{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,4]],"date-time":"2026-06-04T13:56:18Z","timestamp":1780581378682,"version":"3.54.1"},"reference-count":37,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2022,5,5]],"date-time":"2022-05-05T00:00:00Z","timestamp":1651708800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2022,5,5]],"date-time":"2022-05-05T00:00:00Z","timestamp":1651708800000},"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":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2023,2]]},"DOI":"10.1007\/s12190-022-01743-w","type":"journal-article","created":{"date-parts":[[2022,5,5]],"date-time":"2022-05-05T14:16:37Z","timestamp":1651760197000},"page":"251-272","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["An approach based on fractional-order Lagrange polynomials for the numerical approximation of fractional order non-linear Volterra-Fredholm integro-differential equations"],"prefix":"10.1007","volume":"69","author":[{"given":"Saurabh","family":"Kumar","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0571-1488","authenticated-orcid":false,"given":"Vikas","family":"Gupta","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2022,5,5]]},"reference":[{"key":"1743_CR1","doi-asserted-by":"publisher","first-page":"479","DOI":"10.1016\/j.advwatres.2012.04.005","volume":"51","author":"DA Benson","year":"2013","unstructured":"Benson, D.A., Meerschaert, M.M., Revielle, J.: Fractional calculus in hydrologic modeling: a numerical perspective. Adv. Water Resour. 51, 479\u2013497 (2013)","journal-title":"Adv. Water Resour."},{"issue":"1","key":"1743_CR2","doi-asserted-by":"publisher","first-page":"15","DOI":"10.1115\/1.3101682","volume":"50","author":"YA Rossikhin","year":"1997","unstructured":"Rossikhin, Y.A., Shitikova, M.V.: Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Appl. Mech. Rev. 50(1), 15\u201367 (1997)","journal-title":"Appl. Mech. Rev."},{"key":"1743_CR3","doi-asserted-by":"publisher","first-page":"17659","DOI":"10.1007\/s00521-021-06354-3","volume":"33","author":"S Kumar","year":"2021","unstructured":"Kumar, S., Gupta, V.: An application of variational iteration method for solving fuzzy time-fractional diffusion equations. Neural Comput. Appl. 33, 17659\u201317668 (2021)","journal-title":"Neural Comput. Appl."},{"issue":"1","key":"1743_CR4","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1016\/0304-4076(95)01732-1","volume":"73","author":"RT Baillie","year":"1996","unstructured":"Baillie, R.T.: Long memory processes and fractional integration in econometrics. J. Econom. 73(1), 5\u201359 (1996)","journal-title":"J. Econom."},{"issue":"2","key":"1743_CR5","doi-asserted-by":"publisher","first-page":"105","DOI":"10.1615\/CritRevBiomedEng.v32.i2.10","volume":"32","author":"RL Magin","year":"2004","unstructured":"Magin, R.L.: Fractional calculus in bioengineering, part 2. Crit. Rev. Biomed. Eng. 32(2), 105\u2013193 (2004)","journal-title":"Crit. Rev. Biomed. Eng."},{"issue":"3","key":"1743_CR6","doi-asserted-by":"publisher","first-page":"447","DOI":"10.1002\/mrm.21453","volume":"59","author":"MG Hall","year":"2008","unstructured":"Hall, M.G., Barrick, T.R.: From diffusion-weighted MRI to anomalous diffusion imaging. Magn. Reson. Med. 59(3), 447\u2013455 (2008)","journal-title":"Magn. Reson. Med."},{"issue":"9\u201310","key":"1743_CR7","doi-asserted-by":"publisher","first-page":"1487","DOI":"10.1177\/1077546307087435","volume":"14","author":"GW Bohannan","year":"2008","unstructured":"Bohannan, G.W.: Analog fractional order controller in temperature and motor control applications. J. Vib. Control 14(9\u201310), 1487\u20131498 (2008)","journal-title":"J. Vib. Control"},{"key":"1743_CR8","unstructured":"Mainardi, F.: Fractional calculus: Some basic problems in continuum and statistical mechanics, arXiv preprint arXiv:1201.0863"},{"issue":"9","key":"1743_CR9","doi-asserted-by":"publisher","first-page":"2340","DOI":"10.1016\/j.sigpro.2005.10.017","volume":"86","author":"R Panda","year":"2006","unstructured":"Panda, R., Dash, M.: Fractional generalized splines and signal processing. Signal Process. 