Solution of fuzzy differential equations

Abstract

Hybrid system is a dynamic system that exhibits both continuous and discrete dynamic behavior‎. ‎The hybrid differential equations have a wide range of applications in science and engineering‎. ‎The hybrid systems are devoted to modeling‎, ‎design‎, ‎and validation of interactive systems of computer programs and continuous systems‎. ‎Hybrid fuzzy differential equations (HFDEs) is considered by Kim et al.‎ [11]. ‎In the present paper it is shown that the example presented by Kim et al‎. ‎in the Case I is not very accurate and in the Case II‎, ‎is incorrect.‎ ‎Namely‎, ‎the exact solution proposed by the authors in the Case II are not solutions of the given HFDE‎. ‎The correct exact solution is also presented here‎, ‎together with some results for characterizing solutions of FDEs under‎ ‎Hukuhara differentiability by an equivalent system of ODEs‎. ‎Then‎, ‎the homotopy analysis method (HAM) is applied to obtained the series solution of the HFDEs‎. ‎Finally‎, ‎we illustrate‎ ‎our approach by a numerical ‎example.‎