A numerical method on Bakhvalov-Shishkin mesh for Volterra integro-differential equations with a boundary layer

Abstract

We construct a finite difference scheme for a first-order linear singularly perturbed Volterra integro-differential equation(SPVIDE) on Bakhvalov-Shishkin mesh. For the discretization of the problem, we use the integral identities and deal with the emerging integrals terms with interpolating quadrature rules which also yields remaining terms. The stability bound and the error estimates of the approximate solution are established. Further, we demonstrate that the scheme on Bakhvalov-Shishkin mesh is $N\left(O^{-1}\right)$uniformly convergent, where N is the mesh parameter. The numerical results are also provided for a couple of examples.

Keywords

• Singularly perturbed
• VIDE;
• difference schemes
• uniform convergence
• error estimates

References

1. Amiraliyev, G. M., Mamedov, Y. D. H., Difference schemes on the uniform mesh for singular perturbed pseudo-parabolic equations, Turk. J. Math., 19 (1995), 207–222.
2. Amiraliyev, G. M., Sevgin, S., Uniform difference method for singularly perturbed Volterra integro-differential equations, Appl. Math. Comput., 179 (2006), 731–741, https://dx.doi.org/10.1016/j.amc.2005.11.155.
3. Amiraliyev, G. M., Yapman, O., A novel second-order fitted computational method for a singularly perturbed Volterra integro-differential equation, International Journal of Computer Mathematics, 97-6 (2020), 1293–1302, https://dx.doi.org/10.1080/00207160.2019.1614565.
4. Amiraliyev, G. M., Yilmaz, B., Fitted difference method for a singularly perturbed initial value problem, Int. J. Math. Comput., 22 (2014), 1–10.
5. Angell, J. S., Olmstead, W. E., Singular perturbation analysis of an integro-differential equation modelling filament stretching, J. Appl. Math. Phys. (ZAMP), 36 (1985), 487–490, https://dx.doi.org/10.1007/BF00944639.
6. Angell, J. S., Olmstead, W. E., Singularly perturbed Volterra integral equations, SIAM J. Appl. Math., 47 (1987), 1150–1162, https://dx.doi.org/10.1137/0147001.
7. Angell, J. S., Olmstead, W. E., Singularly perturbed Volterra integral equations ii, SIAM J. Appl. Math., 47 (1987), 1–14, https://dx.doi.org/10.1137/0147077.
8. Anjum, N., He, C. H., He, J. H., Two-scale fractal theory for the population dynamics, Fractals, https://dx.doi.org/10.1142/S0218348X21501826.

