{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,20]],"date-time":"2026-03-20T19:07:52Z","timestamp":1774033672921,"version":"3.50.1"},"reference-count":42,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2019,3,21]],"date-time":"2019-03-21T00:00:00Z","timestamp":1553126400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2019,6]]},"DOI":"10.1007\/s40314-019-0838-0","type":"journal-article","created":{"date-parts":[[2019,3,22]],"date-time":"2019-03-22T10:03:59Z","timestamp":1553249039000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["A reproducing kernel Hilbert space approach in meshless collocation method"],"prefix":"10.1007","volume":"38","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0284-084X","authenticated-orcid":false,"given":"Babak","family":"Azarnavid","sequence":"first","affiliation":[]},{"given":"Mahdi","family":"Emamjome","sequence":"additional","affiliation":[]},{"given":"Mohammad","family":"Nabati","sequence":"additional","affiliation":[]},{"given":"Saeid","family":"Abbasbandy","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2019,3,21]]},"reference":[{"key":"838_CR1","doi-asserted-by":"publisher","first-page":"789","DOI":"10.1016\/j.cam.2015.10.035","volume":"296","author":"S Abbasbandy","year":"2016","unstructured":"Abbasbandy S, Azarnavid B (2016) Some error estimates for the reproducing kernel Hilbert spaces method. J Comput Appl Math 296:789\u2013797","journal-title":"J Comput Appl Math"},{"issue":"6","key":"838_CR2","doi-asserted-by":"publisher","first-page":"885","DOI":"10.1016\/j.enganabound.2013.03.006","volume":"37","author":"S Abbasbandy","year":"2013","unstructured":"Abbasbandy S, Roohani Ghehsareh H, Hashim I (2013) A meshfree method for the solution of two-dimensional cubic nonlinear Schrodinger equation. Eng Anal Bound Elem 37(6):885\u2013898","journal-title":"Eng Anal Bound Elem"},{"key":"838_CR3","doi-asserted-by":"publisher","first-page":"293","DOI":"10.1016\/j.cam.2014.11.014","volume":"279","author":"S Abbasbandy","year":"2015","unstructured":"Abbasbandy S, Azarnavid B, Alhuthali MS (2015) A shooting reproducing kernel Hilbert space method for multiple solutions of nonlinear boundary value problems. J Comput Appl Math 279:293\u2013305","journal-title":"J Comput Appl Math"},{"key":"838_CR4","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1155\/2015\/158134","volume":"2015","author":"Ali Akg\u00fcl","year":"2015","unstructured":"Akg\u00fcl A (2015) New reproducing kernel functions. Math Probl Eng 2015:158134. \n                    https:\/\/doi.org\/10.1155\/2015\/158134","journal-title":"Mathematical Problems in Engineering"},{"issue":"1","key":"838_CR5","doi-asserted-by":"publisher","first-page":"112","DOI":"10.11121\/ijocta.01.2017.00368","volume":"7","author":"A Akg\u00fcl","year":"2017","unstructured":"Akg\u00fcl A, Baleanu D (2017) On solutions of variable-order fractional differential equations. Int J Optim Control Theor Appl (IJOCTA) 7(1):112\u2013116","journal-title":"Int J Optim Control Theor Appl (IJOCTA)"},{"issue":"1","key":"838_CR6","doi-asserted-by":"publisher","first-page":"220","DOI":"10.1186\/s13662-015-0558-8","volume":"2015","author":"A Akg\u00fcl","year":"2015","unstructured":"Akg\u00fcl A, Karatas E, Baleanu D (2015) Numerical solutions of fractional differential equations of Lane-Emden type by an accurate technique. Adv Differ Equ 2015(1):220","journal-title":"Adv Differ Equ"},{"key":"838_CR7","first-page":"137","volume":"291","author":"M Al-Smadi","year":"2016","unstructured":"Al-Smadi M, Arqub OA, Shawagfeh N, Momani S (2016) Numerical investigations for systems of second-order periodic boundary value problems using reproducing kernel method. Appl Math Comput 291:137\u2013148","journal-title":"Appl Math Comput"},{"key":"838_CR8","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1090\/S0002-9947-1950-0051437-7","volume":"68","author":"N Aronszajn","year":"1950","unstructured":"Aronszajn N (1950) Theory of reproducing kernels. Trans Am Math Soc 68:337\u2013404","journal-title":"Trans Am Math Soc"},{"issue":"3","key":"838_CR9","doi-asserted-by":"publisher","first-page":"231","DOI":"10.3233\/FI-2016-1384","volume":"146","author":"OA