Exponentially fitted difference scheme for singularly perturbed mixed integro-differential equations. (English) Zbl 1491.65167
Georgian Math. J. 29, No. 2, 193-203 (2022); correction ibid. 30, No. 1, 159 (2023).
Summary: In this study, singularly perturbed mixed integro-differential equations (SPMIDEs) are taken into account. First, the asymptotic behavior of the solution is investigated. Then, by using interpolating quadrature rules and an exponential basis function, the finite difference scheme is constructed on a uniform mesh. The stability and convergence of the proposed scheme are analyzed in the discrete maximum norm. Some numerical examples are solved, and numerical outcomes are obtained.
MSC:
| 65R20 | Numerical methods for integral equations |
| 45J05 | Integro-ordinary differential equations |
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
| 65L11 | Numerical solution of singularly perturbed problems involving ordinary differential equations |
| 65L12 | Finite difference and finite volume methods for ordinary differential equations |
| 65L20 | Stability and convergence of numerical methods for ordinary differential equations |
Keywords:
difference scheme; error estimate; Fredholm integro-differential equation; singular perturbation; uniform mesh; Volterra integro-differential equationReferences:
| [1] | O. Abu Arqub, An iterative method for solving fourth-order boundary value problems of mixed type integro-differential equations, J. Comput. Anal. Appl. 18 (2015), no. 5, 857-874. · Zbl 1329.65176 |
| [2] | M. Al-Smadi, O. Abu Arqub and S. Momani, A computational method for two-point boundary value problems of fourth-order mixed integrodifferential equations, Math. Probl. Eng. 2013 (2013), Article ID 832074. · Zbl 1299.65297 |
| [3] | G. M. Amiraliyev, M. E. Durmaz and M. Kudu, Uniform convergence results for singularly perturbed Fredholm integro-differential equation, J. Math. Anal. 9 (2018), no. 6, 55-64. |
| [4] | G. M. Amiraliyev, M. E. Durmaz and M. Kudu, Fitted second order numerical method for a singularly perturbed Fredholm integro-differential equation, Bull. Belg. Math. Soc. Simon Stevin 27 (2020), no. 1, 71-88. · Zbl 1442.65136 |
| [5] | G. M. Amiraliyev and Y. D. Mamedov, Difference schemes on the uniform mesh for singular perturbed pseudo-parabolic equations, Turkish J. Math. 19 (1995), no. 3, 207-222. · Zbl 0859.65088 |
| [6] | S. Amiri, M. Hajipour and D. Baleanu, A spectral collocation method with piecewise trigonometric basis functions for nonlinear Volterra-Fredholm integral equations, Appl. Math. Comput. 370 (2020), Article ID 124915. · Zbl 1433.65347 |
| [7] | E. Banifatemi, M. Razzaghi and S. Yousefi, Two-dimensional Legendre wavelets method for the mixed Volterra-Fredholm integral equations, J. Vib. Control 13 (2007), no. 11, 1667-1675. · Zbl 1182.65199 |
| [8] | M. S. Bani Issa, A. A. Hamoud, K. P. Ghadle and Giniswamy, Hybrid method for solving nonlinear Volterra-Fredholm integro differential equations, J. Math. Comput. Sci. 7 (2017), no. 4, 625-641. |
| [9] | H. Beiglo and M. Gachpazan, Numerical solution of nonlinear mixed Volterra-Fredholm integral equations in complex plane via PQWs, Appl. Math. Comput. 369 (2020), Article ID 124828. · Zbl 1433.65348 |
| [10] | N. Bellomo, B. Firmani and L. Guerri, Bifurcation analysis for a nonlinear system of integro-differential equations modelling tumor-immune cells competition, Appl. Math. Lett. 12 (1999), no. 2, 39-44. · Zbl 0932.92016 |
