{"title":"Development of Extended Trapezoidal Method for Numerical Solution of Volterra Integro-Differential Equations","authors":"Fuziyah Ishak, Siti Norazura Ahmad","volume":119,"journal":"International Journal of Mathematical and Computational Sciences","pagesStart":579,"pagesEnd":583,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/10005777","abstract":"

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.<\/p>\r\n","references":"\tA. M. Wazwaz, A First Course in Integral Equations. Singapore: World Scientific Publishing Company, 1997.\r\n\tJ. T. Day, \u201cNote on the numerical solution of integro-differential equations,\u201d Computer Journal, vol. 9 no. 4, pp. 394\u2013395, 1967.\r\n\tP. Linz, \u201cLinear multistep methods for Volterra integro-differential equations,\u201d Journal of the Association for Comp. Machinery, vol. 16 no. 2, pp. 295\u2013301, 1969.\r\n\tP. J. van der Houwen, and H. J. J. Riele, \u201cLinear multistep methods for Volterra integral and integro-differential equations,\u201d Mathematics of Computation, vol. 45 no. 172, pp. 439\u2013461, 1985.\r\n\tK. Maleknejad, and M. Shahrezaee, \u201cUsing Runge-Kutta method for numerical solution of the system of Volterra integral equations,\u201dApplied Mathematics and Computation, vol. 149 no.2, pp. 399\u2013410, 2004.\r\n\tM. Gachpazan, \u201cNumerical scheme to solve integro-differential equations system,\u201d Journal of Advanced Research in Scientific Computing, vol. 1 no. 1, pp. 11\u201321, 2009.\r\n\tS. K. Vanani, and A. Aminataei, \u201cNumerical solution of Volterra integro-differential equations,\u201d J. of Comp. Analysis and Applications, vol. 13, pp. 654\u2013662, 2011.\r\n\tF. Mazzia, and A. M. Nagy, \u201cSolving Volterra integro-differential equations by variable stepsize bock BS methods: properties and implementation techniques,\u201d Applied Mathematics and Computation, vol. 239, pp. 198\u2013210, 2014.\r\n\tA. Filiz, \u201cNumerical solution of linear Volterra integro-differential equations using Runge-Kutta-Fehlberg method,\u201d Applied and Computational Mathematics, vol. 3 no. 1, pp. 9\u201314, 2014.\r\n\tL. Zhang, and F. Ma, \u201cPouzet-Runge-Kutta-Chebyshev method for Volterra equations of the second kind,\u201d Journal of Computational and Applied Mathematics, vol. 288, pp. 323\u2013331, 2015.\r\n\tR. A. Usmani, and R. P. Agarwal, \u201cAn A-stable extended trapezoidal rule for the integration of ordinary differential equations,\u201d Computer and Mathematics with Applications, vol. 11 no. 12, pp. 1183\u20131191, 1985.\r\n\tM. M. Chawla, and M. A. Al-Zanaidi, and M. G. Al-Aslab, \u201cA class of stabilized extended one-step methods for the numerical solution of ordinary differential equations,\u201d Computers and Mathematics with Applications, vol. 29 no. 10, pp. 79\u201384, 1995.\r\n\tF. Ibrahim, A. A. Salama, A. Quazzi, and S. Turek, \u201cExtended one-step methods for solving delay differential equations,\u201d Applied Mathematics, vol. 8 no. 3, pp. 941\u2013948, 2014.\r\n\tF. Ibrahim, A. A. Salama, and S. Turek, \u201cA class of extended one-step methods for solving delay differential equations,\u201d Applied Mathematics, vol. 9 no. 2, pp. 593\u2013602, 2015.\r\n\tF. Ibrahim, F. A. Rihan, and S. Turek, \u201cExtended one-step schemes for stiff and non-stiff delay differential equations,\u201d extracted from http:\/\/www.mathematik.tu-dortmund.de\/papers\/IbrahimRihanTurek2015.pdf\r\n\tI. B. Jacques, \u201cExtended one-step methods for the numerical solution of ordinary differential equations,\u201d Int. Journal of Computer Mathematics, vol. 29 no. 2-4, pp. 247\u2013255, 1989.\r\n\tA. M. Wazwaz, Linear and Nonlinear Integral Equations. London: Springer Publishing Company, 2011.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 119, 2016"}