# Numerical solution of Volterra integro-differential equation with delay

Volume 20, Issue 3, pp 255--263
Publication Date: February 13, 2020 Submission Date: July 28, 2019 Revision Date: December 05, 2019 Accteptance Date: December 09, 2019
• 550 Views

### Authors

Erkan Cimen - Department of Mathematics, Faculty of Education, Van Yuzuncu Yil University, 65080, Van, Turkey. Sabahattin Yatar - Department of Mathematics, Institute of Pure and Applied Sciences, Van Yuzuncu Yil University, 65080, Van, Turkey.

### Abstract

We consider an initial value problem for a linear first-order Volterra delay integro-differential equation. We develop a novel difference scheme for the approximate solution of this problem via a finite difference method. The method is based on the fitted difference scheme on a uniform mesh which is achieved by using the method of integral identities which includes the exponential basis functions and applying to interpolate quadrature formulas that contain the remainder term in integral form. Also, the method is proved to be first-order convergent in the discrete maximum norm. Furthermore, a numerical experiment is performed to verify the theoretical results. Finally, the proposed scheme is compared with the implicit Euler scheme.

### Share and Cite

##### ISRP Style

Erkan Cimen, Sabahattin Yatar, Numerical solution of Volterra integro-differential equation with delay, Journal of Mathematics and Computer Science, 20 (2020), no. 3, 255--263

##### AMA Style

Cimen Erkan, Yatar Sabahattin, Numerical solution of Volterra integro-differential equation with delay. J Math Comput SCI-JM. (2020); 20(3):255--263

##### Chicago/Turabian Style

Cimen, Erkan, Yatar, Sabahattin. "Numerical solution of Volterra integro-differential equation with delay." Journal of Mathematics and Computer Science, 20, no. 3 (2020): 255--263

### Keywords

• Volterra delay integro-differential equation
• finite difference method
• error estimate

•  34K28
•  65L10
•  65L20
•  65L70
•  65R20

### References

• [1] A. Abdi, J.-P. Berrut, S. A. Hosseini, The linear barycentric rational method for a class of delay Volterra integro-differential equations, J. Sci. Comput., 75 (2018), 1757--1775

• [2] G. M. Amiraliyev, Y. D. Mamedov, Difference schemes on the uniform mesh for singular perturbed pseudo-parabolic equations, Turkish J. Math., 19 (1995), 207--222

• [3] A. Bellen, M. Zennaro, Numerical methods for delay differential equations, Oxford University Press, New York (2003)

• [4] A. Bellour, M. Bousselsal, Numerical solution of delay integro-differential equations by using Taylor collocation method, Math. Methods Appl. Sci., 37 (2014), 1491--1506

• [5] G. A. Bocharov, F. A. Rihan, Numerical modelling in biosciences with delay differential equations, J. Comput. Appl. Math., 125 (2000), 183--199

• [6] H. Brunner, Collocation methods for Volterra integral and related functional differential equations, Cambridge University Press, Cambridge (2004)

• [7] H. Brunner, W. Zhang, Primary discontinuities in solutions for delay integro-differential equations, Methods Appl. Anal., 6 (1999), 525--533

• [8] V. A. Caus, Delay integral equations, An. Univ. Oradea Fasc. Mat., 9 (2002), 109--112

• [9] J. M. Cushing, Integrodifferential equations and delay models in population dynamics, Springer-Verlag, New York (1977)

• [10] S. Gan, Dissipativity of $\theta$-methods for nonlinear Volterra delay-integro-differential equations, J. Comput. Appl. Math., 206 (2007), 898--907

• [11] C. M. Huang, Stability of linear multistep methods for delay integro-differential equations, Comput. Math. Appl., 55 (2008), 2830--2838

• [12] A. J. Jerri, Introduction to integral equations with applications, Wiley-Interscience, New York (1999)

• [13] V. Kolmanovskii, A. Myshkis, Introduction to the theory and applications of functional differential equations, Kluwer Academic Publishers, Dordrecht (1999)

• [14] T. Koto, Stability of Runge-Kutta methods for delay integro-differential equations, J. Comput. Appl. Math., 145 (2002), 483--492

• [15] K. Kuang, Delay differential equations with applications in population dynamics, Academic Press, Boston (1993)

• [16] M. Kudu, I. Amirali, G. M. Amiraliyev, A finite-difference method for a singularly perturbed delay integro-differential equation, J. Comput. Appl. Math., 308 (2016), 379--390

• [17] P. Markowich, M. Renardy, A nonlinear Volterra integro-differential equation describing the stretching of polymeric liquids, SIAM J. Math. Anal., 14 (1983), 66--97

• [18] F. A. Rihan, E. H. Doha, M. I. Hassan, N. M. Kamel, Numerical treatments for Volterra delay integro-differential equations, Comput. Methods Appl. Math., 9 (2009), 292--308

• [19] M. Shakourifar, W. Enright, Superconvergent interpolants for collocation methods applied to Volterra integro-differential equations with delay, BIT Numer. Math., 52 (2012), 725--740

• [20] Ö. Yapman, G. M. Amiraliyev, 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

• [21] C. J. Zhang, S. Vandewalle, Stability analysis of Volterra delay-integro-differential equations and their backward differentiation time discretization, J. Comput. Appl. Math., 164/165 (2004), 797--814

• [22] J. J. Zhao, Y. Cao, Y. Xu, Sinc numerical solution for pantograph Volterra delay-integro-differential equation, Int. J. Comput. Math., 94 (2017), 853--865

• [23] J. J. Zhao, Y. Fan, Y. Xu, Delay-dependent stability analysis of symmetric boundary value methods for linear delay integro-differential equations, Numer. Algorithms, 65 (2014), 125--151