Parameter uniform second-order numerical approximation for the integro-differential equations involving boundary layers

Abstract

The work handles a Fredholm integro-differential equation involving boundary layers. A fitted second-order difference scheme has been created on a uniform mesh utilizing interpolating quadrature rules and exponential basis functions. The stability and convergence of the proposed discretization technique are analyzed and one example is solved to display the advantages of the presented technique.

Keywords

• Finite difference method
• Fredholm integro-differential equation
• singular perturbation
• uniform convergence

References

1. Amiraliyev, G. M., Durmaz, M. E., Kudu, M., Uniform convergence results for singularly perturbed Fredholm integro-differential equation, J. Math. Anal., 9(6) (2018), 55–64.
2. Amiraliyev, G. M., Durmaz, M. E., Kudu, M., Fitted second order numerical method for a singularly perturbed Fredholm integro-differential equation, Bull. Belg. Math. Soc. Simon Steven., 27(1) (2020), 71–88. https://doi.org/10.36045/bbms/1590199305
3. 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) (2021), 37–48. https://doi.org/10.18514/MMN.2021.2930
4. Amiraliyev, G. M., Mamedov, Y. D., Difference schemes on the uniform mesh for singularly perturbed pseudo-parabolic equations, Turk. J. Math., 19 (1995), 207–222.
5. Brunner, H., Numerical Analysis and Computational Solution of Integro-Differential Equations, Contemporary Computational Mathematics-A Celebration of the 80th Birthday of Ian Sloan (J. Dick et al., eds.), Springer, Cham, 2018, 205–231. https://doi.org 10.1007/978-3-319-72456-0 11
6. Chen, J., He, M., Zeng, T., A multiscale Galerkin method for second-order boundary value problems of Fredholm integro differential equation II: Efficient algorithm for the discrete linear system, J. Vis. Commun. Image R., 58 (2019), 112–118. https://doi.org/10.1016/j.jvcir.2018.11.027
7. Chen, J., He, M., Huang, Y., A fast multiscale Galerkin method for solving second order linear Fredholm integro-differential equation with Dirichlet boundary conditions, J. Comput. Appl. Math., 364 (2020), 112352. https://doi.org/10.1016/j.cam.2019.112352
8. Dehghan, M., Chebyshev finite difference for Fredholm integro-differential equation, Int. J. Comput. Math., 85 (1) (2008), 123–130. https://doi.org/10.1080/00207160701405436

9. Doolan, E. R., Miller, J. J. H., Schilders, W. H. A., Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole Press, Dublin, 1980.
10. Durmaz, M. E., Amiraliyev, G. M., A robust numerical method for a singularly perturbed Fredholm integro-differential equation, Mediterr. J. Math., 18(24) (2021), 1–17. https://doi.org/10.1007/s00009-020-01693-2
11. Durmaz, M. E., Amiraliyev, G. M., Kudu, M., Numerical solution of a singularly perturbed Fredholm integro differential equation with Robin boundary condition, Turk. J. Math., 46(1) (2022), 207–224. https://doi.org/10.3906/mat-2109-11
12. Farrell, P. A., Hegarty, A. F., Miller, J. J. H., O’Riordan, E., Shishkin, G. I., Robust Computational Techniques for Boundary Layers, Chapman Hall/CRC, New York, 2000. https://doi.org/10.1201/9781482285727
13. Jalilian, R., Tahernezhad, T., Exponential spline method for approximation solution of Fredholm integro-differential equation, Int. J. Comput. Math., 97(4) (2020), 791–801. https://doi.org/10.1080/00207160.2019.1586891
14. Jalius, C., Majid, Z. A., Numerical solution of second-order Fredholm integro differential equations with boundary conditions by quadrature-difference method, J. Appl. Math., (2017). https://doi.org/10.1155/2017/2645097
15. Kadalbajoo, M. K., Gupta, V., A brief survey on numerical methods for solving singularly perturbed problems, Appl. Math. Comput., 217 (2010), 3641–3716. https://doi.org/10.1016/j.amc.2010.09.059
16. Karim, M. F., Mohamad, M., Rusiman, M. S., Che-him, N., Roslan, R., Khalid, K., ADM for solving linear second-order Fredholm integro-differential equations, Journal of Physics, (2018), 995. https://doi.org/10.1088/1742-6596/995/1/012009
17. Kudu, M., Amirali, I., Amiraliyev, G. M., A finite-difference method for a singularly perturbed delay integro-differential equation, J. Comput. Appl. Math., 308 (2016), 379–390. https://doi.org/10.1016/j.cam.2016.06.018
18. Miller, J. J. H., O’Riordan, E., Shishkin, G. I., Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, Singapore, 1996.
19. Nayfeh, A. H., Introduction to Perturbation Techniques, Wiley, New York, 1993.
20. O’Malley, R. E., Singular Perturbations Methods for Ordinary Differential Equations, Springer, New York, 1991. https://doi.org/10.1007/978-1-4612-0977-5
21. Roos, H. G., Stynes, M., Tobiska, L., Numerical Methods for Singularly Perturbed Differential Equations, Springer-Verlaq, Berlin, 1996. https://doi.org/10.1007/978-3-662-03206-0
22. Samarskii, A. A., The Theory of Difference Schemes(1st ed.), CRC Press, 2001. https://doi.org/10.1201/9780203908518
23. Shahsavaran, A., On the convergence of Lagrange interpolation to solve special type of second kind Fredholm integro differential equations, Appl. Math. Sci., 6(7) (2012), 343–348.
24. Yapman, Ö., 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(2019), 301309. https://doi.org/10.1016/j.cam.2019.01.026
25. Yapman, Ö., Amiraliyev, G. M., A novel second–order fitted computational method for a singularly perturbed Volterra integro-differential equation, Int. J. Comput. Math., 97(6) (2020), 1293–1302. https://doi.org/10.1080/00207160.2019.1614565
26. Xue, Q., Niu, J., Yu, D., Ran, C., An improved reproducing kernel method for Fredholm integro-differential type two-point boundary value problems, Int. J. Comput. Math., 95(5) (2018), 1015–1023. https://doi.org/10.1080/00207160.2017.1322201