An efficient numerical method for a singularly perturbed Volterra-Fredholm integro-differential equation

Abstract

The scope of this study is to establish an effective approximation method for linear first order singularly perturbed Volterra-Fredholm integro-differential equations. The finite difference scheme is constructed on Shishkin mesh by using appropriate interpolating quadrature rules and exponential basis function. The recommended method is second order convergent in the discrete maximum norm. Numerical results illustrating the preciseness and computationally attractiveness of the proposed method are presented.

Keywords

• finite difference methods
• integro-differential equation
• Shishkin mesh
• singular perturbation
• uniform convergence

References

1. [1] G.M. Amiraliyev, M.E. Durmaz and M. Kudu, Uniform convergence results for singularly perturbed Fredholm integro-differential equation, J. Math. Anal. 9 (6), 55-64, 2018.
2. [2] G.M. Amiraliyev, M.E. Durmaz and M. Kudu, A numerical method for a second order singularly perturbed Fredholm integro-differential equation, Miskolc Math. Notes 22 (1), 37-48, 2021.
3. [3] G.M. Amiraliyev and Y.D. Mamedov, Difference schemes on the uniform mesh for singularly perturbed pseudo-parabolic equations, Turkish J. Math. 19, 207-222, 1995.
4. [4] G.M. Amiraliyev, Ö. Yapman and M. Kudu, A fitted approximate method for a Volterra delay-integro-differential equation with initial layer, Hacet. J. Math. Stat. 48 (5), 1417-1429, 2019.
5. [5] G.M. Amiraliyev and Ö. Yapman, On the Volterra delay-integro-differential equation with layer behavior and its numerical solution, Miskolc Math. Notes 20 (1), 75-87, 2019.
6. [6] E. Banifatemi, M. Razzaghi and S. Youse, Two-dimensional Legendre wavelets method for the mixed Volterra-Fredholm integral equations, J. Vib. Control 13 (11), 1667-1675, 2007.
7. [7] H. Brunner, Numerical Analysis and Computational Solution of Integro-Differential Equations, In: Dick J., Kuo F., Woniakowski H. (eds) Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan. Springer, Cham. 2018.
8. [8] 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. 21, 158-163, 2020.

9. [9] L. Dawood, A. Sharif and A. Hamoud, Solving higher-order integro differential equations by VIM AND MHPM, Int. J. Appl. Math. 33 (2), 253-264, 2020.
10. [10] O. Diekmann, Thresholds and travelling waves for the geographical spread of infection, J. Math. Biol. 6 (2), 109-130, 1978.
11. [11] E.R. Doolan, J.J.H. Miller, W.H.A. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers, Dublin, Boole Press, 1980.
12. [12] M.E. Durmaz and G.M. Amiraliyev, A robust numerical method for a singularly perturbed Fredholm integro-differential equation, Mediterr. J. Math. 18, 1-17, 2021.
13. [13] P.A. Farrell, A.F. Hegarty, J.J.H. Miller, E. O’Riordan and G.I. Shishkin, Robust Computational Techniques for Boundary Layers, Chapman Hall/CRC, New York, 2000.
14. [14] M. Gülsu, Y. Öztürk and M. Sezer, A new collocation method for solution of mixed linear integro-differential-difference equations, Appl. Math. Comput. 216 (7), 2183- 2198, 2010.
15. [15] A.A. Hamoud, L.A. Dawood, K.P. Ghadle and S.M. Atshan, Usage of the modified variational iteration technique for solving Fredholm integro-differential equations, International Journal of Mechanical and Production Engineering Research and Development (IJMPERD) 9 (2), 895-902, 2019.
16. [16] 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. 11 (6), 692- 701, 2018.
17. [17] A.A. Hamoud and K.P. Ghadle, Homotopy analysis method for the first order fuzzy Volterra-Fredholm integro-differential equations, Indones. J. Electr. Eng. Comput. Sci. 11 (3), 857-867, 2018.
18. [18] A.A. Hamoud, N.M. Mohammed and K.P. Ghadle, Solving mixed Volterra-Fredholm integro differential equations by using HAM, Turk. J. Math. Comput. Sci. 12 (1), 18-25, 2020.
19. [19] M.S.B. Issa, A.A. Hamoud and K.P. Ghadle, Numerical solutions of fuzzy integrodifferential equations of the second kind, J. Math. Computer Sci. 23, 67-74, 2021.
20. [20] M.K. Kadalbajoo and V. Gupta, A brief survey on numerical methods for solving singularly perturbed problems, Appl. Math. Comput. 217, 3641-3716, 2010.
21. [21] 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, 379-390, 2016.
22. [22] A.H. Mahmood and L.H. Sadoon, Existence of a solution of a certain Volterra- Fredholm integro differential equations, J. Educ. Sci. 25 (3), 62-67, 2012.
23. [23] D.A. Maturi and E.A.M. Simbawa, The modified decomposition method for solving Volterra Fredholm integro-differential equations using maple, Int. J. GEOMATE 18 (67), 84-89, 2020.
24. [24] 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. 59 (4), 2441-2447, 2020.
25. [25] J.J.H. Miller, E. O’Riordan and G.I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems: Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions (Rev. Ed.), World Scientific, Singapore, 2012.
26. [26] H.K. Mishra, S. Saini, Various numerical methods for singularly perturbed boundary value problems, Am. J. Appl. Math. Stat. 2 (3), 129-142, 2014.
27. [27] A.H. Nayfeh, Introduction to Perturbation Techniques, Wiley, New York, 1993.
28. [28] R.E. O’Malley, Singular Perturbations Methods for Ordinary Differential Equations, Springer, New York, 2013.
29. [29] E.H. Ouda, S. Shihab and M. Rasheed, Boubaker wavelet functions for solving higher order integro-differential equations, J. Southwest Jiaotong Univ. 55 (2), 1-12, 2020.
30. [30] B. Raftari, Numerical solutions of the linear Volterra integro-differential equations: Homotopy perturbation method and finite difference method, World Appl. Sci. J. 9, 7-12, 2010.
31. [31] M. Ramadan and M. Ali, Numerical solution of Volterra-Fredholm integral equations using hybrid orthonormal Bernstein and Block-Pulse functions, Asian Res. J. Math. 4 (4), 1-14, 2017.
32. [32] H.G. Roos, M. Stynes and L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations, Springer-Verlag, Berlin Heidelberg, 2008.
33. [33] A.A. Samarskii, The Theory of Difference Schemes, Marcell Dekker, Inc. New York, 2001.
34. [34] Ö. Yapman and G.M. Amiraliyev, Convergence analysis of the homogeneous second order difference method for a singularly perturbed Volterra delay-integro-differential equation, Chaos Solitons Fractals 150, 111100, 2021.
35. [35] 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, 301-309, 2019.