{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,28]],"date-time":"2026-01-28T19:08:43Z","timestamp":1769627323615,"version":"3.49.0"},"reference-count":39,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2023,7,12]],"date-time":"2023-07-12T00:00:00Z","timestamp":1689120000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,7,12]],"date-time":"2023-07-12T00:00:00Z","timestamp":1689120000000},"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":[[2023,10]]},"DOI":"10.1007\/s12190-023-01895-3","type":"journal-article","created":{"date-parts":[[2023,7,12]],"date-time":"2023-07-12T20:31:11Z","timestamp":1689193871000},"page":"3601-3624","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["A numerical approach for singularly perturbed reaction diffusion type Volterra-Fredholm integro-differential equations"],"prefix":"10.1007","volume":"69","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6216-1032","authenticated-orcid":false,"given":"Muhammet Enes","family":"Durmaz","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,7,12]]},"reference":[{"issue":"1","key":"1895_CR1","doi-asserted-by":"publisher","first-page":"37","DOI":"10.18514\/MMN.2021.2930","volume":"22","author":"GM Amiraliyev","year":"2021","unstructured":"Amiraliyev, G.M., Durmaz, M.E., Kudu, M.: A numerical method for a second order singularly perturbed Fredholm integro-differential equation. Miskolc Math. Notes 22(1), 37\u201348 (2021). https:\/\/doi.org\/10.18514\/MMN.2021.2930","journal-title":"Miskolc Math. Notes"},{"key":"1895_CR2","doi-asserted-by":"publisher","first-page":"1485","DOI":"10.1016\/j.camwa.2008.10.097","volume":"57","author":"D Bahuguna","year":"2009","unstructured":"Bahuguna, D., Ujlayan, A., Pandey, D.N.: A comparative study of numerical methods for solving an integro-differential equation. Comput. Math. Appl. 57, 1485\u20131493 (2009). https:\/\/doi.org\/10.1016\/j.camwa.2008.10.097","journal-title":"Comput. Math. Appl."},{"issue":"2","key":"1895_CR3","doi-asserted-by":"publisher","first-page":"39","DOI":"10.1016\/S0893-9659(98)00146-3","volume":"12","author":"N Bellomo","year":"1999","unstructured":"Bellomo, N., Firmani, B., Guerri, L.: Bifurcation analysis for a nonlinear system of integro-differential equations modelling tumor-immune cells competition. Appl. Math. Lett. 12(2), 39\u201344 (1999). https:\/\/doi.org\/10.1016\/S0893-9659(98)00146-3","journal-title":"Appl. Math. Lett."},{"key":"1895_CR4","doi-asserted-by":"publisher","first-page":"52","DOI":"10.1016\/j.cam.2012.09.020","volume":"252","author":"MI Berenguer","year":"2013","unstructured":"Berenguer, M.I., G\u00e1mez, D., L\u00f3pezLinares, A.J.: Fixed point techniques and Schauder bases to approximate the solution of the first order nonlinear mixed Fredholm-Volterra integro-differential equation. J. Comput. Appl. Math. 252, 52\u201361 (2013). https:\/\/doi.org\/10.1016\/j.cam.2012.09.020","journal-title":"J. Comput. Appl. Math."},{"key":"1895_CR5","doi-asserted-by":"publisher","first-page":"205","DOI":"10.1007\/978-3-319-72456-0_11","volume-title":"Contemporary Computational Mathematics-A Celebration of the 80th Birthday of Ian Sloan","author":"H Brunner","year":"2018","unstructured":"Brunner, H.: Numerical analysis and computational solution of integro-differential equations. In: Dick, J., et al. (eds.) Contemporary Computational Mathematics-A Celebration of the 80th Birthday of Ian Sloan, pp. 205\u2013231. Springer, Cham (2018)"},{"issue":"2","key":"1895_CR6","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1515\/gmj-2022-2213","volume":"29","author":"M Cakir","year":"2022","unstructured":"Cakir, M., Gunes, B.: Exponentially fitted difference scheme for singularly perturbed mixed integro-differential equations. Georgian Math. J. 29(2), 193\u2013203 (2022). https:\/\/doi.org\/10.1515\/gmj-2022-2213","journal-title":"Georgian Math. J."