Arqub","year":"2016","unstructured":"Arqub OA (2016) Approximate solutions of DASs with nonclassical boundary conditions using novel reproducing kernel algorithm. Fundam Inform 146(3):231\u2013254","journal-title":"Fundam Inform"},{"issue":"15","key":"838_CR10","doi-asserted-by":"publisher","first-page":"4549","DOI":"10.1002\/mma.3884","volume":"39","author":"OA Arqub","year":"2016","unstructured":"Arqub OA (2016) The reproducing kernel algorithm for handling differential algebraic systems of ordinary differential equations. Math Methods Appl Sci 39(15):4549\u20134562","journal-title":"Math Methods Appl Sci"},{"issue":"6","key":"838_CR11","doi-asserted-by":"publisher","first-page":"1243","DOI":"10.1016\/j.camwa.2016.11.032","volume":"73","author":"OA Arqub","year":"2017","unstructured":"Arqub OA (2017) Fitted reproducing kernel Hilbert space method for the solutions of some certain classes of time-fractional partial differential equations subject to initial and Neumann boundary conditions. Comput Math Appl 73(6):1243\u20131261","journal-title":"Comput Math Appl"},{"key":"838_CR12","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1016\/j.cam.2017.07.015","volume":"328","author":"B Azarnavid","year":"2018","unstructured":"Azarnavid B, Parand K (2018) An iterative reproducing kernel method in Hilbert space for the multi-point boundary value problems. J Comput Appl Math 328:151\u2013163","journal-title":"J Comput Appl Math"},{"issue":"6","key":"838_CR13","doi-asserted-by":"publisher","first-page":"754","DOI":"10.3846\/13926292.2015.1111953","volume":"20","author":"B Azarnavid","year":"2015","unstructured":"Azarnavid B, Parvaneh F, Abbasbandy S (2015) Picard-reproducing Kernel Hilbert space method for solving generalized singular nonlinear Lane-Emden type equations. Math Model Anal 20(6):754\u2013767","journal-title":"Math Model Anal"},{"issue":"02","key":"838_CR14","doi-asserted-by":"publisher","first-page":"1850020","DOI":"10.1142\/S0219633618500207","volume":"17","author":"B Azarnavid","year":"2018","unstructured":"Azarnavid B, Shivanian E, Parand K, Nikmanesh Soudabeh (2018) Multiplicity results by shooting reproducing kernel Hilbert space method for the catalytic reaction in a flat particle. J Theor Comput Chem 17(02):1850020","journal-title":"J Theor Comput Chem"},{"key":"838_CR15","doi-asserted-by":"publisher","first-page":"544","DOI":"10.1016\/j.cnsns.2017.12.002","volume":"59","author":"B Azarnavid","year":"2018","unstructured":"Azarnavid B, Parand K, Abbasbandy S (2018) An iterative kernel based method for fourth order nonlinear equation with nonlinear boundary condition. Commun Nonlinear Sci Numer Simul 59:544\u2013552","journal-title":"Commun Nonlinear Sci Numer Simul"},{"key":"838_CR16","unstructured":"Azarnavid B, Parand K (2016) Imposing various boundary conditions on radial basis functions. arXiv preprint \n                    arXiv:1611.07292"},{"key":"838_CR17","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1016\/S0893-9659(99)00052-X","volume":"12","author":"HN Caglar","year":"1999","unstructured":"Caglar HN, Caglar SH, Twizell EH (1999) The numerical solution of fifth-order boundary-value problems with sixth-degree B-spline functions. Appl Math Lett 12:25\u201330","journal-title":"Appl Math Lett"},{"key":"838_CR18","volume-title":"Nonlinear numerical analysis in the reproducing kernel space","author":"MG Cui","year":"2009","unstructured":"Cui MG, Lin Y (2009) Nonlinear numerical analysis in the reproducing kernel space. Nova Science, New York"},{"key":"838_CR19","first-page":"489","volume":"202","author":"YL Duan","year":"2008","unstructured":"Duan YL, Liu RX, Jiang YQ (2008) Lattice Boltzmann model for the modified Burgers\u2019 equation. Appl Math Comput 202:489\u2013497","journal-title":"Appl Math Comput"},{"key":"838_CR20","doi-asserted-by":"publisher","unstructured":"Emamjome M, Azarnavid B, Roohani Ghehsareh H (2017) A reproducing kernel Hilbert space pseudospectral method for numerical investigation of a two-dimensional capillary formation model in tumor angiogenesis problem. Neural Comput Appl. \n                    https:\/\/doi.org\/10.1007\/s00521-017-3184-4","DOI":"10.1007\/s00521-017-3184-4"},{"key":"838_CR21","doi-asserted-by":"publisher","unstructured":"Fasshauer