86(9), 2340\u20132350 (2006)","journal-title":"Signal Process."},{"key":"1743_CR10","unstructured":"Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, 1st Edition, Vol. 204 of North-Holland Mathematics Studies, Elsevier (North-Holland) Science, Amsterdam, (2006)"},{"key":"1743_CR11","unstructured":"Podlubny, I.: Fractional Differential Equations, 1st Edition, Vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego (1999)"},{"key":"1743_CR12","volume-title":"An introduction to the fractional calculus and fractional differential equations","author":"KS Miller","year":"1993","unstructured":"Miller, K.S., Ross, B.: An introduction to the fractional calculus and fractional differential equations. Wiley, New York, NY, USA (1993)"},{"key":"1743_CR13","first-page":"11","volume":"1","author":"NH Abel","year":"1823","unstructured":"Abel, N.H.: Solution de quelques problemesa laide d\u2019integrales d\u00e9finies, ed. Oeuvres Completes 1, 11\u201327 (1823)","journal-title":"Oeuvres Completes"},{"issue":"4","key":"1743_CR14","doi-asserted-by":"publisher","first-page":"5594","DOI":"10.3934\/math.2022309","volume":"7","author":"S Singh","year":"2022","unstructured":"Singh, S., Kumar, S., Metwali, M.M.A., Aldosary, S.F., Nisar, K.S.: An existence theorem for nonlinear functional Volterra integral equations via Petryshyn\u2019s fixed point theorem. AIMS Math. 7(4), 5594\u20135604 (2022)","journal-title":"AIMS Math."},{"key":"1743_CR15","doi-asserted-by":"publisher","DOI":"10.1002\/num.22697","author":"WK Williams","year":"2020","unstructured":"Williams, W.K., Vijaykumar, V., Udhayakumar, R., Panda, S.K., Nisar, K.S.: Existence and controllability of nonlocal mixed Volterra-Fredholm type fractional delay integro-differential equations of order $$1 < r < 2$$. Numer. Methods Partial Differ. Equ. (2020). https:\/\/doi.org\/10.1002\/num.22697","journal-title":"Numer. Methods Partial Differ. Equ."},{"key":"1743_CR16","doi-asserted-by":"publisher","DOI":"10.1002\/num.22772","author":"V Vijaykumar","year":"2021","unstructured":"Vijaykumar, V., Ravichandran, C., Nisar, K.S., Kucche, K.D.: New discussion on approximate controllability results for fractional Sobolev type Volterra-Fredholm integro-differential systems of order $$1 < r < 2$$. Numer. Partial Differ. Equ. (2021). https:\/\/doi.org\/10.1002\/num.22772","journal-title":"Numer. Partial Differ. Equ."},{"issue":"1","key":"1743_CR17","first-page":"754","volume":"182","author":"S Momani","year":"2006","unstructured":"Momani, S., Noor, M.A.: Numerical methods for fourth-order fractional integro-differential equations. Appl. Math. Comput. 182(1), 754\u2013760 (2006)","journal-title":"Appl. Math. Comput."},{"issue":"4","key":"1743_CR18","doi-asserted-by":"publisher","first-page":"1295","DOI":"10.1016\/j.cnsns.2008.01.010","volume":"14","author":"SS Ray","year":"2009","unstructured":"Ray, S.S.: Analytical solution for the space fractional diffusion equation by two-step Adomian decomposition method. Commun. Nonlinear Sci. Numer. Simul. 14(4), 1295\u20131306 (2009)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"issue":"5","key":"1743_CR19","first-page":"137","volume":"4","author":"MM Mustafa","year":"2014","unstructured":"Mustafa, M.M., Ghanim, I.N.: Numerical solution of linear Volterra-Fredholm integral equations using Lagrange polynomials. Math. Theory Model. 4(5), 137\u2013146 (2014)","journal-title":"Math. Theory Model."},{"issue":"49","key":"1743_CR20","first-page":"2443","volume":"5","author":"A Shahsavaran","year":"2011","unstructured":"Shahsavaran, A.: Lagrange functions method for solving nonlinear Hammerstein Fredholm-volterra integral equations. Appl. Math. Sci. 5(49), 2443\u20132450 (2011)","journal-title":"Appl. Math. Sci."