},{"key":"1895_CR7","doi-asserted-by":"publisher","first-page":"113070","DOI":"10.1016\/j.cam.2020.113070","volume":"382","author":"H Dehestani","year":"2021","unstructured":"Dehestani, H., Ordokhani, Y., Razzaghi, M.: Combination of Lucas wavelets with Legendre-Gauss quadrature for fractional Fredholm-Volterra integro-differential equations. J. Comput. Appl. Math. 382, 113070 (2021). https:\/\/doi.org\/10.1016\/j.cam.2020.113070","journal-title":"J. Comput. Appl. Math."},{"key":"1895_CR8","doi-asserted-by":"publisher","first-page":"24","DOI":"10.1007\/s00009-020-01693-2","volume":"18","author":"ME Durmaz","year":"2021","unstructured":"Durmaz, M.E., Amiraliyev, G.M.: A robust numerical method for a singularly perturbed Fredholm integro-differential equation. Mediterr. J. Math. 18, 24 (2021). https:\/\/doi.org\/10.1007\/s00009-020-01693-2","journal-title":"Mediterr. J. Math."},{"key":"1895_CR9","doi-asserted-by":"publisher","first-page":"505","DOI":"10.1007\/s12190-022-01757-4","volume":"69","author":"ME Durmaz","year":"2023","unstructured":"Durmaz, M.E., Amirali, I., Amiraliyev, G.M.: An efficient numerical method for a singularly perturbed Fredholm integro-differential equation with integral boundary condition. J. Appl. Math. Comput. 69, 505\u2013528 (2023). https:\/\/doi.org\/10.1007\/s12190-022-01757-4","journal-title":"J. Appl. Math. Comput."},{"key":"1895_CR10","doi-asserted-by":"publisher","first-page":"114327","DOI":"10.1016\/j.cam.2022.114327","volume":"412","author":"ME Durmaz","year":"2022","unstructured":"Durmaz, M.E., Cakir, M., Amirali, I., Amiraliyev, G.M.: Numerical solution of singularly perturbed Fredholm integro-differential equations by homogeneous second order difference method. J. Comput. Appl. Math. 412, 114327 (2022). https:\/\/doi.org\/10.1016\/j.cam.2022.114327","journal-title":"J. Comput. Appl. Math."},{"issue":"2","key":"1895_CR11","doi-asserted-by":"publisher","first-page":"326","DOI":"10.15672\/hujms.1050505","volume":"52","author":"ME Durmaz","year":"2023","unstructured":"Durmaz, M.E., Yapman, O., Kudu, M., Amiraliyev, G.M.: An efficient numerical method for a singularly perturbed Volterra-Fredholm integro-differential equation. Hacettepe J. Math. Stat. 52(2), 326\u2013339 (2023). https:\/\/doi.org\/10.15672\/hujms.1050505","journal-title":"Hacettepe J. Math. Stat."},{"key":"1895_CR12","doi-asserted-by":"publisher","DOI":"10.1201\/9781482285727","volume-title":"Robust Computational Techniques for Boundary Layers","author":"PA Farrell","year":"2000","unstructured":"Farrell, P.A., Hegarty, A.F., Miller, J.J.H., O\u2019Riordan, E., Shishkin, G.I.: Robust Computational Techniques for Boundary Layers. Chapman Hall\/CRC, New York (2000)"},{"key":"1895_CR13","doi-asserted-by":"publisher","first-page":"1865","DOI":"10.1007\/s12190-022-01817-9","volume":"69","author":"B Ghosh","year":"2023","unstructured":"Ghosh, B., Mohapatra, J.: Analysis of finite difference schemes for Volterra integro-differential equations involving arbitrary order derivatives. J. Appl. Math. Comput. 69, 1865\u20131886 (2023). https:\/\/doi.org\/10.1007\/s12190-022-01817-9","journal-title":"J. Appl. Math. Comput."},{"key":"1895_CR14","doi-asserted-by":"publisher","first-page":"851","DOI":"10.1007\/s40995-023-01446-2","volume":"47","author":"B Ghosh","year":"2023","unstructured":"Ghosh, B., Mohapatra, J.: An iterative scheme for solving arbitrary-order nonlinear Volterra integro-differential equations involving delay. Iran J Sci. 47, 851\u2013861 (2023). https:\/\/doi.org\/10.1007\/s40995-023-01446-2","journal-title":"Iran J Sci."