GE (2005) RBF collocation methods as pseudospectral methods. WIT Trans Model Simul. \n                    https:\/\/doi.org\/10.2495\/BE050051","DOI":"10.2495\/BE050051"},{"key":"838_CR22","first-page":"7685","volume":"219","author":"Y Gao","year":"2013","unstructured":"Gao Y, Le LH, Shi BC (2013) Numerical solution of Burgers\u2019 equation by lattice Boltzmann method. Appl Math Comput 219:7685\u20137692","journal-title":"Appl Math Comput"},{"key":"838_CR23","first-page":"177","volume":"119","author":"YC Hon","year":"2001","unstructured":"Hon YC, Schaback R (2001) On unsymmetric collocation by radial basis functions. Appl Math Comput 119:177\u2013186","journal-title":"Appl Math Comput"},{"key":"838_CR24","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1155\/2012\/984682","volume":"2012","author":"Mustafa Inc","year":"2012","unstructured":"Inc M, Akg\u00fcl A, Kili\u00e7man A (2012) Explicit solution of telegraph equation based on reproducing kernel method. J Funct Sp Appl 2012:984682. \n                    https:\/\/doi.org\/10.1155\/2012\/984682","journal-title":"Journal of Function Spaces and Applications"},{"issue":"3","key":"838_CR25","first-page":"217","volume":"69","author":"A Iske","year":"2011","unstructured":"Iske A (2011) Scattered data approximation by positive definite kernel functions. Rendiconti del Seminario Matematico 69(3):217\u2013246","journal-title":"Rendiconti del Seminario Matematico"},{"key":"838_CR26","first-page":"6680","volume":"219","author":"R Jiwari","year":"2013","unstructured":"Jiwari R, Mittal RC, Sharma KK (2013) A numerical scheme based on weighted average differential quadrature method for the numerical solution of Burgers\u2019 equation. Appl Math Comput 219:6680\u20136691","journal-title":"Appl Math Comput"},{"key":"838_CR27","doi-asserted-by":"publisher","first-page":"251","DOI":"10.1016\/S0955-7997(02)00081-4","volume":"27","author":"J Li","year":"2003","unstructured":"Li J, Cheng AHD, Chen CS (2003) A comparison of efficiency and error convergence of multiquadric collocation method and finite element method. Eng Anal Bound Elem 27:251\u2013257","journal-title":"Eng Anal Bound Elem"},{"key":"838_CR28","doi-asserted-by":"publisher","first-page":"459","DOI":"10.1016\/j.cam.2018.01.027","volume":"339","author":"J Lin","year":"2018","unstructured":"Lin J, Reutskiy SY, Lu J (2018) A novel meshless method for fully nonlinear advection-diffusion-reaction problems to model transfer in anisotropic media. Appl Math Comput 339:459\u2013476","journal-title":"Appl Math Comput"},{"issue":"2","key":"838_CR29","doi-asserted-by":"publisher","first-page":"322","DOI":"10.4208\/aamm.OA-2016-0187","volume":"10","author":"J Lin","year":"2018","unstructured":"Lin J, Zhang C, Sun L, Lu J (2018) Simulation of seismic wave scattering by embedded cavities in an elastic half-plane using the novel singular boundary method. Adv Appl Math Mech 10(2):322\u2013342","journal-title":"Adv Appl Math Mech"},{"key":"838_CR30","doi-asserted-by":"publisher","first-page":"1551","DOI":"10.1016\/j.cam.2010.02.036","volume":"234","author":"X Lv","year":"2010","unstructured":"Lv X, Cui MG (2010) An efficient computational method for linear fifth-order two-point boundary value problems. J Comput Appl Math 234:1551\u20131558","journal-title":"J Comput Appl Math"},{"key":"838_CR31","first-page":"7839","volume":"218","author":"RC Mittal","year":"2012","unstructured":"Mittal RC, Jain RK (2012) Numerical solutions of nonlinear Burgers\u2019 equation with modified cubic B-splines collocation method. Appl Math Comput 218:7839\u20137855","journal-title":"Appl Math Comput"},{"key":"838_CR32","doi-asserted-by":"publisher","first-page":"551","DOI":"10.1016\/S0898-1221(01)00305-4","volume":"43","author":"H Power","year":"2002","unstructured":"Power H, Barraco V, Palermo UDS (2002) A comparison analysis between unsymmetric and symmetric radial basis function collocation method for the numerical solution of partial differential equations. Comput Math Appl 43:551\u2013583","journal-title":"Comput Math Appl"},{"key":"838_CR33","doi-asserted-by":"crossref","unstructured":"Rahman