A New Approach to Fuzzy Differential Equations using Logarithmic Mean

In this study, we examine fuzzy differential equations by using a new and dynamic approach. A novel scheme based on logarithmic mean is discussed in detail and comparison with harmonic mean is also performed with error analysis. Obtained solutions reveal that one gets very accurate and effective results by applying this scheme to solve the fuzzy differential equations.


Introduction
Modeling dynamical systems using differential equations is usually abstruse or incomplete.They usually deal with the parameter values, initial and boundary conditions.Analytic and numerical methods for solving initial value problems are only utilized for establishing a selected behavior of systems [1][2][3][4].However, it is impossible to describe the whole system behaviors compatible with our partial knowledge.In order to cope with this task, fuzzy differential equations have been rapidly developed recent years.The concept of the fuzzy derivative was first given by Chang [5].After him, Prade defined the extension principle in their approach [6].Other methods on fuzzy systems and problems have been scrutinized by many other researchers [7][8][9][10][11][12][13][14][15].The numerical methods for solving fuzzy differential equations were introduced by Abbasbandy et.al.and Allahviranloo et.al. in [16][17][18].Buckley and Feuring introduced two analytical methods for solving nth -order linear differential equations with fuzzy initial value conditions.Ivaz et.al developed a numerical method namely trapezoidal rule for fuzzy differential equations and hybrid fuzzy differential equations [19].Bildik et al. studied on the numerical solution of initial value problems for non-linear trapezoidal formula with different types [20].http://www.ispacs.com/journals/jfsva/2018/jfsva-00417/International Scientific Publications and Consulting Services In this study, we implement logarithmic mean for solving first order fuzzy initial value problems.Logarithmic rule is a simple and powerful method to solve related ODEs numerically in engineering and physical problems.Our results also prove that this rule has a higher convergence order when compared to other one step methods.
denotes(explicitly the   level set of u .The following theorem shows when , uu    is a valid   level set [19].
Theorem 2.1.The sufficient conditions for , uu    to define the parametric form of a fuzzy number are as follows [8]: (i) u  is a(bounded(monotonic ()increasing() (non-decreasing) left-continuous function on (0,1] and right continuous for 0   , (ii) u  is a bounded monotonic decreasing (non-increasing) left-continuous function on (0,1] and right continuous for , where      ()  gt  are differentiable functions and (2.4)

Fuzzy Differential Equations
Consider the first order fuzzy differential equation ( , )  .In some cases, this system can be solved analytically.In most cases analytical solutions cannot be obtained, and a numerical approach must be considered.Some numerical methods such as Euler method, Nyström method, and predictor-corrector method, etc. for these types of problems presented in [23][24][25].
In the next section, we solve first-order fuzzy initial value problem by using non-linear logarithmic mean is given as follows: ,  , [11].
The exact solution of the given differential equation for 01   is given by  1 and also sketched as in Figure 1.We also give the results obtained with harmonic mean for comparison in Table 2      It goes without saying that the errors may not be taken into consideration for logarithmic mean.Also one cannot easily see the difference between exact and approximate solution plotted in Figure 2.  5 and also sketched as in Figure 3, and compared with harmonic mean which given in Table 6.

Conclusions
In this paper, we proposed logarithmic and harmonic mean for numerical solution of first-order fuzzy differential equations.One can easily see that using logarithmic mean gives very good results for fuzzy differential equations.Even, it gives exact solutions for Example 4.1 and 4.3.Harmonic mean is also used to compare our approximate solutions.It is shown that one can obtain very approximate solutions when we choose fuzzy numbers and fuzzy initial values which are appropriate for the harmonic function.Tables and figures also represent that absolute error for logarithmic mean is smaller, when exists, than those of harmonic means.We finally conclude that using logarithmic mean gives the best results to the solution of fuzzy differential equation when comparing with the other means in the literature.

Figure 3 :
Figure 3: • and hollow circle shows the values of logarithmic and harmonic mean respectively.
Standards *Authors declare that they have no conflict of interest.*Ethical approval: This article does not contain any studies with human participants or animals performed by any of the authors.

Definition 2.2. Let ,
[21,22] is the usual product between a scalar and a subset of R .Let I be a real interval.A mapping :F y I R is called a fuzzy process and its   level set is denoted derivative is very restrictive.Some authors introduced a more general definition of derivative for fuzzy number-valued function[21,22].Definition 2.4.Let : .(2.2) Theorem 2.2.Let : ( , ) where y is a fuzzy function of and the fuzzy variable y , and y is Hukuhara derivative of t y .If an initial value 00 () F y t y R is given then a fuzzy Cauchy problem of first order will be acquired as follows:( ) ( , ( )), ( ) ,yt f t y t y t y t t T      .(3.5) http://www.ispacs.com/journals/jfsva/2018/jfsva-00417/International Scientific Publications and Consulting Services

Table 1 :
. Numerical results of logarithmic mean for Example 4.1.
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Table 2 :
Numerical results of harmonic mean for Example 4.1.One can easily deduce that solutions are the same with exact solution for the logarithmic mean from Table1and Table2.Figure1also supports this claim.
Figure1: • and hollow circle shows the values of logarithmic and harmonic mean respectively.. We do not give here calculations for brevity.A comparison between the exact solution  ; Yt and the approximate solutions by logarithmic mean and harmonic mean   , are given inTable 3,4 and sketched in

Table 3 :
Numerical results of logarithmic mean for Example 4.2.

Table 4 :
Numerical results of harmonic mean for Example 4.2.

Table 5 :
Numerical results of logarithmic mean for Example 4.3.
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Table 6 :
Numerical results of harmonic mean for Example 4.3.