| [11] | M. I. Berenguer, D. Gámez and A. J. López Linares, 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 (2013), 52-61. · Zbl 1288.65183 |
| [12] | J. Biazar, H. Ghazvini and M. Eslami, He’s homotopy perturbation method for systems of integro-differential equations, Chaos Solitons Fractals 39 (2009), no. 3, 1253-1258. · Zbl 1197.65106 |
| [13] | T. A. Burton, Volterra Integral and Differential Equations, 2nd ed., Math. Sci. Eng. 202, Elsevier, Amsterdam, 2005. · Zbl 1075.45001 |
| [14] | E. Cimen, A computational method for Volterra-integro differential equation, Erzincan Univ. J. Sci. Technol. 11 (2018), no. 3, 347-352. |
| [15] | E. Cimen and M. Cakir, A uniform numerical method for solving singularly perturbed Fredholm integro-differential problem, Comput. Appl. Math. 40 (2021), no. 2, Paper No. 42. · Zbl 1476.65336 |
| [16] | E. Cimen and K. Enterili, An alternative method for numerical solution of Fredholm integro differential equation (in Turkish), Erzincan Univ. J. Sci. Technol. 13 (2020), no. 1, 46-53. |
| [17] | R. M. Colombo, M. Garavello, F. Marcellini and E. Rossi, An age and space structured SIR model describing the Covid-19 pandemic, J. Math. Ind. 10 (2020), Paper No. 22. · Zbl 1469.92107 |
| [18] | P. Darania and K. Ivaz, Numerical solution of nonlinear Volterra-Fredholm integro-differential equations, Comput. Math. Appl. 56 (2008), no. 9, 2197-2209. · Zbl 1165.65404 |
| [19] | O. Diekmann, Thresholds and travelling waves for the geographical spread of infection, J. Math. Biol. 6 (1978), no. 2, 109-130. · Zbl 0415.92020 |
| [20] | M. E. Durmaz and G. M. Amiraliyev, A robust numerical method for a singularly perturbed Fredholm integro-differential equation, Mediterr. J. Math. 18 (2021), no. 1, Paper No. 24. · Zbl 1461.65216 |
| [21] | F. S. Fadhel, A. K. O. Mezaal and S. H. Salih, Approximate solution of the linear mixed volterrafredholm integro differential equations of second kind by using variational iteration method, Al-Mustansiriyah J. Sci. 24 (2013), no. 5, 137-146. |
| [22] | A. A. Hamoud and K. P. Ghadle, The combined modified Laplace with Adomian decomposition method for solving the nonlinear Volterra-Fredholm integro differential equations, J. Korean Soc. Ind. Appl. Math. 21 (2017), no. 1, 17-28. · Zbl 1383.65157 |
| [23] | A. A. Hamoud and K. P. Ghadle, Existence and uniqueness of the solution for Volterra-Fredholm integro-differential equations, Zh. Sib. Fed. Univ. Mat. Fiz. 11 (2018), no. 6, 692-701. · Zbl 1532.65123 |
| [24] | M. Inc and Y. Cherruault, A reliable method for obtaining approximate solutions of linear and nonlinear Volterra-Freholm integro-differential equations, Kybernetes 34 (2005), no. 7-8, 1034-1048. · Zbl 1117.65369 |
| [25] | B. C. Iragi and J. B. Munyakazi, New parameter-uniform discretisations of singularly perturbed Volterra integro-differential equations, Appl. Math. Inf. Sci. 12 (2018), no. 3, 517-527. |
| [26] | B. C. Iragi and J. B. Munyakazi, A uniformly convergent numerical method for a singularly perturbed Volterra integro-differential equation, Int. J. Comput. Math. 97 (2020), no. 4, 759-771. · Zbl 1480.65377 |
| [27] | B. S. H. Kashkaria and M. I. Syam, Evolutionary computational intelligence in solving a class of nonlinear Volterra-Fredholm integro-differential equations, J. Comput. Appl. Math. 311 (2017), 314-323. · Zbl 1416.65539 |