K, Helil N, Yimin R (2010) Some new semi-implicit finite difference schemes for numerical solution of Burgers equation. In: International conference on computer application and system modeling (ICCASM 2010)","DOI":"10.1109\/ICCASM.2010.5622119"},{"key":"838_CR34","doi-asserted-by":"publisher","first-page":"795","DOI":"10.1016\/j.chaos.2005.01.054","volume":"26","author":"MA Ramadan","year":"2005","unstructured":"Ramadan MA, El-Danaf TS, Alaal F (2005) A numerical solution of the Burgers\u2019 equation using septic B-splines. Chaos Solitons Fractals 26:795\u2013804","journal-title":"Chaos Solitons Fractals"},{"issue":"1","key":"838_CR35","doi-asserted-by":"publisher","first-page":"39","DOI":"10.1186\/s13662-017-1091-8","volume":"2017","author":"MG Sakar","year":"2017","unstructured":"Sakar MG, Akg\u00fcl A, Baleanu D (2017) On solutions of fractional Riccati differential equations. Adv Differ Equ 2017(1):39","journal-title":"Adv Differ Equ"},{"issue":"3","key":"838_CR36","doi-asserted-by":"publisher","first-page":"597","DOI":"10.1007\/s00211-015-0731-8","volume":"132","author":"R Schaback","year":"2016","unstructured":"Schaback R (2016) All well-posed problems have uniformly stable and convergent discretizations. Numer Math 132(3):597\u2013630","journal-title":"Numer Math"},{"key":"838_CR37","doi-asserted-by":"publisher","first-page":"591","DOI":"10.1016\/j.aml.2006.06.012","volume":"20","author":"SS Siddiqi","year":"2007","unstructured":"Siddiqi SS, Akram G (2007) Sextic spline solutions of fifth order boundary value problems. Appl Math Lett 20:591\u2013597","journal-title":"Appl Math Lett"},{"key":"838_CR38","first-page":"532","volume":"190","author":"SS Siddiqi","year":"2007","unstructured":"Siddiqi SS, Akram G, Malik SA (2007) Nonpolynomial sextic spline method for the solution along with convergence of linear special case fifth-order two-point boundary value problems. Appl Math Comput 190:532\u2013541","journal-title":"Appl Math Comput"},{"key":"838_CR39","volume-title":"Scattered data approximation","author":"H Wendland","year":"2005","unstructured":"Wendland H (2005) Scattered data approximation. Cambridge University Press, Cambridge"},{"issue":"11","key":"838_CR40","doi-asserted-by":"publisher","first-page":"1640","DOI":"10.1016\/j.enganabound.2012.04.012","volume":"36","author":"G Yao","year":"2012","unstructured":"Yao G, \u0160arler B (2012) Assessment of global and local meshless methods based on collocation with radial basis functions for parabolic partial differential equations in three dimensions. Eng Anal Bound Elem 36(11):1640\u20131648","journal-title":"Eng Anal Bound Elem"},{"key":"838_CR41","doi-asserted-by":"publisher","first-page":"333","DOI":"10.1007\/s004660000181","volume":"26","author":"X Zhang","year":"2000","unstructured":"Zhang X, Song KZ, Liu X (2000) Meshless methods based on collocation with radial basis functions. Comput Mech 26:333\u2013343","journal-title":"Comput Mech"},{"key":"838_CR42","first-page":"260","volume":"208","author":"CG Zhu","year":"2009","unstructured":"Zhu CG, Wang RH (2009) Numerical solution of Burgers\u2019 equation by cubic B-spline quasi-interpolation. Appl Math Comput 208:260\u2013272","journal-title":"Appl Math Comput"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-019-0838-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s40314-019-0838-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-019-0838-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,3,20]],"date-time":"2020-03-20T00:14:57Z","timestamp":1584663297000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s40314-019-0838-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,3,21]]},"references-count":42,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2019,6]]}},"alternative-id":["838"],"URL":"https:\/\/doi.org\/10.1007\/s40314-019-0838-0","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,3,21]]},"assertion":[{"value":"14 August 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 March 2019","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 March 2019","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 March 2019","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"72"}}