10005777

Volterra integro-differential equations appear in many models for real life phenomena. Since analytical solutions for this type of differential equations are hard and at times impossible to attain, engineers and scientists resort to numerical solutions that can be made as accurately as possible. Conventionally, numerical methods for ordinary differential equations are adapted to solve Volterra integro-differential equations. In this paper, numerical solution for solving Volterra integro-differential equation using extended trapezoidal method is described. Formulae for the integral and differential parts of the equation are presented. Numerical results show that the extended method is suitable for solving first order Volterra integro-differential equations.

[1] A. M. Wazwaz, A First Course in Integral Equations. Singapore: World Scientific Publishing Company, 1997.

[2] J. T. Day, “Note on the numerical solution of integro-differential equations,” Computer Journal, vol. 9 no. 4, pp. 394–395, 1967.

[3] P. Linz, “Linear multistep methods for Volterra integro-differential equations,” Journal of the Association for Comp. Machinery, vol. 16 no. 2, pp. 295–301, 1969.

[4] P. J. van der Houwen, and H. J. J. Riele, “Linear multistep methods for Volterra integral and integro-differential equations,” Mathematics of Computation, vol. 45 no. 172, pp. 439–461, 1985.

[5] K. Maleknejad, and M. Shahrezaee, “Using Runge-Kutta method for numerical solution of the system of Volterra integral equations,”Applied Mathematics and Computation, vol. 149 no.2, pp. 399–410, 2004.

[6] M. Gachpazan, “Numerical scheme to solve integro-differential equations system,” Journal of Advanced Research in Scientific Computing, vol. 1 no. 1, pp. 11–21, 2009.

[7] S. K. Vanani, and A. Aminataei, “Numerical solution of Volterra integro-differential equations,” J. of Comp. Analysis and Applications, vol. 13, pp. 654–662, 2011.

[8] F. Mazzia, and A. M. Nagy, “Solving Volterra integro-differential equations by variable stepsize bock BS methods: properties and implementation techniques,” Applied Mathematics and Computation, vol. 239, pp. 198–210, 2014.

[9] A. Filiz, “Numerical solution of linear Volterra integro-differential equations using Runge-Kutta-Fehlberg method,” Applied and Computational Mathematics, vol. 3 no. 1, pp. 9–14, 2014.

[10] L. Zhang, and F. Ma, “Pouzet-Runge-Kutta-Chebyshev method for Volterra equations of the second kind,” Journal of Computational and Applied Mathematics, vol. 288, pp. 323–331, 2015.

[11] R. A. Usmani, and R. P. Agarwal, “An A-stable extended trapezoidal rule for the integration of ordinary differential equations,” Computer and Mathematics with Applications, vol. 11 no. 12, pp. 1183–1191, 1985.

[12] M. M. Chawla, and M. A. Al-Zanaidi, and M. G. Al-Aslab, “A class of stabilized extended one-step methods for the numerical solution of ordinary differential equations,” Computers and Mathematics with Applications, vol. 29 no. 10, pp. 79–84, 1995.

[13] F. Ibrahim, A. A. Salama, A. Quazzi, and S. Turek, “Extended one-step methods for solving delay differential equations,” Applied Mathematics, vol. 8 no. 3, pp. 941–948, 2014.

[14] F. Ibrahim, A. A. Salama, and S. Turek, “A class of extended one-step methods for solving delay differential equations,” Applied Mathematics, vol. 9 no. 2, pp. 593–602, 2015.

[15] F. Ibrahim, F. A. Rihan, and S. Turek, “Extended one-step schemes for stiff and non-stiff delay differential equations,” extracted from http://www.mathematik.tu-dortmund.de/papers/IbrahimRihanTurek2015.pdf

[16] I. B. Jacques, “Extended one-step methods for the numerical solution of ordinary differential equations,” Int. Journal of Computer Mathematics, vol. 29 no. 2-4, pp. 247–255, 1989.

[17] A. M. Wazwaz, Linear and Nonlinear Integral Equations. London: Springer Publishing Company, 2011.