},{"issue":"6","key":"1895_CR15","doi-asserted-by":"publisher","first-page":"692","DOI":"10.17516\/1997-1397-2018-11-6-692-701","volume":"11","author":"AA Hamoud","year":"2018","unstructured":"Hamoud, A.A., Ghadle, K.P.: Existence and uniqueness of the solution for Volterra Fredholm integro-differential equations. J. Sib. Fed. Univ. Math. Phys. 11(6), 692\u2013701 (2018). https:\/\/doi.org\/10.17516\/1997-1397-2018-11-6-692-701","journal-title":"J. Sib. Fed. Univ. Math. Phys."},{"key":"1895_CR16","doi-asserted-by":"publisher","first-page":"122","DOI":"10.1016\/j.apnum.2018.10.003","volume":"136","author":"E Hesameddini","year":"2019","unstructured":"Hesameddini, E., Shahbazi, M.: Solving multipoint problems with linear Volterra-Fredholm integro-differential equations of the neutral type using Bernstein polynomials method. Appl. Numer. Math. 136, 122\u2013138 (2019). https:\/\/doi.org\/10.1016\/j.apnum.2018.10.003","journal-title":"Appl. Numer. Math."},{"key":"1895_CR17","doi-asserted-by":"publisher","first-page":"3641","DOI":"10.1016\/j.amc.2010.09.059","volume":"217","author":"MK Kadalbajoo","year":"2010","unstructured":"Kadalbajoo, M.K., Gupta, V.: A brief survey on numerical methods for solving singularly perturbed problems. Appl. Math. Comput. 217, 3641\u20133716 (2010). https:\/\/doi.org\/10.1016\/j.amc.2010.09.059","journal-title":"Appl. Math. Comput."},{"key":"1895_CR18","doi-asserted-by":"publisher","DOI":"10.13140\/RG.2.2.10845.44000","author":"A Keimer","year":"2020","unstructured":"Keimer, A., Pflug, L.: Modeling infectious diseases using integro-differential equations: optimal control strategies for policy decisions and Applications in COVID-19. Res. Gate (2020). https:\/\/doi.org\/10.13140\/RG.2.2.10845.44000","journal-title":"Res. Gate"},{"key":"1895_CR19","doi-asserted-by":"publisher","first-page":"379","DOI":"10.1016\/j.cam.2016.06.018","volume":"308","author":"M Kudu","year":"2016","unstructured":"Kudu, M., Amirali, I., Amiraliyev, G.M.: A finite-difference method for a singularly perturbed delay integro-differential equation. J. Comput. Appl. Math. 308, 379\u2013390 (2016). https:\/\/doi.org\/10.1016\/j.cam.2016.06.018","journal-title":"J. Comput. Appl. Math."},{"issue":"3","key":"1895_CR20","doi-asserted-by":"publisher","first-page":"62","DOI":"10.33899\/EDUSJ.2012.59190","volume":"25","author":"AH Mahmood","year":"2012","unstructured":"Mahmood, A.H., Sadoon, L.H.: Existence of a solution of a certain Volterra-Fredholm integro differential equations. J. Educ. Sci. 25(3), 62\u201367 (2012). https:\/\/doi.org\/10.33899\/EDUSJ.2012.59190","journal-title":"J. Educ. Sci."},{"key":"1895_CR21","doi-asserted-by":"publisher","DOI":"10.3390\/axioms11020069","author":"E Messina","year":"2022","unstructured":"Messina, E., Pezzella, M., Vecchio, A.: Positive numerical approximation of an integro-differential epidemic model. Axioms (2022). https:\/\/doi.org\/10.3390\/axioms11020069","journal-title":"Axioms"},{"key":"1895_CR22","doi-asserted-by":"publisher","DOI":"10.1142\/8410","volume-title":"Fitted Numerical Methods for Singular Perturbation Problems: Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions","author":"JJH Miller","year":"2012","unstructured":"Miller, J.J.H., O\u2019Riordan, E., Shishkin, G.I.: Fitted Numerical Methods for Singular Perturbation Problems: Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions, Rev World Scientific, Singapore (2012)","edition":"Rev"},{"key":"1895_CR23","volume-title":"Introduction to Perturbation Techniques","author":"AH Nayfeh","year":"1993","unstructured":"Nayfeh, A.H.: Introduction to Perturbation Techniques. Wiley, New York (1993)"},{"key":"1895_CR24","volume-title":"Singular Perturbations Methods for Ordinary Differential Equations","author":"RE O\u2019Malley","year":"2013","unstructured":"O\u2019Malley, R.E.: Singular Perturbations Methods for Ordinary Differential Equations. Springer, New York (2013)"},{"key":"1895_CR25","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1007\/s00009-021-01873-8","volume":"18","author":"A Panda","year":"2021","unstructured":"Panda, A., Mohapatra, J., Amirali, I.: A second-order post-processing technique for singularly perturbed Volterra integro-differential equations. Mediterr. J. Math. 18, 231 (2021). https:\/\/doi.org\/10.1007\/s00009-021-01873-8","journal-title":"Mediterr. J. Math."