{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,26]],"date-time":"2025-09-26T13:24:40Z","timestamp":1758893080846,"version":"3.37.3"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2023,5,25]],"date-time":"2023-05-25T00:00:00Z","timestamp":1684972800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,5,25]],"date-time":"2023-05-25T00:00:00Z","timestamp":1684972800000},"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":[[2023,8]]},"DOI":"10.1007\/s12190-023-01873-9","type":"journal-article","created":{"date-parts":[[2023,5,25]],"date-time":"2023-05-25T13:02:21Z","timestamp":1685019741000},"page":"3171-3188","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Numerical solution of It\u00f4\u2013Volterra integral equations by the QR factorization method"],"prefix":"10.1007","volume":"69","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5635-9325","authenticated-orcid":false,"given":"M.","family":"Ahmadinia","sequence":"first","affiliation":[]},{"given":"H.","family":"Afshariarjmand","sequence":"additional","affiliation":[]},{"given":"M.","family":"Salehi","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,5,25]]},"reference":[{"key":"1873_CR1","doi-asserted-by":"crossref","unstructured":"Qi, H., Meng, X. Dynamics of a stochastic predator-prey model with fear effect and hunting cooperation. J. Appl. Math. Comput. 69, 2077\u20132103 (2023)","DOI":"10.1007\/s12190-022-01746-7"},{"issue":"1","key":"1873_CR2","doi-asserted-by":"publisher","first-page":"437","DOI":"10.1007\/s12190-017-1114-3","volume":"57","author":"C Lu","year":"2018","unstructured":"Lu, C., Chen, J., Fan, X., Zhang, L.: Dynamics and simulations of a stochastic predator-prey model with infinite delay and impulsive perturbations. J. Appl. Math. Comput. 57(1), 437\u2013465 (2018)","journal-title":"J. Appl. Math. Comput."},{"issue":"1","key":"1873_CR3","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1007\/s12190-020-01490-w","volume":"67","author":"J Danane","year":"2021","unstructured":"Danane, J.: Stochastic predator-prey L\u00e9vy jump model with Crowley\u2013Martin functional response and stage structure. J. Appl. Math. Comput. 67(1), 41\u201367 (2021)","journal-title":"J. Appl. Math. Comput."},{"issue":"1","key":"1873_CR4","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1007\/s12190-013-0725-6","volume":"45","author":"M Liu","year":"2014","unstructured":"Liu, M.: Dynamics of a stochastic Lotka\u2013Volterra model with regime switching. J. Appl. Math. Comput. 45(1), 327\u2013349 (2014)","journal-title":"J. Appl. Math. Comput."},{"issue":"1","key":"1873_CR5","doi-asserted-by":"publisher","first-page":"785","DOI":"10.1007\/s12190-021-01504-1","volume":"67","author":"D Shangguan","year":"2021","unstructured":"Shangguan, D., Liu, Z., Wang, L., Tan, R.: A stochastic epidemic model with infectivity in incubation period and homestead-isolation on the susceptible. J. Appl. Math. Comput. 67(1), 785\u2013805 (2021)","journal-title":"J. Appl. Math. Comput."},{"issue":"1","key":"1873_CR6","doi-asserted-by":"publisher","first-page":"471","DOI":"10.1007\/s12190-020-01364-1","volume":"64","author":"W Yang","year":"2020","unstructured":"Yang, W., Lu, C.: Long time behavior of stochastic Lotka\u2013Volterra competitive system with general L\u00e9vy jumps. J. Appl. Math. Comput. 64(1), 471\u2013486 (2020)","journal-title":"J. Appl. Math. Comput."},{"key":"1873_CR7","doi-asserted-by":"crossref","unstructured":"Platen, E., Bruti-Liberati, N.: Numerical Solution of Stochastic Differential Equations with Jumps in Finance. Springer (2010)","DOI":"10.1007\/978-3-642-13694-8"},{"issue":"4\u20135","key":"1873_CR8","doi-asserted-by":"publisher","first-page":"593","DOI":"10.1515\/jiip.2011.057","volume":"19","author":"M Ehler","year":"2011","unstructured":"Ehler, M.: Shrinkage rules for variational minimization problems and applications to analytical ultracentrifugation. J. Inverse Ill-Posed Probl. 