{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,20]],"date-time":"2026-01-20T12:01:50Z","timestamp":1768910510921,"version":"3.49.0"},"reference-count":24,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2024,8,3]],"date-time":"2024-08-03T00:00:00Z","timestamp":1722643200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,8,3]],"date-time":"2024-08-03T00:00:00Z","timestamp":1722643200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11271100"],"award-info":[{"award-number":["11271100"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2024,12]]},"DOI":"10.1007\/s12190-024-02210-4","type":"journal-article","created":{"date-parts":[[2024,8,3]],"date-time":"2024-08-03T15:02:09Z","timestamp":1722697329000},"page":"5979-5992","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A sixth order kernel functions approach for nonlinear fourth order boundary value problems"],"prefix":"10.1007","volume":"70","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2103-5678","authenticated-orcid":false,"given":"F. Z.","family":"Geng","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"C. N.","family":"Li","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"X. Y.","family":"Wu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,8,3]]},"reference":[{"issue":"2","key":"2210_CR1","first-page":"1929","volume":"189","author":"MA Noor","year":"2007","unstructured":"Noor, M.A., Mohyud-Din, S.T.: Variational iteration technique for solving higher order boundary value problems. Appl. Math. Comput. 189(2), 1929\u20131942 (2007)","journal-title":"Appl. Math. Comput."},{"key":"2210_CR2","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1155\/2007\/98602","volume":"2007","author":"ST Mohyud-Din","year":"2007","unstructured":"Mohyud-Din, S.T., Noor, M.A.: Homotopy perturbation method for solving fourth-order boundry value problems. Math. Probl. Eng. 2007, 1\u201315 (2007)","journal-title":"Math. Probl. Eng."},{"issue":"3","key":"2210_CR3","doi-asserted-by":"publisher","first-page":"345","DOI":"10.1080\/00207160211928","volume":"79","author":"AM Wazwaz","year":"2002","unstructured":"Wazwaz, A.M.: The numerical solution of special fourth-order boundary value problems by the modified decomposition method. Int. J. Comput. Math. 79(3), 345\u2013356 (2002)","journal-title":"Int. J. Comput. Math."},{"key":"2210_CR4","doi-asserted-by":"publisher","first-page":"427","DOI":"10.1007\/s11075-017-0264-6","volume":"76","author":"QA Dang","year":"2017","unstructured":"Dang, Q.A., Dang, Q.L., Ngo, T.K.Q.: A novel efficient method for nonlinear boundary value problems. Numer. Algorithms 76, 427\u2013439 (2017)","journal-title":"Numer. Algorithms"},{"key":"2210_CR5","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2023.108813","volume":"146","author":"QA Dang","year":"2023","unstructured":"Dang, Q.A., Nguyen, T.T.H.: Numerical method of sixth order convergence for solving a fourth order nonlinear boundary value problem. Appl. Math. Lett. 146, 108813 (2023)","journal-title":"Appl. Math. Lett."},{"key":"2210_CR6","first-page":"8447","volume":"217","author":"WK Zahra","year":"2011","unstructured":"Zahra, W.K.: A smooth approximation based on exponential spline solutions for nonlinear fourth order two point boundary value problems. Appl. Math. Comput. 217, 8447\u20138457 (2011)","journal-title":"Appl. Math. Comput."},{"key":"2210_CR7","doi-asserted-by":"publisher","first-page":"1167","DOI":"10.1016\/j.jmaa.2006.05.011","volume":"327","author":"FZ Geng","year":"2007","unstructured":"Geng, F.Z., Cui, M.G.: Solving a nonlinear system of second order boundary value problems. J. Math. Anal. Appl. 327, 1167\u20131181 (2007)","journal-title":"J. Math. Anal. Appl."},{"key":"2210_CR8","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2021.107525","volume":"122","author":"FZ Geng","year":"2021","unstructured":"Geng, F.Z., Wu, X.Y.: Reproducing kernel functions based univariate spline interpolation. Appl. Math. Lett. 122, 107525 (2021)","journal-title":"Appl. Math. Lett."},{"key":"2210_CR9","volume":"397","author":"FZ Geng","year":"2021","unstructured":"Geng, F.Z., Wu, X.Y.: Reproducing kernel function-based Filon and Levin methods for solving highly oscillatory integral. Appl. Math. Comput. 397, 125980 (2021)","journal-title":"Appl. Math. Comput."},{"key":"2210_CR10","doi-asserted-by":"publisher","first-page":"1269","DOI":"10.1002\/num.22578","volume":"37","author":"FZ Geng","year":"2021","unstructured":"Geng, F.Z., Wu, X.Y.: A Kernel functions-based approach for distributed order diffusion equations. Numer. Methods. Part. Differ. Equ. 37, 1269\u20131281 (2021)","journal-title":"Numer. Methods. Part. Differ. Equ."},{"key":"2210_CR11","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2022.108318","volume":"134","author":"FZ Geng","year":"2022","unstructured":"Geng, F.Z., Wu, X.Y.: A novel kernel functions algorithm for solving impulsive boundary value problems. Appl. Math. Lett. 134, 108318 (2022)","journal-title":"Appl. Math. Lett."},{"key":"2210_CR12","doi-asserted-by":"publisher","first-page":"20","DOI":"10.1007\/s10092-023-00516-6","volume":"60","author":"FZ Geng","year":"2023","unstructured":"Geng, F.Z., Wu, X.Y.: Reproducing kernel-based piecewise methods for efficiently solving oscillatory systems of second-order initial value. Calcolo 60, 20 (2023)","journal-title":"Calcolo"},{"key":"2210_CR13","doi-asserted-by":"publisher","first-page":"206","DOI":"10.1016\/j.apnum.2021.10.008","volume":"172","author":"XY Li","year":"2022","unstructured":"Li, X.Y., Wang, H.L., Wu, B.Y.: A stable and efficient technique for linear boundary value problems by applying kernel functions. Appl. Numer. Math. 172, 206\u2013214 (2022)","journal-title":"Appl. Numer. Math."