| [28] | J.-P. Kauthen, Continuous time collocation methods for Volterra-Fredholm integral equations, Numer. Math. 56 (1989), no. 5, 409-424. · Zbl 0662.65116 |
| [29] | A. A. Khajehnasiri, Numerical solution of nonlinear 2D Volterra-Fredholm integro-differential equations by two-dimensional triangular function, Int. J. Appl. Comput. Math. 2 (2016), no. 4, 575-591. · Zbl 1420.65138 |
| [30] | F. Köhler-Rieper, C. H. F. Röhl and E. De Micheli, A novel deterministic forecast model for the Covid-19 epidemic based on a single ordinary integro-differential equations, Eur. Phys. J. Plus 135 (2020), no. 7, 1-19. |
| [31] | M. Kudu, I. Amirali and G. M. Amiraliyev, A finite-difference method for a singularly perturbed delay integro-differential equation, J. Comput. Appl. Math. 308 (2016), 379-390. · Zbl 1346.65076 |
| [32] | K. Maleknejad, B. Basirat and E. Hashemizadeh, A Bernstein operational matrix approach for solving a system of high order linear Volterra-Fredholm integro-differential equations, Math. Comput. Modelling 55 (2012), no. 3-4, 1363-1372. · Zbl 1255.65245 |
| [33] | D. A. Maturi and E. A. Simbawa, The modified decomposition method for solving Volterra-Fredholm integro-differential equations using Maple, Int. J. GEOMATE 18 (2020), no. 67, 84-89. |
| [34] | N. A. Mbroh, S. C. O. Noutchie and R. Y. M. Massoukou, A second order finite difference scheme for singularly perturbed Volterra integro-differential equation, Alexandria Engrg. J. 59 (2020), no. 4, 2441-2447. |
| [35] | E. Najafi, Nyström-quasilinearization method and smoothing transformation for the numerical solution of nonlinear weakly singular Fredholm integral equations, J. Comput. Appl. Math. 368 (2020), Article ID 112538. · Zbl 1446.65209 |
| [36] | S. Y. Reutskiy, The backward substitution method for multipoint problems with linear Volterra-Fredholm integro-differential equations of the neutral type, J. Comput. Appl. Math. 296 (2016), 724-738. · Zbl 1342.65163 |
| [37] | N. Rohaninasab, K. Maleknejad and R. Ezzati, Numerical solution of high-order Volterra-Fredholm integro-differential equations by using Legendre collocation method, Appl. Math. Comput. 328 (2018), 171-188. · Zbl 1427.65425 |
| [38] | M. Safavi and A. A. Khajehnasiri, Numerical solution of nonlinear mixed Volterra-Fredholm integro-differential equations by two-dimensional block-pulse functions, Cogent Math. Stat. 5 (2018), no. 1, Article ID 1521084. · Zbl 1426.65219 |
| [39] | X. Tao and Y. Zhang, The coupled method for singularly perturbed Volterra integro-differential equations, Adv. Difference Equ. 2019 (2019), Paper No. 217. · Zbl 1459.65248 |
| [40] | M. Turkyilmazoglu, High-order nonlinear Volterra-Fredholm-Hammerstein integro-differential equations and their effective computation, Appl. Math. Comput. 247 (2014), 410-416. · Zbl 1338.45007 |
| [41] | A.-M. Wazwaz, Linear and Nonlinear Integral Equations. Methods and Applications, Higher Education Press, Beijing, 2011. · Zbl 1227.45002 |
| [42] | Ö. Yapman and G. M. Amiraliyev, A novel second-order fitted computational method for a singularly perturbed Volterra integro-differential equation, Int. J. Comput. Math. 97 (2020), no. 6, 1293-1302. · Zbl 1480.65383 |
| [43] | Ö. Yapman, G. M. Amiraliyev and I. Amirali, Convergence analysis of fitted numerical method for a singularly perturbed nonlinear Volterra integro-differential equation with delay, J. Comput. Appl. Math. 355 (2019), 301-309. · Zbl 1415.65170 |
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