19(4\u20135), 593\u2013614 (2011)","journal-title":"J. Inverse Ill-Posed Probl."},{"key":"1873_CR9","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2020.113071","volume":"382","author":"AD Khalaf","year":"2021","unstructured":"Khalaf, A.D., Abouagwa, M., Mustafa, A., Wang, X.: Stochastic Volterra integral equations with jumps and the strong superconvergence of the Euler Maruyama approximation. J. Comput. Appl. Math. 382, 113071 (2021)","journal-title":"J. Comput. Appl. Math."},{"key":"1873_CR10","volume":"410","author":"F Mirzaee","year":"2021","unstructured":"Mirzaee, F., Solhi, E., Naserifar, S.: Approximate solution of stochastic Volterra integro-differential equations by using moving least squares scheme and spectral collocation method. Appl. Math. Comput. 410, 126447 (2021)","journal-title":"Appl. Math. Comput."},{"key":"1873_CR11","doi-asserted-by":"publisher","first-page":"275","DOI":"10.1016\/j.apnum.2020.11.013","volume":"161","author":"F Mirzaee","year":"2021","unstructured":"Mirzaee, F., Solhi, E., Samadyar, N.: Moving least squares and spectral collocation method to approximate the solution of stochastic Volterra\u2013Fredholm integral equations. Appl. Numer. Math. 161, 275\u2013285 (2021)","journal-title":"Appl. Numer. Math."},{"key":"1873_CR12","doi-asserted-by":"publisher","first-page":"369","DOI":"10.1016\/j.matcom.2023.01.009","volume":"207","author":"E Solhi","year":"2023","unstructured":"Solhi, E., Mirzaee, F., Naserifar, S.: Approximate solution of two dimensional linear and nonlinear stochastic It\u00f4\u2013Volterra integral equations via meshless scheme. Math. Comput. Simul. 207, 369\u2013387 (2023)","journal-title":"Math. Comput. Simul."},{"key":"1873_CR13","doi-asserted-by":"publisher","first-page":"74","DOI":"10.1016\/j.cam.2017.09.035","volume":"333","author":"M Saffarzadeh","year":"2018","unstructured":"Saffarzadeh, M., Loghmani, G.B., Heydari, M.: An iterative technique for the numerical solution of nonlinear stochastic It\u00f4\u2013Volterra integral equations. J. Comput. Appl. Math. 333, 74\u201386 (2018)","journal-title":"J. Comput. Appl. Math."},{"key":"1873_CR14","doi-asserted-by":"publisher","first-page":"182","DOI":"10.1016\/j.apnum.2019.07.010","volume":"146","author":"M Saffarzadeh","year":"2019","unstructured":"Saffarzadeh, M., Heydari, M., Loghmani, G.: Convergence analysis of an iterative numerical algorithm for solving nonlinear stochastic It\u00f4\u2013Volterra integral equations with $$m$$-dimensional Brownian motion. Appl. Numer. Math. 146, 182\u2013198 (2019)","journal-title":"Appl. Numer. Math."},{"key":"1873_CR15","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2020.113153","volume":"384","author":"F Mirzaee","year":"2021","unstructured":"Mirzaee, F., Alipour, S.: Quintic b-spline collocation method to solve n-dimensional stochastic It\u00f4\u2013Volterra integral equations. J. Comput. Appl. Math. 384, 113153 (2021)","journal-title":"J. Comput. Appl. Math."},{"issue":"1","key":"1873_CR16","doi-asserted-by":"publisher","first-page":"384","DOI":"10.1002\/mma.5890","volume":"43","author":"F Mirzaee","year":"2020","unstructured":"Mirzaee, F., Alipour, S.: An efficient cubic b-spline and bicubic b-spline collocation method for numerical solutions of multidimensional nonlinear stochastic quadratic integral equations. Math. Methods Appl. Sci. 43(1), 384\u2013397 (2020)","journal-title":"Math. Methods Appl. Sci."},{"issue":"1","key":"1873_CR17","volume":"33","author":"N Samadyar","year":"2020","unstructured":"Samadyar, N., Mirzaee, F.: Orthonormal Bernoulli polynomials collocation approach for solving stochastic It\u00f4\u2013Volterra integral equations of Abel type. Int. J. Numer.l Model.: Electron. Netw. Dev. Fields 33(1), e2688 (2020)","journal-title":"Int. J. Numer.l Model.: Electron. Netw. Dev. Fields"},{"key":"1873_CR18","doi-asserted-by":"publisher","first-page":"246","DOI":"10.1016\/j.enganabound.2018.05.006","volume":"100","author":"F Mirzaee","year":"2019","unstructured":"Mirzaee, F., Samadyar, N.: On the numerical solution of fractional stochastic integro-differential equations via meshless discrete collocation method based on radial basis functions. Eng. Anal. Bound. Elem. 100, 246\u2013255 (2019)","journal-title":"Eng. Anal. Bound. Elem."