},{"key":"1895_CR26","doi-asserted-by":"publisher","first-page":"198","DOI":"10.1007\/s00009-023-02402-5","volume":"20","author":"A Panda","year":"2023","unstructured":"Panda, A., Mohapatra, J.: A robust finite difference method for the solutions of singularly perturbed Fredholm integro-differential equations. Mediterr. J. Math. 20, 198 (2023). https:\/\/doi.org\/10.1007\/s00009-023-02402-5","journal-title":"Mediterr. J. Math."},{"key":"1895_CR27","doi-asserted-by":"publisher","first-page":"160","DOI":"10.1016\/j.anucene.2014.12.009","volume":"78","author":"B Quintero-Leyva","year":"2015","unstructured":"Quintero-Leyva, B.: Numerical solution of the integro-differential equation of the point kinetics of nuclear reactors as an ODE. Ann. Nucl. Energy 78, 160\u2013165 (2015). https:\/\/doi.org\/10.1016\/j.anucene.2014.12.009","journal-title":"Ann. Nucl. Energy"},{"key":"1895_CR28","doi-asserted-by":"publisher","first-page":"299","DOI":"10.1007\/s00780-005-0153-z","volume":"9","author":"C Rama","year":"2005","unstructured":"Rama, C., Ekaterina, V.: Integro-differential equations for option prices in exponential Levy models. Finan. Stochast. 9, 299\u2013325 (2005). https:\/\/doi.org\/10.1007\/s00780-005-0153-z","journal-title":"Finan. Stochast."},{"key":"1895_CR29","doi-asserted-by":"publisher","first-page":"485","DOI":"10.1016\/j.amc.2003.08.021","volume":"156","author":"MT Rashed","year":"2004","unstructured":"Rashed, M.T.: Numerical solution of functional differential integral and integro-differential equations. Appl. Numer. Math. 156, 485\u2013492 (2004). https:\/\/doi.org\/10.1016\/j.amc.2003.08.021","journal-title":"Appl. Numer. Math."},{"key":"1895_CR30","doi-asserted-by":"publisher","first-page":"724","DOI":"10.1016\/j.cam.2015.10.013","volume":"296","author":"SY Reutskiy","year":"2016","unstructured":"Reutskiy, S.Y.: The backward substitution method for multipoint problems with linear Volterra-Fredholm integro-differential equations of the neutral type. J. Comput. Appl. Math. 296, 724\u2013738 (2016). https:\/\/doi.org\/10.1016\/j.cam.2015.10.013","journal-title":"J. Comput. Appl. Math."},{"key":"1895_CR31","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1016\/j.amc.2018.01.032","volume":"328","author":"N Rohaninasab","year":"2018","unstructured":"Rohaninasab, N., Maleknejad, K., Ezzati, R.: Numerical solution of high-order Volterra-Fredholm integro-differential equations by using Legendre collocation method. Appl. Math. Comput. 328, 171\u2013188 (2018). https:\/\/doi.org\/10.1016\/j.amc.2018.01.032","journal-title":"Appl. Math. Comput."},{"key":"1895_CR32","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-34467-4","volume-title":"Robust Numerical Methods for Singularly Perturbed Differential Equations","author":"HG Roos","year":"2008","unstructured":"Roos, H.G., Stynes, M., Tobiska, L.: Robust Numerical Methods for Singularly Perturbed Differential Equations. Springer-Verlag, Berlin Heidelberg (2008). https:\/\/doi.org\/10.1007\/978-3-540-34467-4"},{"key":"1895_CR33","doi-asserted-by":"publisher","DOI":"10.1201\/9780203908518","volume-title":"The Theory of Difference Schemes","author":"A A Samarskii","year":"2001","unstructured":"Samarskii, A. A.: The Theory of Difference Schemes. Marcell Dekker, Inc., New York (2001). https:\/\/doi.org\/10.1201\/9780203908518"},{"key":"1895_CR34","doi-asserted-by":"publisher","first-page":"410","DOI":"10.1016\/j.amc.2014.08.074","volume":"247","author":"M Turkyilmazoglu","year":"2014","unstructured":"Turkyilmazoglu, M.: High-order nonlinear Volterra-Fredholm-Hammerstein integro-differential equations and their effective computation. Appl. Math. Comput. 