Numerical investigation of the hybrid fuzzy differential equations using He's homotopy perturbation method

S. Sekar; A. Sakthivel

doi:10.26637/mjm502/026

Abstract

This paper presents an efficient method namely He's Homotopy Perturbation Method (HHPM) is introduced for solving hybrid fuzzy differential equations based on Seikkala derivative with initial value problem [2]. The proposed method is tested on hybrid fuzzy differential equations. The discrete solutions obtained through He's Homotopy Perturbation Method are compared with Leapfrog method [13]. The applicability of the He's Homotopy Perturbation Method is more suitable to solve the hybrid fuzzy differential equations. Error graphs are presented to highlight the efficiency of the He's Homotopy Perturbation Method.

Keywords:

Fuzzy differential equations, Fuzzy initial value problems, Hybrid Fuzzy Differential Equations, Leapfrog method

Mathematics Subject Classification:

Mathematics

Authors Imprint References Funding Related Articles Downloads

S. Sekar Department of Mathematics, Government Arts College (Autonomous), Salem – 636 007, Tamil Nadu, India.
A. Sakthivel Department of Mathematics, Bharathiyar College of Engineering and Technology, Karaikal – 609 609, India.

Pages: 475-482
Date Published: 01-04-2017
Vol. 5 No. 02 (2017): Malaya Journal of Matematik (MJM)

R. Goetschel and W. Voxman, Elementary calculus, Fuzzy Sets and Systems, 18 (1986), 31-43. DOI: https://doi.org/10.1016/0165-0114(86)90026-6

T. Jayakumar and K. Kanakarajan, Numerical Solution for Hybrid Fuzzy System by Improved Euler Method, International Journal of Applied Mathematical Science, 38 (2012), 1847-1862.

T. Jayakumar and K. Kanagarajan, Numerical Solution of Hybrid Fuzzy Differential Equations by Adams Fifth Order Predictor-Corrector Method, International Journal of Mathematical Trends and Technology, 9 $(2014), 70-83$. DOI: https://doi.org/10.14445/22315373/IJMTT-V9P507

T. Jayakumar, K. Kanagarajan and S. Indrakumar, Numerical Solution of Nth Order Fuzzy Differential Equations by Runge Kutta Method of Order Five, International Journal of Mathematical Analysis, 6(58) $(2012), 2885-2896$.

K. Kanagarajan and S. Muthukumar, Extended Runge-Kutta method of order four for hybrid fuzzy differential Equations, International Journal of Applied Mathematics and Computation, 4(3)(2012), 312320.

K. Kanagarajan and M. Sambath, Numerical Solution Hybrid Fuzzy Differential Equations by Improved Predictor- Corrector method, Nonlinear Studies, 19(2012), 171-185.

H. Kim and R. Sakthivel, Numerical solution of hybrid fuzzy differential equations using improved predictor-corrector method, Communications in Nonlinear Science and Numerical Simulation, 17(10)(2012), 37883794. DOI: https://doi.org/10.1016/j.cnsns.2012.02.003

V. Lakshmikantham and R. N. Mohapatra, Theory of Fuzzy Differential Equations and Inclusions, Taylor and Francis, London, UK, (2003). DOI: https://doi.org/10.1201/9780203011386

V. Lakshmikantham and X. Z. Liu, Impulsive Hybrid systems and stability theory, International Journal of Nonlinear Differential Equations, 5(1999), 9-17.

S. Pederson and M. Sambandham, Numerical solution to hybrid fuzzy systems, Mathematical and Computer Modelling, 45(2007), 11331144. DOI: https://doi.org/10.1016/j.mcm.2006.09.014

M. Sambandham, Perturbed Lyapunov-like functions and Hybrid Fuzzy Differential Equations, International Journal of Hybrid Systems, 2(2002), 23-34.

S. Seikkala, On the fuzzy initial value problem, Fuzzy Sets and Systems, 24(3)(1987), 319330. DOI: https://doi.org/10.1016/0165-0114(87)90030-3

S. Sekar and K. Prabhavathi, Numerical Investigations of the Hybrid fuzzy differential equations using Leapfrog Method, International Journal of Pure and Applied Mathematics, 103(3)(2015), 385-394. DOI: https://doi.org/10.12732/ijpam.v103i3.1

S. Sekar and A. Sakthivel, Numerical Investigations of linear first order Fuzzy Differential Equations using He's Homotopy Perturbation Method, IOSR Journal of Mathematics, 11(2015),33-38.

S. Sekar and A. Sakthivel, Numerical Investigations of linear second order fuzzy differential equations using He's Homotopy Perturbation Method, International Journal of Advanced Science and Engineering Research, 1(2016), 1112-1117.

S. Sekar and A. S. Thirumurugan, Numerical Investigations of integro-differential equations using He's Homotopy Perturbation Method, International Journal of Advanced Science and Engineering Research, 1(2016), 1006-1011.

C. X. Wu and M. Ma. Embedding problem of fuzzy number space Part I, Fuzzy Sets and Systems, 44(1991), 33-38. DOI: https://doi.org/10.1016/0165-0114(91)90030-T