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## Numerical Solution Of First Order Fuzzy Differential Equation By Fourth Order Runge Kutta Method Based On Linear Combination Of Arithmetic Mean, Harmonic Mean And Geometric Mean

. Dr.R.Gethsi Sharmila *1, S.Suvitha *2

Abstract

Fuzzy Differential Equations (FDEs) have a wide range of applications in many branches of engineering and medicine. Chang S.L. and Zadeh L.A. in  first introduced the concept of fuzzy derivative. It was followed up by Dubosis.D and Prade.who used extension principle in their approach. The term “fuzzy differential Equation” was introduced in1987 by Kandel.A and Byatt.W.J. They have given many suggestions for the definition of fuzzy derivative to study “fuzzy differential Equation”. In the litreture, there are several approaches to study fuzzy differential equations. The first and most popular one is Hukuhara derivative made by Puri.M.L Ralesu.D.A . Seikkala  rigorously treated the FDE and the initial value problem (Cauchy problem). Here the solution of fuzzy differential equation becomes fuzzier as time goes on. This approach does not reproduce the rich and varied behaviour of ordinary differential equations. Bede.B and Gal.S.G [6,7] have introduced another concept of derivatives called the generalized Hukukara derivative. Under this interpretation, the solution of a fuzzy differential equation becomes less fuzzier as time goes on. A strong generalized derivative is defined for a large class of fuzzy number valued function than the Hukuhara derivative. This case of a fuzzy differential equation is not unique. it has two solution locally. The analytical methods for solving Nth - order linear differential equations with fuzzy initial value conditions was introduced by Buckley and Feuring . Recently some Mathematicians have studied fuzzy differential equations by Numerical methods. Numerical Solution of fuzzy differential equations has been introduced by M.Ma, M.Friedman, and Kandel.A in  through Euler method. Taylors method and Runge –Kutta methods have also been studied by authors in ,. The existence of solutions for fuzzy differential equations have been extensively studied by several other authors in [3,4]. It is difficult to obtain an exact solution for fuzzy differential equations and hence several numerical methods were proposed in [10,11,12,17].