247, 410\u2013416 (2014). https:\/\/doi.org\/10.1016\/j.amc.2014.08.074","journal-title":"Appl. Math. Comput."},{"key":"1895_CR35","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1016\/S0096-3003(99)00225-8","volume":"118","author":"AM Wazwaz","year":"2001","unstructured":"Wazwaz, A.M.: A reliable algorithm for solving boundary value problems for higher-order integro-differential equations. Appl. Math. Comput. 118, 327\u2013342 (2001). https:\/\/doi.org\/10.1016\/S0096-3003(99)00225-8","journal-title":"Appl. Math. Comput."},{"key":"1895_CR36","doi-asserted-by":"crossref","unstructured":"Wazwaz, A. M. (Ed.): Volterra-Fredholm integro-differential equations. In: Linear and Nonlinear Integral Equations, pp. 285\u2013309. Springer, Berlin\/Heidelberg (2011)","DOI":"10.1007\/978-3-642-21449-3_9"},{"issue":"6","key":"1895_CR37","doi-asserted-by":"publisher","first-page":"1293","DOI":"10.1080\/00207160.2019.1614565","volume":"97","author":"O Yapman","year":"2020","unstructured":"Yapman, O., Amiraliyev, G.M.: A novel second-order fitted computational method for a singularly perturbed Volterra integro-differential equation. Int. J. Comput. Math. 97(6), 1293\u20131302 (2020). https:\/\/doi.org\/10.1080\/00207160.2019.1614565","journal-title":"Int. J. Comput. Math."},{"key":"1895_CR38","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1016\/j.cam.2019.01.026","volume":"355","author":"O Yapman","year":"2019","unstructured":"Yapman, O., Amiraliyev, G.M., Amirali, I.: Convergence analysis of fitted numerical method for a singularly perturbed nonlinear Volterra integro-differential equation with delay. J. Comput. Appl. Math. 355, 301\u2013309 (2019). https:\/\/doi.org\/10.1016\/j.cam.2019.01.026","journal-title":"J. Comput. Appl. Math."},{"issue":"9\u201310","key":"1895_CR39","doi-asserted-by":"publisher","first-page":"5349","DOI":"10.1016\/j.apm.2015.12.029","volume":"40","author":"S Yuzbasi","year":"2016","unstructured":"Yuzbasi, S.: Improved Bessel collocation method for linear Volterra integro-differential equations with piecewise intervals and application of a Volterra population model. Appl. Math. Model. 40(9\u201310), 5349\u20135363 (2016). https:\/\/doi.org\/10.1016\/j.apm.2015.12.029","journal-title":"Appl. Math. Model."}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-023-01895-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-023-01895-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-023-01895-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,27]],"date-time":"2023-09-27T14:31:48Z","timestamp":1695825108000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-023-01895-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,12]]},"references-count":39,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2023,10]]}},"alternative-id":["1895"],"URL":"https:\/\/doi.org\/10.1007\/s12190-023-01895-3","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"value":"1598-5865","type":"print"},{"value":"1865-2085","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,7,12]]},"assertion":[{"value":"18 May 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 June 2023","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 June 2023","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 July 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 author declares that there is no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}