},{"key":"1873_CR19","doi-asserted-by":"publisher","first-page":"238","DOI":"10.1016\/j.matcom.2019.03.005","volume":"165","author":"R Zeghdane","year":"2019","unstructured":"Zeghdane, R.: Numerical solution of stochastic integral equations by using Bernoulli operational matrix. Math. Comput. Simul. 165, 238\u2013254 (2019)","journal-title":"Math. Comput. Simul."},{"key":"1873_CR20","doi-asserted-by":"publisher","DOI":"10.1016\/j.rinam.2022.100260","volume":"14","author":"F Sharafi","year":"2022","unstructured":"Sharafi, F., Basirat, B.: Numerical solution of nonlinear stochastic It\u00f4\u2013Volterra integral equation by stochastic modified hat function operational matrices. Results Appl. Math. 14, 100260 (2022)","journal-title":"Results Appl. Math."},{"key":"1873_CR21","doi-asserted-by":"publisher","first-page":"402","DOI":"10.1016\/j.jcp.2014.03.064","volume":"270","author":"MH Heydari","year":"2014","unstructured":"Heydari, M.H., Hooshmandasl, M.R., Maalek, F.M., Cattani, C.: A computational method for solving stochastic It\u00f4\u2013Volterra integral equations based on stochastic operational matrix for generalized hat basis functions. J. Comput. Phys. 270, 402\u2013415 (2014)","journal-title":"J. Comput. Phys."},{"key":"1873_CR22","doi-asserted-by":"publisher","first-page":"148","DOI":"10.1016\/j.jcp.2014.11.042","volume":"283","author":"MH Heydari","year":"2015","unstructured":"Heydari, M.H., Hooshmandasl, M.R., Cattani, C., Ghaini, F.M.M.: An efficient computational method for solving nonlinear stochastic It\u00f4 integral equations: application for stochastic problems in physics. J. Comput. Phys. 283, 148\u2013168 (2015)","journal-title":"J. Comput. Phys."},{"issue":"1","key":"1873_CR23","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1515\/gmj-2018-0009","volume":"27","author":"MH Heydari","year":"2020","unstructured":"Heydari, M.H., Hooshmandasl, M.R., Cattani, C.: Wavelets method for solving nonlinear stochastic It\u00f4\u2013Volterra integral equations. Georg. Math. J. 27(1), 81\u201395 (2020)","journal-title":"Georg. Math. J."},{"issue":"1","key":"1873_CR24","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00009-016-0820-7","volume":"14","author":"B Hashemi","year":"2017","unstructured":"Hashemi, B., Khodabin, M., Maleknejad, K.: Numerical solution based on hat functions for solving nonlinear stochastic It\u00f4\u2013Volterra integral equations driven by fractional Brownian motion. Mediterr. J. Math. 14(1), 1\u201315 (2017)","journal-title":"Mediterr. J. Math."},{"key":"1873_CR25","doi-asserted-by":"publisher","first-page":"254","DOI":"10.1016\/j.jcp.2015.05.051","volume":"289","author":"F Mohammadi","year":"2015","unstructured":"Mohammadi, F.: A wavelet-based computational method for solving stochastic It\u00f4\u2013Volterra integral equations. J. Comput. Phys. 289, 254\u2013265 (2015)","journal-title":"J. Comput. Phys."},{"issue":"3","key":"1873_CR26","doi-asserted-by":"publisher","first-page":"575","DOI":"10.1108\/MMMS-04-2018-0075","volume":"15","author":"F Mirzaee","year":"2019","unstructured":"Mirzaee, F., Samadyar, N.: Application of Bernoulli wavelet method for estimating a solution of linear stochastic It\u00f4\u2013Volterra integral equations. Multidiscip. Model. Mater. Struct. 15(3), 575\u2013598 (2019)","journal-title":"Multidiscip. Model. Mater. Struct."},{"issue":"4","key":"1873_CR27","doi-asserted-by":"publisher","first-page":"1410","DOI":"10.1002\/mma.4671","volume":"41","author":"F Mirzaee","year":"2018","unstructured":"Mirzaee, F., Samadyar, N.: Numerical solution of nonlinear stochastic It\u00f4\u2013Volterra integral equations driven by fractional Brownian motion. Math. Methods Appl. Sci. 41(4), 1410\u20131423 (2018)","journal-title":"Math. Methods Appl. Sci."