Numerical Simulation of Fuzzy Volterra Integro-differential ‎Equation using Improved Runge-Kutta Method

Document Type : Research Paper


1 School of Engineering, Monash University Malaysia, 47500, Selangor, Malaysia

2 Institute of Mathematical Research, Universiti Putra Malaysia, 43400, Selangor, Malaysia‎

3 Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China,‎ Chengdu, 610054, Sichuan, PR China‎

4 Faculty of Mechanical and Industrial Engineering, Quchan University of Technology, Quchan, Iran

5 Department of Mathematics, University of Malakand, Dir(L), 18000, Khyber Pakhtunkhwa, Pakistan

6 Department of Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran‎

7 Department of Information Technology, Malaysia University of Science and Technology, 47810, Selangor, Malaysia‎


In this research, fourth-order Improved Runge-Kutta method with three stages for solving fuzzy Volterra integro-differential (FVID) equations of the second kind under the concept of generalized Hukuhara differentiability is proposed. The advantage of the proposed method in this study compared with the same order classic Runge-Kutta method is, Improved Runge-Kutta (IRK) method uses a fewer number of stages in each step which causes less computational cost in total. Here, the integral part is approximated by applying the combination of Lagrange interpolation polynomials and Simpson’s rule. The numerical results are compared with some existing methods such as the fourth-order Runge-Kutta (RK) method, variational iteration method (VIM), and homotopy perturbation method (HPM) to prove the efficiency of IRK method. Based on the obtained results, it is clear that the fourth-order Improved Runge-Kutta method with higher accuracy and less number of stages which leads the less computational cost is more efficient than other existing methods for solving FVID equations.


Main Subjects

Publisher’s Note Shahid Chamran University of Ahvaz remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

[1] Akin, O. Khaniyev, T., Oruc, O., Turksen, I.B., Some possible fuzzy solutions for second order fuzzy initial value problems involving forcing terms, Appl. Comput. Math., 13(2), 2014, 239-249.
[2] Abu Arqub, O., Adaptation of reproducing kernel algorithm for solving fuzzy Fredholm-Volterra integrodifferential equations, Neural. Comput. Applic., 28, 2017, 1591–1610.
[3] Bede, B., Gal, S.G., Generalizations of the differentiability of fuzzy number value functions with applications to fuzzy differential equations, Fuzzy Sets and Systems, 151(3), 2005, 581-599.
[4] Bede, B., Mathematics of fuzzy sets and fuzzy logic, Springer, Berlin, 2013.
[5] Ma, M., Friedman, M., Kandel, A., Numerical solutions of fuzzy differential equations, Fuzzy Sets and Systems, 105(1), 1999, 133-138.
[6] Abbasbandy, S., Viranloo, T.A., Numerical solution of fuzzy differential equation by Runge-Kutta method, J. Sci. Teacher Training University, 1(3-4), 2001-2002, 31-43.
[7] Ghanaie, Z.A., Moghadam, M.M., Solving fuzzy differential equations by Runge-Kutta method, Journal of Mathematics and Computer Science, 2(2), 2011, 208-211.
[8] Ghazanfari, B., Shakerami, A., Numerical solutions of fuzzy differential equations by extended RungeKutta-like formulae of order 4, Fuzzy Sets and Systems, 189(1), 2012, 74-91.
[9] Allahviranloo, T., Ahmady, N., Ahmady, E., Numerical solution of fuzzy differential equations by predictor-corrector method, Information Sciences, 177(7), 2007, 1633-1647.
[10] Matinfar, M., Ghanbari, M., Nuraei, R., Numerical solution of linear fuzzy Volterra integro-differential equations by variational iteration method, Intelligence and Fuzzy System, 24(3), 2013, 575-586.
[11] Allahviranloo, T., Ghanbari, M., Nuraei, R., An application of a semi-analytical method on linear fuzzy Volterra integral equations, Fuzzy Set Valued Analysis, 2014, 2014, 1-15.
[12] Rabiei, F., Abd Hamid, F., Rashidi, M. M., Ismail, F., Numerical simulation of fuzzy differential equations using general linear method and B-series, Advances in Mechanical Engineering, 9(9), 2017, 1-16.
[13] Hajighasemi, S., Allahviranloo, T., Khazerloo, M., Khorasany, M., Salahshour, S., Existence and uniqueness of solutions of fuzzy Volterra integro-differential equations, Information Processing and Management of Uncertainty in Knowledge-Based Systems, 2010, 491-500.
[14] Zeinali, M., Shahmorad, S., Mirnia, K., Fuzzy integro-diffrential equations: discrete solution and error estimation, Iranian Journal of Fuzzy Systems, 10(1), 2013, 107-122.
[15] Allahviranloo, T., Abbasbandy, S., Hashemzehi, S., Approximating the solution of the linear and nonlinear fuzzy Volterra integro-differential equations using expansion method, Abstract and Applied Analysis, 2014, Article ID 713892.
[16] Rabiei, F., Ismail, F., Suleiman, M., Improved Runge-Kutta methods for solving ordinary differential equations, Sains Malaysiana, 42(11), 2013, 107-122.
[17] Rabiei, F., Ismail, F., Ahmadian, A., Salahshour, S., Numerical Solution of Second-Order Fuzzy Differential Equation Using Improved Runge-Kutta Nystrom Method, Mathematical Problems in Engineering, 2013, Article ID 803462.
[18] Rabiei, F., Ismail, F., Numerical solution of fuzzy differential equation using improved Runge-Kutta method, GSTF Journal of Mathematics, Statistics and Operations Research, 2(2), 2014, 78-83.
[19] Linz, P. Analytical and Numerical Methods for Volterra Equations, SIAM, Philadelphia, 1985.
[20] Butcher, J.C., Numerical Methods or Ordinary Differential Equations, Second Edition, John Wiley and Sons, 2008.
[21] Mikaelvand, N., Khakrangin, S., Allahviranloo, T., Solving fuzzy Volterra integro-differential equation by fuzzy differential transform method, Proceedings of the 7th conference of the European Society for Fuzzy Logic and Technology, August 2011.
[22] Rashidi, M.M., Mohimanian Pour, S.A., Analytic Approximate Solutions for Unsteady Boundary-Layer Flow and Heat Transfer due to a Stretching Sheet by Homotopy Analysis Method, Nonlinear Analysis: Modelling and Control, 15(1), 2010, 83–95.
[23] Sarwar, S., Rashidi, M.M., Approximate Solution of Two-Term Fractional-Order Diffusion, Wave-Diffusion, and Telegraph Models Arising in Mathematical Physics Using Optimal Homotopy Asymptotic Method, Waves in Random and Complex Media, 26(3), 2016, 365–382.
[24] Rashidi, M.M., Shooshtari, A., Anwar Bég, O., Homotopy Perturbation Study of Nonlinear Vibration of Von Karman Rectangular Plates, Computers and Structures, 106–107, 2012, 46–55.
[25] Rashidi, M.M., Ferdows, M., Uddin, J., Beg, O., Rahimzadeh, N., Group Theory and Differential Transform Analysis of Mixed Convective Heat and Mass Transfer from a Horizontal Surface with Chemical Reaction Effects, Chemical Engineering Communications, 199(8), 2012, 1012–1043.