A fitted approximate method for a Volterra delay-integro-differential equation with initial layer

Year 2019, Volume: 48 Issue: 5, 1417 - 1429, 08.10.2019

Gabil M. Amiraliyev , Ömer Yapman , Mustafa Kudu

Abstract

This study is concerned with the finite-difference solution of singularly perturbed initial value problem for a linear first order Volterra integro-differential equation with delay. The method is based on the method of integral identities with the use of exponential basis functions and interpolating quadrature rules with the weight and remainder terms in integral form. The emphasis is on the convergence of numerical method. It is shown that the method displays uniform convergence in respect to the perturbation parameter. Numerical results are also given.

Keywords

Volterra delay-integro-differential equation, singular perturbation, finite difference, uniform convergence

References

• [1] G.M. Amiraliyev and F. Erdoğan, Uniform Numerical Method for Singularly Per- turbed Delay Differential Equations, Comput. Math. Appl. 53, 1251–1259, 2007.
• [2] G.M. Amiraliyev and S. Şevgin, Uniform Difference Method for Singularly Perturbed Volterra Integro-Differential Equations, Appl. Math. Comput. 179, 731–741, 2006.
• [3] G.M. Amiraliyev and B. Yilmaz, Fitted Difference Method for a Singularly Perturbed Initial Value Problem, Int. J. Math. Comput. 22, 1–10, 2014.
• [4] A. Bellour and M. Bousselsal, Numerical solution of delay integro-differential equa- tions by using Taylor collocation method, Math. Meth. Appl. Sci. 37, 1491–1506, 2014.
• [5] A.M. Bijura, Singularly Perturbed Volterra Integro-differential Equations, Quaest. Math. 25 (2), 229–248, 2002.
• [6] A.A. Bobodzhanov and V.F. Safonov, Singularly Perturbed Integro-Differential Equa- tions with Diagonal Degeneration of the Kernel in Reverse Time, Differ. Equ. 40 (1), 120–127, 2004.
• [7] G.A. Bocharov and F.A. Rihan, Numerical modelling in biosciences with delay dif- ferential equations, J. Comput. Appl. Math. 125, 183–199, 2000.
• [8] H. Brunner and P.J. van der Houwen, The Numerical Solution of Volterra Equations, in: CWI Monographs 3, North-Holland, Amsterdam, 1986.
• [9] A. De Gaetano and O. Arino, Mathematical modelling of the intravenous glucose tolerance test, J. Math. Biol. 40, 136–168, 2000.
• [10] E.R. Doolan, J.J.H. Miller and W.H.A. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole Press, Dublin, 1980.
• [11] P.A. Farrel, 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.
• [12] S. Gan, Dissipativity of -methods for nonlinear Volterra delay-integro-differential equations, J. Comput. Appl. Math. 206, 898–907, 2007.
• [13] D. He and L. Xu, Integrodifferential Inequality for Stability of Singularly Perturbed Impulsive Delay Integrodifferential Equations, J. Inequal. Appl. ID 369185, 1–11, 2009.
• [14] C. Huang, Stability of linear multistep methods for delay integro-differential equations, Comput. Math. Appl. 55, 2830–2838, 2008.
• [15] A. Jerri, Introduction to Integral Equations with Applications, Wiley, New York, 1999.
• [16] M.K. Kadalbajoo and V. Gupta, A brief survey on numerical methods for solving singularly perturbed problems, Appl. Math. Comput. 217, 3641–3716, 2010.
• [17] J.P. Kauthen, A survey on singularly perturbed Volterra equations, Appl. Numer. Math. 24, 95–114, 1997.
• [18] A.H. Khater, A.B. Shamardan, D.K. Callebaut and M.R.A. Sakran, Numerical solu- tions of integral and integro-differential equations using Legendre polynomials, Numer. Algor. 46, 195–218, 2007.
• [19] T. Koto, Stability of Runge-Kutta methods for delay integro-differential equations, J. Comput. Appl. Math. 145, 483–492, 2002.
• [20] M. Kudu, I. Amirali and G.M. Amiraliyev, A finite-difference method for singularly perturbed delay integro-differential equation, J. Comput. Appl. Math. 308, 379–390, 2016.
• [21] A.S. Lodge, J.B. McLeod and J.A. Nohel, A nonlinear singularly perturbed Volterra integrodifferential equation occurring in polymer rheology, Proc. Roy. Soc. Edinburgh, Sect. A, 80, 99–137, 1978.
• [22] 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, 261–288, 2007.
• [23] J.J.H. Miller, E. O’Riordan and G.I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems, Rev. ed. World Scientific, Singapore, 2012.
• [24] H.K. Mishra and S. Saini, Various Numerical Methods for Singularly Perturbed Boundary Value problems, Amer. J. Appl. Math. Stat. 2, 129–142, 2014.
• [25] A.H. Nayfeh, Introduction to Perturbation Techniques, Wiley, New York, 1993.
• [26] R.E. O’Malley, Singular Perturbation Methods for Ordinary Differential Equations, Springer-Verlag, New York, 1991.
• [27] J.I. Ramos, Exponential techniques and implicit Runge Kutta method for singularly perturbed Volterra integro differential equations, Neural Parallel Sci. Comput. 16, 387–404, 2008.
• [28] H.G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations, Springer-Verlag, Berlin, 1996.
• [29] A.A. Salama and S.A. Bakr, Difference schemes of exponential type for singularly perturbed Volterra integro-differential problems, Appl. Math. Model. 31, 866–879, 2007.
• [30] A.A. Samarskii, The Theory of Difference Schemes, Marcel Dekker, Inc., New York, 2001.
• [31] M. Shakourifar and W. Enright, Superconvergent interpolants for collocation methods applied to Volterra integro-differential equations with delay, Bit. Numer. Math. 52, 725–740, 2012.
• [32] Y. Song and C.T.H. Baker, Qualitative behaviour of numerical approximations to Volterra integro-differential equations, J. Comput. Appl. Math. 172, 101–115, 2004.
• [33] S. Şevgin, Numerical solution of a singularly perturbed Volterra integro-differential equation, Adv. Differ. Equ. 171, 2014.
• [34] M. Turkyilmazoglu, Series solution of nonlinear two-point singularly perturbed bound- ary layer problems, Comput. Math. Appl. 60 (7), 2109–2114, 2010.
• [35] M. Turkyilmazoglu, An effective approach for numerical solutions of high-order Fred- holm integro-differential equations, Appl. Math. Comput. 227, 384–398, 2014.
• [36] M. Turkyilmazoglu, High-order nonlinear Volterra-Fredholm-Hammerstein integro- differential equations and their effective computation, Appl. Math. Comput. 247, 410–416, 2014.
• [37] S.Wu and S. Gan, Errors of linear multistep methods for singularly perturbed Volterra delay-integro-differential equations, Math. Comput. Simulat. 79, 3148–3159, 2009.
• [38] C. Zhang and S. Vandewalle, Stability analysis of Volterra delay-integro-differential equations and their backward differentiation time discretization, J. Comput. Appl. Math. 164-165, 797–814, 2004.
• [39] J. Zhao, Y. Cao and Y. Xu, Sinc numerical solution for pantograph Volterra delay- integro-differential equation, Int. J. Comput. Math. 94 (5), 853–865, 2017.
• [40] J. Zhao, Y. Fan and Y. Xu, Delay-dependent stability analysis of symmetric boundary value methods for linear delay integro-differential equations, Numer. Algor. 65, 125– 151, 2014.