},{"key":"1873_CR28","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1016\/j.cam.2018.09.040","volume":"349","author":"F Mirzaee","year":"2019","unstructured":"Mirzaee, F., Alipour, S., Samadyar, N.: Numerical solution based on hybrid of block-pulse and parabolic functions for solving a system of nonlinear stochastic It\u00f4\u2013Volterra integral equations of fractional order. J. Comput. Appl. Math. 349, 157\u2013171 (2019)","journal-title":"J. Comput. Appl. Math."},{"key":"1873_CR29","doi-asserted-by":"publisher","first-page":"574","DOI":"10.1016\/j.cam.2017.09.005","volume":"330","author":"F Mirzaee","year":"2018","unstructured":"Mirzaee, F., Samadyar, N., Hoseini, S.F.: Euler polynomial solutions of nonlinear stochastic It\u00f4\u2013Volterra integral equations. J. Comput. Appl. Math. 330, 574\u2013585 (2018)","journal-title":"J. Comput. Appl. Math."},{"key":"1873_CR30","doi-asserted-by":"publisher","first-page":"783","DOI":"10.1016\/j.matcom.2021.02.003","volume":"185","author":"SAS Hashemi","year":"2021","unstructured":"Hashemi, S.A.S., Saeedi, H.: ADM-TF hybrid method for nonlinear It\u00f4\u2013Volterra integral equations. Math. Comput. Simul. 185, 783\u2013798 (2021)","journal-title":"Math. Comput. Simul."},{"issue":"2","key":"1873_CR31","doi-asserted-by":"publisher","first-page":"591","DOI":"10.1007\/s11075-019-00770-2","volume":"84","author":"M Ahmadinia","year":"2020","unstructured":"Ahmadinia, M., Afshariarjmand, H., Heydari, M.: Numerical solution of It\u00f4\u2013Volterra integral equation by least squares method. Numer. Algorithms 84(2), 591\u2013602 (2020)","journal-title":"Numer. Algorithms"},{"key":"1873_CR32","unstructured":"Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations, Applications of Mathematics. Springer (1999)"},{"key":"1873_CR33","doi-asserted-by":"crossref","unstructured":"Mason, J.C., Handscomb, D.: Chebyshev Polynomials. A CRC Press Company (2002)","DOI":"10.1201\/9781420036114"},{"issue":"3","key":"1873_CR34","doi-asserted-by":"publisher","first-page":"525","DOI":"10.1137\/S0036144500378302","volume":"43","author":"DJ Higham","year":"2001","unstructured":"Higham, D.J.: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43(3), 525\u2013546 (2001)","journal-title":"SIAM Rev."},{"key":"1873_CR35","doi-asserted-by":"crossref","unstructured":"Suli, E., Mayers, D.: An Introduction to Numerical Analysis. Cambridge University Press (2003)","DOI":"10.1017\/CBO9780511801181"},{"key":"1873_CR36","doi-asserted-by":"crossref","unstructured":"Klebaner, F.C.: Introduction to Stochastic Calculus with Applications. Imperial College Press (1998)","DOI":"10.1142\/p110"}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-023-01873-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-023-01873-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-023-01873-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,7,19]],"date-time":"2023-07-19T19:25:25Z","timestamp":1689794725000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-023-01873-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,25]]},"references-count":36,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2023,8]]}},"alternative-id":["1873"],"URL":"https:\/\/doi.org\/10.1007\/s12190-023-01873-9","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"type":"print","value":"1598-5865"},{"type":"electronic","value":"1865-2085"}],"subject":[],"published":{"date-parts":[[2023,5,25]]},"assertion":[{"value":"25 January 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 April 2023","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 May 2023","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 May 2023","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 report no declarations of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}