Numerical algorithm for solving second order nonlinear fuzzy initial value problems

Received Aug 23, 2019 Revised May 25, 2020 Accepted Jun 5, 2020 The purpose of this analysis would be to provide a computational technique for the numerical solution of second-order nonlinear fuzzy initial value (FIVPs). The idea is based on the reformulation of the fifth order Runge Kutta with six stages (RK56) from crisp domain to the fuzzy domain by using the definitions and properties of fuzzy set theory to be suitable to solve second order nonlinear FIVP numerically. It is shown that the second order nonlinear FIVP can be solved by RK56 by reducing the original nonlinear equation intoa system of couple first order nonlinear FIVP. The findings indicate that the technique is very efficient and simple to implement and satisfy the Fuzzy solution properties. The method’s potential is demonstrated by solving nonlinear second-order FIVP.


INTRODUCTION
As a conceptual model, several complex real-life problems can be articulated. Such models' equations may either be constructed as an ordinary or partial differential equation [1,2]. These equations can be solved via several method numerical or approximated method [3][4][5][6]. The FDEs are helpful in modelling a complex structure when knowledge is insufficient on its behavior. The FIVPs arises as such models have not been developed and are unpredictable in nature. These types of models can be defined by mathematical dynamic systems models under vagueness of FDEs. Real words, phenomena such as population physics and medicine take into account of fuzzy problems [5][6][7][8][9]. Ordinary differential. The efficacy of the methods proposed can be tested on linear FIVPs as simplest consideration. The dynamics of the Range Kutta methods depend on increasing number calculations in the methods function per each step that we can gain the method's ability to solve such problems [10].
The explicit fourth order Runge-Kutta methods have been widely used in the solution for first and high order FIVPs [11,12]. Various types of RungeKutte methods have been used in the last ducat to solve linear second order FIVPs, including the second order RungeKutta method [10], the third order Runge Kutta methods [13], and the fifth order Runge Kutta method of five stages in [14]. The focus is to develop and evaluate RK56's use to provide the non-linear second order FIVP numerical solution. This is the first attempt, to the best of our knowledge, to solve the nonlinear form of second order FIVP using RK56. The present article shall be arranged accordingly. We start with certain standard concepts for fuzzy numbers in section 2. The second order FIVP fuzzy analysis is described in section 3. We analyze and develop RK56 for the solution of the second order of FIVP in section 4. For the nonlinear second order FIVP solution, section 5 uses RK56 for a test example and display results in the form of table and figures. Finally, in section 6, we give the conclusion of this study.

PRELIMINARIES
Some definitions and most basic notes are standard and popular (see [15][16][17][18][19] for details), in this section, which will be used in this paper, are introduced Definition 2.1([20]): "The triangular fuzzy are linked to degrees of membership which state how true it is to say if something belongs or not to a determined set. As shown if Figure 1 this number is defined by three parameters  <  <  where the graph of μ(x) is a triangle with the base on the interval [, ] and vertex at x =  and its membership function has the following form:

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The r-level sets of a fuzzy number are much more effective as representation forms of fuzzy set than the above. Fuzzy sets can be defined by the families of their r-level sets based on the resolution identity theorem." Definition 2.5( [18]): Each function f: X → Y induces another function f: F(X) → F(Y) defined for each fuzzy interval U in X by: This is called the Zadeh extension principle. Definition 2.6 ([4]): "Considerx, ỹ ∈ Ẽ. If there exists z ∈ Ẽsuch thatx = ỹ + z, then zis called the H-difference (Hukuhara difference) of x and y and is denoted byz = x ⊝ ỹ." exists and equal to f (n) and for n = 2 we have second order Hukuharaderivative."

ANALYSIS OF FUZZY RK56
Here we take into account the six steps Runge Kutta of order five process as a framework for obtaining a numerical solution of second-order FIVP. In [21][22][23] the structure of this method to solve the crisp initial values problems of was introduced. Also in [24] RK56 to solve first order linear FIVPs with fuzzy intial conditions. In addition, we must defuzzify RKM56 in order to solve nonlinear second order FIVPs. We consider (1) in section 3 and since RK56 is numerical method we need to reduce (1) in to system of first order nonlinear FIVPs for all ∈ [0,1] : where ∈ [ 0 , ], and ̃0, ̃0 are fuzzy numbers for all ∈ [0,1] such that From (11)(12)(13), system (10) can be written with following assumptions for = 1,2 where ( where ( 0 ) ( ) = [ ,0 ( )] and Now we show that under the assumption for the fuzzy functions ̃, for = 1,2 solution for system (11) for each r-level set can obtain a unique fuzzy [11]. Now, for the analysis of fuzzy RK56 with a view to finding a numerical solution of system (11), we first the followings: The RK5 numerical solution for all r-level sets of (17-18) is as follows

NUMERICAL EXAMPLE
Consider the second-order fuzzy nonlinear differential equation [25]: From [25] the exact solution of (25) is: According to section 4, we apply RK5 to system (26) with selected ℎ = 0.1 which is enough to obtain convergence results as follows. According to RK56 in section 4, we have: From (27-28) we numerical solution of (25) as displayed in Table 1 and Figures 3 and 4: From Table 1, we can note that the high accuracy of the suggested method in ten step size only and the results or the lower and upper bounds of (25) flow the fuzzy solution as in definition 2.1. Additionally, Figure 3 shows that the RK56 solution over the all (25) given interval and for all the fuzzy level sets from 0 to 1 flow the fuzzy solution as in definition 2.1. Overall, as displayed in Table 1  6505 differential equations solution in the form of triangular fuzzy numbers. Figure 4 displayed the compression of the RK56 and exact solutions of (25).

CONCLUSION
In this paper a computational approach has been presented and developed for solving fuzzy ordinary differential equations. An approach is based on RK6 for solving nonlinear second order FIVP. A full fuzzy analysis is presented to analyze RK56 to be suitable for solving the proposed problem. Numerical illustration of non-linear second-order FIVP under fuzzy initial conditions demonstrate that the RKM56 is a capable and accurate process with few steps. The findings also by RK56 satisfy the characteristics of fuzzy numbers in triangular shape.