9. Anjum, N., He, J.-H., Homotopy perturbation method for N/MEMS oscillators, Math Meth Appl Sci., 1–15, https://dx.doi.org/10.1002/mma.6583.
10. Arslan, D., A uniformly convergent numerical study on Bakhvalov-Shishkin mesh for singularly perturbed problem, Communications in Mathematics and Applications, 11-1 (2020), 161–171.
11. Cakir, M., Gunes, B., Duru, H., A novel computational method for solving nonlinear Volterra integro-differential equation, Kuwait J. Sci, 48 (1) (2021), 1–9, https://dx.doi.org/10.48129/kjs.v48i1.9386.
12. Cen, Z., Le, A., Xu, A., Parameter-uniform hybrid difference scheme for solutions and derivatives in singularly perturbed initial value problems, J. Comput. Appl. Math., 320 (2017), 176–192, https://dx.doi.org/10.1016/j.cam.2017.02.009.
13. Cushing, J. M., Integro-differential Equations and Delay Models in Population dynamics, Springer-Verlag, New York, 1992.
14. De Gaetano, A., Arino, O., Mathematical modelling of the intravenous glucose tolerance test, J. Math. Biol., 40 (2000), 136–168, https://dx.doi.org/10.1007/s002850050007.
15. Doolan, E. P., Miller, J. J. H., Schilders, H. A., Uniform Numerical Method for Problems with Initial and Boundary Layers, Boole Press, Dublin, 1980.
16. Farrel, 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://dx.doi.org/10.1201/9781482285727.
17. He, J., El-Dib, Y. O., Homotopy perturbation method for Fangzhu oscillator, J. of Math. Chem., 58 (2020), 2245–2253, https://dx.doi.org/10.1007/s10910-020-01167-6.
18. He, J.-H., A simple approach to Volterra-Fredholm integral equations, Journal of Applied and Computational Mechanics, 6 (Special Issue) (2020), 1184–1186, https://dx.doi.org/10.22055/jacm.2020.34653.2451.
19. Horvat, V., Rogina, M., Tension spline collocation methods for singularly perturbed Volterra integro-differential and Volterra integral equations, J. Comput. Appl. Math., 140 (7) (2002), 381–402.
20. Jordan, G. . S., Some nonlinear singularly perturbed Volterra integro-differential equations, in: Volterra Equations (Proc. Helsinki Sympos. Integral Equations, Otaniemi, 1978), Lecture Notes in Mathematics, 737 (1979), 107–119, https://dx.doi.org/10.1007/BFb0064501.
21. Jordan, G. S., A nonlinear singularly perturbed Volterra integro-differential equation of nonconvolution type, Proc. Roy. Soc. Edinburgh Sect. A, 80 (1978), 235–247, https://dx.doi.org/10.1017/S030821050001026X.
22. Kadalbajoo, M. K., Gupta, V., A brief survey on numerical methods for solving singularly perturbed problems, Appl. Math. Comput., 217 (2010), 3641–3716, https://dx.doi.org/10.1016/j.amc.2010.09.059.
23. Kauthen, J. P., Implicit Runge–Kutta methods for some integro-differential–algebraic equations, Appl. Numer. Math., 13 (1993), 125–134.
24. Kauthen, J. P., Implicit Runge–Kutta methods for singularly perturbed integro-differential systems, Appl. Numer. Math., 18 (1995), 201–210, https://dx.doi.org/https://doi.org/10.1016/0168-9274(95)00053-W.
25. Kauthen, J. P., A survey of singularly perturbed Volterra equations, Appl.Numer.Math, 24 (1997), 95–114.
26. Khater, A. H., Shamardan, A. B., Callebaut, D. K. Sakran, M. R. A., Numerical solutions of integral and integro- differential equations using Legendre polynomials, Numer. Algorithms, 46 (2007), 195–218, https://dx.doi.org/10.1007/s11075-007-9130-2.
27. 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://dx.doi.org/10.1016/j.cam.2016.06.018.
28. Kumar, M., Singh, P., Misra, H. K., A recent survey on computational techniques for solving singularly perturbed boundary value problems, Int. J. Comput. Math., 84(10) (2007), 1439–1463, https://dx.doi.org/10.1080/00207160701295712.
29. Kumar, V., Srinivasan, B., An adaptive mesh strategy for singularly perturbed convection diffusion problems, Applied Mathematical Modelling,, 39 (7) (2015), 2081–2091, https://dx.doi.org/10.1016/j.apm.2014.10.019.
30. Linss, T., Analysis of a Galerkin finite element method on a Bakhvalov-Shishkin mesh for a linear convection-diffusion problem, IMA Journal of Numerical Analysis, 20 (2000), 621–632, https://dx.doi.org/10.1093/imanum/20.4.621.
31. Linss, T., Layer-adapted meshes for convection–diffusion problems, Computer Methods in Applied Mechanics and Engineering, 192 (9-10) (2003), 1061–1105, https://dx.doi.org/10.1007/978-3-642-05134-0.
32. Lodge, A. S., McLeod, J. B., Nohel, J. A., A nonlinear singularly perturbed Volterra integrodifferential equation occurring in polymer rheology, Proc. Roy. Soc. Edinburgh Sect. A, 80 (1978), 99–137, https://dx.doi.org/10.1017/S0308210500010167.
33. Marino, S., Beretta, E., Kirschner, D. E., The role of delays in innate and adaptive immunity to intracellular bacterial infection, Math. Biosci. Eng., 4 (2007), 261–286, https://dx.doi.org/10.3934/mbe.2007.4.261.
34. Miller, J. J. H., O’Riordan, Shishkin, G. I., Fitted Numerical Methods for Singular Perturbation Problems, Revised Edition, World Scientific, 2012, https://dx.doi.org/10.1142/8410.
35. Mishra, H. K., S. S., Various numerical methods for singularly perturbed boundary value problems, Am. J. Appl. Math. Statist., 2 (2014), 129–142, https://dx.doi.org/10.12691/ajams-2-3-7.
36. Ramos, J. I., Exponential techniques and implicit Runge-Kutta method for singularly perturbed Volterra integro- differential equations, Neural Parallel Sci. Comput., 16 (2008), 387–404.
37. Reddingius, J., Notes on the mathematical theory of epidemics, Acta Biotheor., 20 (1971), 125–135, https://dx.doi.org/10.1007/BF01556687.
38. Roos, H. G., Stynes, M., Tobiska, L., Numerical Methods for Singularly Perturbed Differential Equations, Springer- Verlag, Berlin, 1996, https://dx.doi.org/10.1007/978-3-662-03206-0.
39. Salama, A. A., Bakr, S. A., Difference schemes of exponential type for singularly perturbed Volterra integro-differential problems, Appl. Math. Model., 31 (2007), 866–879, https://dx.doi.org/10.1016/j.apm.2006.02.007.
40. Wazwaz, A., Linear and Nonlinear Integral Equations, Springer, 2011, https://dx.doi.org/10.1007/978-3-642-21449-3.