},{"key":"2210_CR14","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2024.109073","volume":"153","author":"XY Li","year":"2024","unstructured":"Li, X.Y., Wu, B.Y.: Mittag\u2013Leffler kernel-based oversampling collocation method for fractional initial value problems with contaminated data. Appl. Math. Lett. 153, 109073 (2024)","journal-title":"Appl. Math. Lett."},{"key":"2210_CR15","doi-asserted-by":"publisher","DOI":"10.1088\/1402-4896\/ad1eab","volume":"99","author":"T Amoozad","year":"2024","unstructured":"Amoozad, T., Allahviranloo, T., Abbasbandy, S., Malkhalifeh, M.R.: Application of the reproducing kernel method for solving linear Volterra integral equations with variable coefficients. Phys. Scr. 99, 025246 (2024)","journal-title":"Phys. Scr."},{"key":"2210_CR16","first-page":"583","volume":"361","author":"H Sahihi","year":"2019","unstructured":"Sahihi, H., Allahviranloo, T., Abbasbandy, S.: Computational method based on reproducing kernel for solving singularly perturbed differential\u2013difference equations with a delay. Appl. Math. Comput. 361, 583\u2013598 (2019)","journal-title":"Appl. Math. Comput."},{"key":"2210_CR17","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1080\/00207160.2021.1897111","volume":"98","author":"Z Chen","year":"2021","unstructured":"Chen, Z., Jiang, W., Du, H.: A new reproducing kernel method for Duffing equations. Int. J. Comput. Math. 98, 1\u201313 (2021)","journal-title":"Int. J. Comput. Math."},{"key":"2210_CR18","doi-asserted-by":"crossref","unstructured":"Abu Arqub, O., Mezghiche, R., Maayah, B.: Fuzzy M-fractional integrodifferential models: theoretical existence and uniqueness results, and approximate solutions utilizing the Hilbert reproducing kernel algorithm. Front. Phys. 11, 1252919 (2023)","DOI":"10.3389\/fphy.2023.1252919"},{"key":"2210_CR19","doi-asserted-by":"publisher","DOI":"10.1016\/j.rinp.2022.105225","volume":"35","author":"N Attia","year":"2022","unstructured":"Attia, N., Akgul, A., Seba, D., Nour, A., Riaz, M.B.: Reproducing kernel Hilbert space method for solving fractal fractional differential equations. Results Phys. 35, 105225 (2022)","journal-title":"Results Phys."},{"key":"2210_CR20","doi-asserted-by":"publisher","first-page":"3631","DOI":"10.1007\/s12190-021-01685-9","volume":"68","author":"J Zhang","year":"2022","unstructured":"Zhang, J., Niu, J.: Lobatto-reproducing kernel method for solving a linear system of second order boundary value problems. J. Appl. Math. Comput. 68, 3631\u20133653 (2022)","journal-title":"J. Appl. Math. Comput."},{"key":"2210_CR21","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1002\/mma.9640","volume":"47","author":"RM Zhang","year":"2023","unstructured":"Zhang, R.M., Lin, Y.Z.: A new algorithm of boundary value problems based on improved wavelet basis and the reproducing kernel theory. Math. Method Appl. Sci. 47, 47\u201357 (2023)","journal-title":"Math. Method Appl. Sci."},{"key":"2210_CR22","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2022.108475","volume":"136","author":"M Xu","year":"2023","unstructured":"Xu, M., Zhang, L., Tohidi, E.: An efficient method based on least-squares technique for interface problems. Appl. Math. Lett. 136, 108475 (2023)","journal-title":"Appl. Math. Lett."},{"key":"2210_CR23","doi-asserted-by":"publisher","first-page":"2100","DOI":"10.1080\/00207160.2018.1544367","volume":"96","author":"YL Wang","year":"2019","unstructured":"Wang, Y.L., Jia, L.N., Zhang, H.L.: Numerical solution for a class of space-time fractional equation by the piecewise reproducing kernel method. Int. J. Comput. Math. 96, 2100\u20132111 (2019)","journal-title":"Int. J. Comput. Math."},{"key":"2210_CR24","first-page":"163","volume":"213","author":"FZ Geng","year":"2009","unstructured":"Geng, F.Z.: A new reproducing kernel Hilbert space method for solving nonlinear fourth-order boundary value problems. Appl. Math. Comput. 213, 163\u2013169 (2009)","journal-title":"Appl. Math. Comput."}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-024-02210-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-024-02210-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-024-02210-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,11,29]],"date-time":"2024-11-29T14:03:48Z","timestamp":1732889028000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-024-02210-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,3]]},"references-count":24,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2024,12]]}},"alternative-id":["2210"],"URL":"https:\/\/doi.org\/10.1007\/s12190-024-02210-4","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"value":"1598-5865","type":"print"},{"value":"1865-2085","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,8,3]]},"assertion":[{"value":"25 April 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 June 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 July 2024","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 August 2024","order":4,"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"}}]}}