},{"issue":"13","key":"1743_CR21","doi-asserted-by":"publisher","first-page":"2050","DOI":"10.1177\/1077546310395977","volume":"17","author":"A Saadatmandi","year":"2011","unstructured":"Saadatmandi, A., Dehghan, M.: A Legendre collocation method for fractional integro-differential equations. J. Vib. Control 17(13), 2050\u20132058 (2011)","journal-title":"J. Vib. Control"},{"issue":"5","key":"1743_CR22","doi-asserted-by":"publisher","first-page":"e1047","DOI":"10.1002\/cmm4.1047","volume":"1","author":"P Das","year":"2019","unstructured":"Das, P., Rana, S., Ramos, H.: Homotopy perturbation method for solving Caputo-type fractional-order Volterra-Fredholm integro-differential equations. Comput. Math. Methods 1(5), e1047 (2019)","journal-title":"Comput. Math. Methods"},{"issue":"3","key":"1743_CR23","doi-asserted-by":"publisher","first-page":"035232","DOI":"10.1063\/5.0032636","volume":"11","author":"S Ahsan","year":"2021","unstructured":"Ahsan, S., Nawaz, R., Akbar, M., Nisar, K.S., Abualnaja, K.M., Mahmoud, E.E., Abdel-Aty, A.H.: Numerical solution of two-dimensional fractional order Volterra integro-differential equations. AIP Adv. 11(3), 035232 (2021)","journal-title":"AIP Adv."},{"key":"1743_CR24","doi-asserted-by":"publisher","first-page":"8845491","DOI":"10.1155\/2020\/8845491","volume":"2020","author":"M Akbar","year":"2020","unstructured":"Akbar, M., Nawaz, R., Ahsan, S., Baleanu, D., Nisar, K.S.: Analytical solution of system of Volterra integral equations using OHAM. J. Math. 2020, 8845491 (2020)","journal-title":"J. Math."},{"key":"1743_CR25","doi-asserted-by":"publisher","first-page":"103453","DOI":"10.1016\/j.rinp.2020.103453","volume":"19","author":"M Akbar","year":"2020","unstructured":"Akbar, M., Nawaz, R., Ahsan, S., Nisar, K.S., Abdel-Aty, A.H., Eleuch, H.: New approach to approximate the solution for the system of fractional order Volterra integro-differential equations. Result Phys. 19, 103453 (2020)","journal-title":"Result Phys."},{"issue":"2","key":"1743_CR26","first-page":"468","volume":"214","author":"\u00dc Lepik","year":"2009","unstructured":"Lepik, \u00dc.: Solving fractional integral equations by the Haar wavelet method. Appl. Math. Comput. 214(2), 468\u2013478 (2009)","journal-title":"Appl. Math. Comput."},{"issue":"3","key":"1743_CR27","doi-asserted-by":"publisher","first-page":"535","DOI":"10.2478\/v10006-011-0042-x","volume":"21","author":"H Saeedi","year":"2011","unstructured":"Saeedi, H., Mollahasani, N., Moghadam, M., Chuev, G.: An operational Haar wavelet method for solving fractional Volterra integral equations. Int. J. Appl. Math. Comput. Sci. 21(3), 535\u2013547 (2011)","journal-title":"Int. J. Appl. Math. Comput. Sci."},{"issue":"6","key":"1743_CR28","doi-asserted-by":"publisher","first-page":"2333","DOI":"10.1016\/j.cnsns.2011.10.014","volume":"17","author":"L Zhu","year":"2012","unstructured":"Zhu, L., Fan, Q.: Solving fractional nonlinear Fredholm integro-differential equations by the second kind Chebyshev wavelet. Commun. Nonlinear Sci. Numer. Simul. 17(6), 2333\u20132341 (2012)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"issue":"3\u20134","key":"1743_CR29","first-page":"409","volume":"18","author":"MK Kadalbajoo","year":"2010","unstructured":"Kadalbajoo, M.K., Gupta, V.: Hybrid finite difference methods for solving modified Burgers and Burgers-Huxley equations. Neural Parallel Sci. Comput. 18(3\u20134), 409\u2013422 (2010)","journal-title":"Neural Parallel Sci. Comput."},{"issue":"4","key":"1743_CR30","doi-asserted-by":"publisher","first-page":"1825","DOI":"10.1016\/j.cnsns.2010.07.020","volume":"16","author":"V Gupta","year":"2011","unstructured":"Gupta, V., Kadalbajoo, M.K.: A singular perturbation approach to solve Burgers-Huxley equation via monotone finite difference scheme on layer-adaptive mesh. Commun. Nonlinear Sci. Numer. Simul. 