalue problems on curved surfaces using the closest point method. J Comput Phys 230(22):7944\u20137956","journal-title":"J Comput Phys"},{"issue":"23","key":"1794_CR22","doi-asserted-by":"crossref","first-page":"9209","DOI":"10.1073\/pnas.1221408110","volume":"110","author":"CB Macdonald","year":"2013","unstructured":"Macdonald CB, Merriman B, Ruuth SJ (2013) Simple computation of reaction-diffusion processes on point clouds. Proc Natl Acad Sci 110(23):9209\u20139214","journal-title":"Proc Natl Acad Sci"},{"issue":"5","key":"1794_CR23","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1016\/S0375-9601(03)01148-4","volume":"316","author":"N Manz","year":"2003","unstructured":"Manz N, Davydov VA, M\u00fcller SC, B\u00e4r M (2003) Dependence of the spiral rotation frequency on the surface curvature of reaction-diffusion systems. Phys Lett A 316(5):311\u2013316","journal-title":"Phys Lett A"},{"issue":"6","key":"1794_CR24","doi-asserted-by":"crossref","first-page":"3303","DOI":"10.1137\/120865537","volume":"50","author":"T M\u00e4rz","year":"2012","unstructured":"M\u00e4rz T, Macdonald CB (2012) Calculus on surfaces with general closest point functions. SIAM J Numer Anal 50(6):3303\u20133328","journal-title":"SIAM J Numer Anal"},{"issue":"2","key":"1794_CR25","doi-asserted-by":"crossref","first-page":"2267","DOI":"10.1016\/j.jcp.2007.03.034","volume":"225","author":"B Merriman","year":"2007","unstructured":"Merriman B, Ruuth SJ (2007) Diffusion generated motion of curves on surfaces. J Comput Phys 225(2):2267\u20132282","journal-title":"J Comput Phys"},{"issue":"4","key":"1794_CR26","doi-asserted-by":"crossref","first-page":"1170","DOI":"10.1002\/mma.4043","volume":"40","author":"H Moghaderi","year":"2017","unstructured":"Moghaderi H, Dehghan M (2017) Mixed two-grid finite difference methods for solving one-dimensional and two-dimensional Fitzhugh-Nagumo equations. Math Methods Appl Sci 40(4):1170\u20131200","journal-title":"Math Methods Appl Sci"},{"key":"1794_CR27","volume-title":"Mathematical biology II: spatial models and biomedical applications, 3","author":"JD Murray","year":"2001","unstructured":"Murray JD (2001) Mathematical biology II: spatial models and biomedical applications, 3. Springer, New York"},{"key":"1794_CR28","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1016\/j.apm.2019.04.022","volume":"74","author":"\u00d6 Oru\u00e7","year":"2019","unstructured":"Oru\u00e7 \u00d6 (2019) Numerical solution to the deflection of thin plates using the two-dimensional Berger equation with a meshless method based on multiple-scale Pascal polynomials. Appl Math Model 74:441\u2013456","journal-title":"Appl Math Model"},{"issue":"4","key":"1794_CR29","doi-asserted-by":"crossref","first-page":"1215","DOI":"10.1007\/s00366-019-00758-5","volume":"36","author":"\u00d6 Oru\u00e7","year":"2019","unstructured":"Oru\u00e7 \u00d6 (2019) A meshless multiple-scale polynomial method for numerical solution of 3d convection-diffusion problems with variable coefficients. Eng Comput 36(4):1215\u20131228","journal-title":"Eng Comput"},{"issue":"10","key":"1794_CR30","doi-asserted-by":"crossref","first-page":"1950080","DOI":"10.1142\/S0219876219500804","volume":"17","author":"\u00d6 Oru\u00e7","year":"2020","unstructured":"Oru\u00e7 \u00d6 (2020) A meshfree computational approach based on multiple-scale pascal polynomials for numerical solution of a 2d elliptic problem with nonlocal boundary conditions. Int J Comput Methods 17(10):1950080","journal-title":"Int J Comput Methods"},{"key":"1794_CR31","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2020.110070","volume":"428","author":"\u00d6 Oru\u00e7","year":"2021","unstructured":"Oru\u00e7 \u00d6 (2021) An efficient meshfree method based on Pascal polynomials and multiple-scale approach for numerical solution of 2-D and 3-D second order elliptic interface problems. J Comput Phys 428:110070","journal-title":"J Comput Phys"},{"key":"1794_CR32","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1016\/j.enganabound.2022.09.009","volume":"146","author":"\u00d6 Oru\u00e7","year":"2023","unstructured":"Oru\u00e7 \u00d6 (2023) A strong-form meshfree computational method for plane elastostatic equations of anisotropic functionally graded materials via multiple-scale Pascal polynomials. Eng Anal Boundary Elem 146:132\u2013145","journal-title":"Eng Anal Boundary Elem"},{"issue":"15","key":"1794_CR33","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1016\/j.cam.2014.09.024","volume":"278","author":"MD Petkovi\u0107","year":"2015","unstructured":"Petkovi\u0107 MD, Petkovi\u0107 MS (2015) Hyper-power methods for the computation of outer inverses. J Comput Appl Math 278(15):110\u2013118","journal-title":"J Comput Appl Math"},{"key":"1794_CR34","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/j.jcp.2018.05.022","volume":"370","author":"A Petras","year":"2018","unstructured":"Petras A, Ling L, Ruuth SJ (2018) An RBF-FD closest point method for solving PDEs on surfaces. J Comput Phys 