# Diagonally Multistep Block Method for Solving Volterra Integro-differential Equation with Delay

### Abstract

Volterra integro-differential equation with delay (VIDED) is solved using a diagonally multistep block method (DMB). This study provides the derivation of the DMB utilizing Taylor series with a constant step size strategy for treating the first order VIDED. In predictor-corrector mode, the DMB method combines the predictor and corrector formulae. It approximates two numerical solutions simultaneously within a block. The algorithm for the approximation solution is developed and the Newton-Cotes formulae are adapted in the DMB method to estimate the solution for an integral component. Theoretically, the consistency and zero stability that led to convergence properties are examined. The stability region also has been plotted. The numerical results indicate that the developed method is superior in terms of the number of steps, accuracy and computation time taken.

### References

Ali, H. A., (2009), Expansion method for solving linear delay integro-differential equation using B-spline functions, Engineering and Technology Journal, 27(10): 1651-1661.

Ali, N., Zaman, G., and Jung, I. H., (2020), Stability analysis of delay integro-differential equations of HIV-1 infection model, Georgian Mathematical Journal, 27(3): 331-340.

Baharum, N. A., Majid, Z. A., & Senu, N. (2022). Boole’s strategy in multistep block method for Volterra integro-differential equation. Malaysian Journal of Mathematical Sciences, 16(2),237-256.

Baharum, N. A., Majid, Z. A., Senu, N., & Rosali, H. (2022). Numerical approach for delay Volterra integro-differential equation. Sains Malaysiana, 51(12), 4125-4144.

Brunner, H., and Zhang, W., (1999), Primary discontinuities in solutions for delay integrodifferential equations, methods and Applications of Analysis, 6(4): 525-534.

Driver, R. D. (1977), Ordinary and Delay Differential Equation, New York: Springer-Verlag.

Ghomanjani, F., Farahi, M. H., and Pariz, N., (2017), A new approach for numerical solution of a linear system with distributed delays, Volterra delay-integro-differential equations and nonlinear Volterra-Fredholm integral equations by Bezier curves, Computational and Applied Mathematics, 36(3): 1349-1365.

Ismail, N. I. N., Majid, Z. A., & Senu, N. (2020). Solving neutral delay differential equation of pantograph type. Malaysian Journal of Mathematical Sciences (ICoAIMS2019), 14(S), 107-121.

Janodi, M. R., Majid, Z. A., Ismail, F., & Senu, N. (2020). Numerical solution of Volterra integrodifferential equations by hybrid block with quadrature rules method. Malaysian Journal of Mathematical Sciences, 14(2), 191-208.

Lambert, J. D. (1973), Computational Methods in Ordinary Differential Equations, New York: John Wiley and Sons.

Mahmoudi, M., Ghovatmand, M., and Noori Skandari, M. H., (2020), A novel numerical method and its convergence for nonlinear delay Volterra integrodifferential equations, Mathematical Methods in the Applied Sciences, 43(5): 2357-2368.

Moghimi, M. B., and Borhanifar, A. (2016), Solving a class of nonlinear delay integrodifferential equations by using differential transformation method, Applied and Computational Mathematics, 5(3): 142-149.

Rihan, F. A., Doha, E. H., Hassan, M. I., and Kamel, N. M., (2009), Numerical treatments for Volterra delay integro-differential equations, Computational Methods in Applied Mathematics, 9(3): 292-318.

Salih, R. K., Hassan, I. H., and Kadhim, A. J., (2014), An approximated solutions for thn order

linear delay integro-differential equations of convolution type using B-spline functions and Weddle Equations, Baghdad Science Journal, 11(1).

**Menemui Matematik (Discovering Mathematics)**, [S.l.], v. 45, n. 2, p. 208-223, nov. 2023. ISSN 0126-9003. Available at: <https://myjms.mohe.gov.my/index.php/dismath/article/view/23137>. Date accessed: 26 july 2024.

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.