Approximation Solution for Initial Value Problem of Singularly Perturbed Integro-Differential Equation

Year 2024, Volume: 13 Issue: 3, 744 - 749, 26.09.2024

Derya Arslan

https://doi.org/10.17798/bitlisfen.1481490

Abstract

Adomian decomposition method (ADM) is used to approximately solve the initial value problem of the singularly perturbed Volterra and Fredholm differential equation. With this method, the desired accurate results are obtained in only a few terms. The approach is simple and effective. An example application is made to demonstrate the effectiveness of ADM. The result obtained is compared with the exact solution. Convergence analysis of the method was performed.

Keywords

ADM, Singularly perturbed equation, Volterra-Fredholm integro-differential equation, Approximate solution

References

• [1] O. Diekmann, “Thresholds and travelling waves for the geographical spread of infection,” J. Math. Biol., vol. 6, pp. 109-130, 1978.
• [2] E.H. Ouda, S. Shihab and M. Rasheed, “Boubaker wavelet functions for solving higher order integro-differential equations,” J. Southwest Jiaotong Univ., vol. 55, pp. 1-12, 2020.
• [3] M.K. Kadalbajoo and V. Gupta, “A brief survey on numerical methods for solving singularly perturbed problems,” Appl. Math. Comput., vol. 217, pp. 3641-3716, 2010.
• [4] M.E. Durmaz, Ö. Yapman, M. Kudu and G. M. Amiraliyev, “An efficient numerical method for a singularly perturbed Volterra-Fredholm integro-differential equation,” Hacettepe Journal of Mathematics & Statistics, vol. 52, pp. 326 – 339, 2023.
• [5] L.A. Dawood, A.A. Hamoud and N.M. Mohammed, “Laplace discrete decomposition method for solving nonlinear Volterra-Fredholm integro-differential equations,” J. Math. Computer Sci., vol. 21, pp. 158-163, 2020.
• [6] N.A. Mbroh, S.C. Oukouomi Noutchie and R.Y. M’pika Massoukou, “A second order finite difference scheme for singularly perturbed Volterra integro-differential equation,” Alex. Eng. J., vol. 59, pp. 2441-2447, 2020.
• [7] M.S.B. Issa, A.A. Hamoud and K.P. Ghadle, “Numerical solutions of fuzzy integro-differential equations of the second kind,” J. Math. Computer Sci., vol. 23, pp. 67-74, 2021.
• [8] G. Adomian, “A review of the decomposition method and some recent results for nonlinear equation,” Math. Comput. Model., vol. 13, pp. 17, 1992.
• [9] G. Adomian and R. Rach, “Analytic solution of nonlinear boundary-value problems in several dimensions by decomposition,” Journal of Mathematical Analysis and Applications, vol.174, pp. 118-137, 1993.
• [10] G. Adomian, Solving Frontier problems of physics: The decomposition method. Kluwer Academic Publishers, Boston: 1994.
• [11] Y. Cherruault and G. Adomian, “Decomposition methods a new proof of convergence,” Mathematical and Computer modelling, vol. 18, pp. 103- 106, 1993.
• [12] Y. Cherruault, “Convergence of Adomian’s method,” Kybernetes, vol. 18, pp. 31–38, 1989.
• [13] I. El-Kalla, Error analysis of Adomian series solution to a class of non-linear differential equations, Appl. Math E-Notes, vol. 7, pp. 214-221, 2007.
• [14] M. Cakir and D. Arslan, “The Adomian decomposition method and the differential transform method for numerical solution of multi-pantograph delay differential equations,” Applied Mathematics, vol. 6, pp. 1332-1343, 2015.
• [15] E. Banifatemi, M. Razzaghi and S. Youse, “Two-dimensional Legendre wavelets method for the mixed Volterra-Fredholm integral equations,” J. Vib. Control, vol.13, pp. 1667-1675, 2007.
• [16] H. Brunner, "Numerical analysis and computational solution of integro-differential equations." Contemporary Computational Mathematics-A Celebration of the 80th Birthday of Ian Sloan, 205-231, 2018.
• [17] M. Gülsu, Y. Öztürk and M. Sezer, “A new collocation method for solution of mixed linear integro-differential-difference equations,” Appl. Math. Comput., vol. 216, pp. 2183- 2198, 2010.
• [18] D.A. Maturi and E.A.M. Simbawa, “The modified decomposition method for solving Volterra Fredholm integro-differential equations using Maple”, Int. J. GEOMATE, vol. 18, pp. 84-89, 2020.
• [19] B. Raftari, “Numerical solutions of the linear Volterra integro-differential equations: Homotopy perturbation method and finite difference method,” World Appl. Sci. J., vol. 9, pp. 7-12, 2010.
• [20] A. M. Dalal, “Adomian decomposition method for solving of Fredholm integral equation of the second kind using matlab,” International Journal of GEOMATE, vol. 11, pp. 2830-2833, 2016.
• [21] A.A. Hamoud and K.P. Ghadle, “Existence and uniqueness of the solution for Volterra-Fredholm integro-differential equations,” J. Sib. Fed. Univ. - Math. Phys., vol. 11, pp. 692-701, 2018.
• [22] A.H. Mahmood and L.H. Sadoon, “Existence of a solution of a certain Volterra-Fredholm integro differential equations,” J. Educ. Sci., vol. 25, pp. 62-67, 2012.
• [23] H.G. Roos, M. Stynes and L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations. Springer-Verlag, Berlin Heidelberg: 2008.
• [24] E.R. Doolan, J. J. H. Miller, and W. H. A. Schilders, Uniform numerical methods for problems with initial and boundary layers. Dublin: Boole Press, 1980.
• [25] G.M. Amiraliyev and I. Amirali, Nümerik Analiz Teori ve Uygulamalarla. Ankara: Seçkin Yayıncılık, 2018.
• [26] P.A. Farrell, A.F. Hegarty, J.J.H. Miller, E. O’Riordan and G.I. Shishkin, Robust Computational Techniques for Boundary Layers. New York: Chapman Hall/CRC, 2000.
• [27] R.O. O’Malley, Singular Perturbations Methods for Ordinary Differential Equations. New York: Springer-Verlag, 1991.
• [28] V. Lakshmikantham, Theory of Integro-differential Equations. CRC press, 1995.