370:43\u201357","journal-title":"J Comput Phys"},{"key":"1794_CR35","doi-asserted-by":"crossref","first-page":"146","DOI":"10.1016\/j.jcp.2018.12.031","volume":"381","author":"A Petras","year":"2019","unstructured":"Petras A, Ling L, Piret C, Ruuth SJ (2019) A least-squares implicit RBF-FD closest point method and applications to PDEs on moving surfaces. J Comput Phys 381:146\u2013161","journal-title":"J Comput Phys"},{"issue":"14","key":"1794_CR36","doi-asserted-by":"crossref","first-page":"4662","DOI":"10.1016\/j.jcp.2012.03.007","volume":"231","author":"C Piret","year":"2012","unstructured":"Piret C (2012) The orthogonal gradients method: a radial basis functions method for solving partial differential equations on arbitrary surfaces. J Comput Phys 231(14):4662\u20134675","journal-title":"J Comput Phys"},{"issue":"3","key":"1794_CR37","doi-asserted-by":"crossref","first-page":"878","DOI":"10.1529\/biophysj.105.073809","volume":"90","author":"IF Sbalzarini","year":"2006","unstructured":"Sbalzarini IF, Hayer A, Helenius A, Koumoutsakos P (2006) Simulations of (an)isotropic diffusion on curved biological surfaces. Biophys J 90(3):878\u2013885","journal-title":"Biophys J"},{"issue":"32","key":"1794_CR38","doi-asserted-by":"crossref","first-page":"11151","DOI":"10.1073\/pnas.0504953102","volume":"102","author":"P Schwartz","year":"2005","unstructured":"Schwartz P, Adalsteinsson D, Colella P, Arkin AP, Onsum M (2005) Numerical computation of diffusion on a surface. Proc Nat Acad Sci 102(32):11151\u201311156","journal-title":"Proc Nat Acad Sci"},{"issue":"3","key":"1794_CR39","doi-asserted-by":"crossref","first-page":"745","DOI":"10.1007\/s10915-014-9914-1","volume":"63","author":"V Shankar","year":"2015","unstructured":"Shankar V, Wright GB, Kirby RM, Fogelson AL (2015) A radial basis function (RBF)-finite difference (FD) method for diffusion and reaction-diffusion equations on surfaces. J Sci Comput 63(3):745\u2013768","journal-title":"J Sci Comput"},{"issue":"3","key":"1794_CR40","first-page":"1943","volume":"227","author":"J Steven","year":"2008","unstructured":"Steven J (2008) Ruuth, Barry Merriman, A simple embedding method for solving partial differential equations on surfaces. J Comput Appl Math 227(3):1943\u20131961","journal-title":"J Comput Appl Math"},{"key":"1794_CR41","doi-asserted-by":"crossref","DOI":"10.1016\/j.aml.2021.107534","volume":"122","author":"Z Tang","year":"2021","unstructured":"Tang Z, Zhuojia F, Chen M, Ling L (2021) A localized extrinsic collocation method for Turing pattern formations on surfaces. Appl Math Lett 122:107534","journal-title":"Appl Math Lett"},{"key":"1794_CR42","doi-asserted-by":"crossref","unstructured":"Turk G (2001) Texture synthesis on surfaces. In: Proceedings of the 28th annual conference on Computer graphics and interactive techniques 347\u2013354","DOI":"10.1145\/383259.383297"},{"issue":"4","key":"1794_CR43","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1145\/127719.122749","volume":"25","author":"G Turk","year":"1991","unstructured":"Turk G (1991) Generating textures on arbitrary surfaces using reaction-diffusion. Acm Siggraph Comput Graph 25(4):289\u2013298","journal-title":"Acm Siggraph Comput Graph"},{"issue":"4","key":"1794_CR44","doi-asserted-by":"crossref","first-page":"4588","DOI":"10.1103\/PhysRevE.60.4588","volume":"60","author":"C Varea","year":"1999","unstructured":"Varea C, Arag\u00f3n JL, Barrio RA (1999) Turing patterns on a sphere. Phys Rev E 60(4):4588\u20134592","journal-title":"Phys Rev E"},{"key":"1794_CR45","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/j.enganabound.2018.01.004","volume":"90","author":"M Wang","year":"2018","unstructured":"Wang M, Watson D, Li M (2018) The method of particular solutions with polynomial basis functions for solving axisymmetric problems. Eng Anal Bound Elem 90:39\u201346","journal-title":"Eng Anal Bound Elem"}],"container-title":["Engineering with Computers"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-023-01794-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00366-023-01794-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-023-01794-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,1]],"date-time":"2024-02-01T02:04:32Z","timestamp":1706753072000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00366-023-01794-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,26]]},"references-count":45,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2024,2]]}},"alternative-id":["1794"],"URL":"https:\/\/doi.org\/10.1007\/s00366-023-01794-y","relation":{},"ISSN":["0177-0667","1435-5663"],"issn-type":[{"value":"0177-0667","type":"print"},{"value":"1435-5663","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,2,26]]},"assertion":[{"value":"5 November 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 February 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 February 2023","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 that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}