Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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A ROBUST NUMERICAL TECHNIQUE FOR SOLVING NON-LINEAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS WITH BOUNDARY LAYER

  • Received : 2021.07.31
  • Accepted : 2021.12.30
  • Published : 2022.07.31
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Abstract

In this paper, we study a first-order non-linear singularly perturbed Volterra integro-differential equation (SPVIDE). We discretize the problem by a uniform difference scheme on a Bakhvalov-Shishkin mesh. The scheme is constructed by the method of integral identities with exponential basis functions and integral terms are handled with interpolating quadrature rules with remainder terms. An effective quasi-linearization technique is employed for the algorithm. We establish the error estimates and demonstrate that the scheme on Bakhvalov-Shishkin mesh is O(N-1) uniformly convergent, where N is the mesh parameter. The numerical results on a couple of examples are also provided to confirm the theoretical analysis.

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References

  1. G. M. Amiraliyev and Ya. D. Mamedov, Difference schemes on the uniform mesh for singular perturbed pseudo-parabolic equations, Turkish J. Math. 19 (1995), no. 3, 207-222.
  2. J. S. Angell and W. E. Olmstead, Singular perturbation analysis of an integro-differential equation modelling filament stretching, Z. Angew. Math. Phys. 36 (1985), no. 3, 487-490. https://doi.org/10.1007/BF00944639
  3. J. S. Angell and W. E. Olmstead, Singularly perturbed Volterra integral equations. II, SIAM J. Appl. Math. 47 (1987), no. 6, 1150-1162. https://doi.org/10.1137/0147077
  4. J. S. Angell and W. E. Olmstead, Singularly perturbed Volterra integral equations, SIAM J. Appl. Math. 47 (1987), no. 1, 1-14. https://doi.org/10.1137/0147001
  5. D. Arslan, A uniformly convergent numerical study on bakhvalov-shishkin mesh for singularly perturbed problem, Commun. Math. Appl. 11 (2020), no. 1, 161-171.
  6. M. Cakir, B. Gunes, and H. Duru, A novel computational method for solving nonlinear Volterra integro-differential equation, Kuwait J. Sci. 48 (2021), no. 1, 1-9.
  7. Z. Cen, A. Le, and A. Xu, Parameter-uniform hybrid difference scheme for solutions and derivatives in singularly perturbed initial value problems, J. Comput. Appl. Math. 320 (2017), 176-192. https://doi.org/10.1016/j.cam.2017.02.009
  8. A. De Gaetano and O. Arino, Mathematical modelling of the intravenous glucose tolerance test, J. Math. Biol. 40 (2000), no. 2, 136-168. https://doi.org/10.1007/s002850050007
  9. E. P. Doolan, J. J. H. Miller, and W. H. A. Schilders, Uniform numerical methods for problems with initial and boundary layers, Boole Press, Dun Laoghaire, 1980.
  10. P. A. Farrell, A. F. Hegarty, J. J. H. Miller, E. O'Riordan, and G. I. Shishkin, Robust computational techniques for boundary layers, Applied Mathematics (Boca Raton), 16, Chapman & Hall/CRC, Boca Raton, FL, 2000.
  11. V. Horvat and M. Rogina, Tension spline collocation methods for singularly perturbed Volterra integro-differential and Volterra integral equations, J. Comput. Appl. Math. 140 (2002), no. 1-2, 381-402. https://doi.org/10.1016/S0377-0427(01)00517-9
  12. G. S. Jordan, A nonlinear singularly perturbed Volterra integro-differential equation of nonconvolution type, Proc. Roy. Soc. Edinburgh Sect. A 80 (1978), no. 3-4, 235-247. https://doi.org/10.1017/S030821050001026X
  13. G. S. Jordan, Some nonlinear singularly perturbed Volterra integro-differential equations, in Volterra equations (Proc. Helsinki Sympos. Integral Equations, Otaniemi, 1978), 107-119, Lecture Notes in Math., 737, Springer, Berlin, 1979.
  14. M. K. Kadalbajoo and V. Gupta, A brief survey on numerical methods for solving singularly perturbed problems, Appl. Math. Comput. 217 (2010), no. 8, 3641-3716. https://doi.org/10.1016/j.amc.2010.09.059