16(4), 1825\u20131844 (2011)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"issue":"2","key":"1743_CR31","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1515\/jnma-2014-0056","volume":"24","author":"V Gupta","year":"2016","unstructured":"Gupta, V., Kadalbajoo, M.K.: Qualitative analysis and numerical solution of Burgers\u2019 equation via B-spline collocation with implicit Euler method on piecewise uniform mesh. J. Numer. Math. 24(2), 73\u201394 (2016)","journal-title":"J. Numer. Math."},{"issue":"1","key":"1743_CR32","doi-asserted-by":"publisher","first-page":"375","DOI":"10.1186\/1687-1847-2013-375","volume":"2013","author":"M Javidi","year":"2013","unstructured":"Javidi, M., Ahmad, B.: Numerical solution of fractional partial differential equations by numerical Laplace inversion technique. Adv. Differ. Equ. 2013(1), 375 (2013)","journal-title":"Adv. Differ. Equ."},{"issue":"1","key":"1743_CR33","first-page":"286","volume":"186","author":"ZM Odibat","year":"2007","unstructured":"Odibat, Z.M., Shawagfeh, N.T.: Generalized Taylor\u2019s formula. Appl. Math. Comput. 186(1), 286\u2013293 (2007)","journal-title":"Appl. Math. Comput."},{"issue":"3","key":"1743_CR34","doi-asserted-by":"publisher","first-page":"333","DOI":"10.12732\/ijam.v31i3.3","volume":"31","author":"AA Hamoud","year":"2018","unstructured":"Hamoud, A.A., Ghadle, K.P., Issa, G. M Sh. B.: Existence and uniqueness theorems for fractional Volterra-Fredholm integro-differential equations. Int. J. Appl. Math. 31(3), 333\u2013348 (2018)","journal-title":"Int. J. Appl. Math."},{"issue":"3","key":"1743_CR35","doi-asserted-by":"publisher","first-page":"3846","DOI":"10.1007\/s40314-017-0547-5","volume":"37","author":"S Sabermahani","year":"2018","unstructured":"Sabermahani, S., Ordokhani, Y., Yousefi, S.: Numerical approach based on fractional-order Lagrange polynomials for solving a class of fractional differential equations. Comput. Appl. Math. 37(3), 3846\u20133868 (2018)","journal-title":"Comput. Appl. Math."},{"key":"1743_CR36","volume-title":"Introductory functional analysis with applications","author":"E Kreyszig","year":"1978","unstructured":"Kreyszig, E.: Introductory functional analysis with applications, vol. 1. Wiley, New York, USA (1978)"},{"issue":"3","key":"1743_CR37","doi-asserted-by":"publisher","first-page":"1154","DOI":"10.1016\/j.cnsns.2010.05.036","volume":"16","author":"H Saeedi","year":"2011","unstructured":"Saeedi, H., Moghadam, M.M., Mollahasani, N., Chuev, G.: A CAS wavelet method for solving nonlinear Fredholm integro-differential equations of fractional order. Commun. Nonlinear Sci. Numer. Simul. 16(3), 1154\u20131163 (2011)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-022-01743-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-022-01743-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-022-01743-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,1,27]],"date-time":"2023-01-27T23:02:26Z","timestamp":1674860546000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-022-01743-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,5]]},"references-count":37,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2023,2]]}},"alternative-id":["1743"],"URL":"https:\/\/doi.org\/10.1007\/s12190-022-01743-w","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"value":"1598-5865","type":"print"},{"value":"1865-2085","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,5,5]]},"assertion":[{"value":"11 October 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 April 2022","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 April 2022","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 May 2022","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"}}]}}