  15. J.-P. Kauthen, Implicit Runge-Kutta methods for some integrodifferential-algebraic equations, Appl. Numer. Math. 13 (1993), no. 1-3, 125-134. https://doi.org/10.1016/0168-9274(93)90136-F
  16. J.-P. Kauthen, Implicit Runge-Kutta methods for singularly perturbed integrodifferential systems, Appl. Numer. Math. 18 (1995), no. 1-3, 201-210. https://doi.org/10.1016/0168-9274(95)00053-W
  17. J.-P. Kauthen, A survey of singularly perturbed Volterra equations, Appl. Numer. Math. 24 (1997), no. 2-3, 95-114. https://doi.org/10.1016/S0168-9274(97)00014-7
  18. A. H. Khater, A. B. Shamardan, D. K. Callebaut, and M. R. A. Sakran, Numerical solutions of integral and integro-differential equations using Legendre polynomials, Numer. Algorithms 46 (2007), no. 3, 195-218. https://doi.org/10.1007/s11075-007-9130-2
  19. M. Kumar, P. Singh, and H. K. Mishra, A recent survey on computational techniques for solving singularly perturbed boundary value problems, Int. J. Comput. Math. 84 (2007), no. 10, 1439-1463. https://doi.org/10.1080/00207160701295712
  20. V. Kumar and B. Srinivasan, An adaptive mesh strategy for singularly perturbed convection diffusion problems, Appl. Math. Model. 39 (2015), no. 7, 2081-2091. https://doi.org/10.1016/j.apm.2014.10.019
  21. T. Linss, Analysis of a Galerkin finite element method on a Bakhvalov-Shishkin mesh for a linear convection-diffusion problem, IMA J. Numer. Anal. 20 (2000), no. 4, 621-632. https://doi.org/10.1093/imanum/20.4.621
  22. T. Linss, Layer-adapted meshes for convection-diffusion problems, Comput. Methods Appl. Mech. Engrg. 192 (2003), no. 9-10, 1061-1105. https://doi.org/10.1016/S0045-7825(02)00630-8
  23. A. S. Lodge, J. B. McLeod, and J. A. Nohel, A nonlinear singularly perturbed Volterra integro-differential equation occurring in polymer rheology, Proc. Roy. Soc. Edinburgh Sect. A 80 (1978), no. 1-2, 99-137. https://doi.org/10.1017/S0308210500010167
  24. S. Marino, E. Beretta, and D. E. Kirschner, The role of delays in innate and adaptive immunity to intracellular bacterial infection, Math. Biosci. Eng. 4 (2007), no. 2, 261-286. https://doi.org/10.3934/mbe.2007.4.261
  25. J. J. H. Miller, E. O'Riordan, and G. I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems, revised edition, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012. https://doi.org/10.1142/9789814390743
  26. H. K. Mishra and S. Saini, Various numerical methods for singularly perturbed boundary value problems, Am. J. Appl. Math. Statist. 2 (2014), 129-142. https://doi.org/10.12691/ajams-2-3-7
  27. A. H. Nayfeh, Introduction to Perturbation Techniques, A Wiley-Interscience Publication, Wiley-Interscience, New York, 1981.
  28. R. E. O'Malley, Jr., Singular perturbation methods for ordinary differential equations, Applied Mathematical Sciences, 89, Springer-Verlag, New York, 1991. https://doi.org/10.1007/978-1-4612-0977-5
  29. J. I. Ramos, Exponential techniques and implicit Runge-Kutta methods for singularlyperturbed Volterra integro-differential equations, Neural Parallel Sci. Comput. 16 (2008), no. 3, 387-404.
  30. H.-G. Roos, M. Stynes, and L. Tobiska, Numerical methods for singularly perturbed differential equations, Springer Series in Computational Mathematics, 24, Springer-Verlag, Berlin, 1996. https://doi.org/10.1007/978-3-662-03206-0
  31. A. A. Salama and S. A. Bakr, Difference schemes of exponential type for singularly perturbed volterra integro-differential problems, Appl. Math. Model. 31 (2007), 866-879. https://doi.org/10.1016/j.apm.2006.02.007
  32. S. Sevgin, Numerical solution of a singularly perturbed Volterra integro-differential equation, Adv. Difference Equ. 2014 (2014), 171, 15 pp. https://doi.org/10.1186/1687-1847-2014-171
  33. Sumit, S. Kumar, and J. Vigo-Aguiar, Analysis of a nonlinear singularly perturbed Volterra integro-differential equation, J. Comput. Appl. Math. 404 (2022), Paper No. 113410. https://doi.org/10.1016/j.cam.2021.113410
  34. O. 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. https://doi.org/10.1016/j.cam.2019.01.026