Numerical Solution of First Order Linear Differential Equations in Fuzzy Environment by Modified Runge-Kutta-Method and Runga-Kutta-Merson-Method under generalized H-differentiability and its Application in Industry

: The paper presents an adaptation of numerical solution of first order linear differential equation in fuzzy environment. The numerical method is re-established and studied with fuzzy concept to estimate its uncertain parameters whose values are not precisely known. Demonstrations of fuzzy solutions of the governing methods are carried out by the approaches, namely Modified Runge Kutta method and Runge Kutta Merson method. The results are compared with the exact solution which is found using generalized Hukuhara derivative (gH-derivative) concepts. Additionally, different illustrative examples and an application in industry of the methods are also undertaken with the useful table and graph to show the usefulness for attained to the proposed approaches.


Fuzzy differential equation
In modeling of real natural phenomena, differential equations play an important role in many areas of discipline, exemplary in economics, biomathematics, science and engineering. Many experts in such areas widely use differential equations in order to make some problems under study more comprehensible. In many cases, information about the physical phenomena related is always immanent with uncertainty.
Today, the study of differential equations with uncertainty is instantaneously growing as a new area in fuzzy analysis. The terms such as "fuzzy differential equation", "fuzzy differential inclusion" are used interchangeably in mention to differential equations with fuzzy initial values or fuzzy boundary values or even differential equations dealing with functions on the space of (ii) The numerical solutions are compared with the exact solutions which are found by using fuzzy derivative (generalized Hukuhara derivative) concepts. (iii) The use of Modified Runge Kutta method and Runge Kutta Merson method for solving fuzzy differential equation. (iv) The solutions are found using different step length for analyze accuracy of the result.
The necessary algorithm for numerical methods for finding the numerical solution are given. (vi) Numerical example and application are taken to show the applicability of the idea.

Structure of the paper
The paper is organized as follows: In Section 2 the preliminary concepts and basic concepts on fuzzy number, fuzzy derivative are written. In Section 3 we give the concept for finding the exact solution of a fuzzy differential equation. In Section 4 we discussed the numerical methods for finding the solution of fuzzy differential equation. Section 5 goes to convergence analysis of the numerical methods on fuzzy concepts. Section 6 refers the algorithm on the said techniques. Numerical examples are given in Section 7. In Section 8 application are given. Remarks from tables are discussed in Section 9. In Section 10 the conclusions of this article are drawn.
Also we say that ( ) is (i)-gH differentiable at if

Definition 2.6: Fuzzy ordinary differential equation (FODE):
Consider a simple 1 st Order Linear Ordinary Differential Equation as follows: The above ODE is called FODE if any one of the following three cases holds: (i) Only the initial condition i.e., is a generalized fuzzy number (Type-I). (ii) Only coefficients i.e., k is a generalized fuzzy number (Type-II). (iii) Both the initial condition and coefficients i.e., k and are generalized fuzzy numbers (Type-III).

Exact solution of Fuzzy Differential Equation:
Consider the fuzzy initial value problem Where is a continuous mapping from into R and ∈ with r-level sets (2) After taking -cut of the given FDE, it transform to system of ordinary differential equation. Then an approximation to the solution of initial value problem is made using higher order Runge-Kutta method of order 4:
The local truncation error at each step can be estimated using the following relation

Runge-Kutta-Merson method for Ordinary (Crisp) differential equation:
Runge-Kutta-Mersion method is an improved version of classical fourth-order Runge-Kutta method with the global error (ℎ ).
It can be written as: The local truncation error at each step can be estimated using the following relation

Convergence of numerical method on fuzzy differential equation:
The solution calculated by grid points at = ≤ ≤ ⋯ … … … … ≤ = and ℎ = = − Therefore, we have Proof: Before we going to the main proof we need to know some results.

Lemma 5.1: Let the sequence of numbers satisfy
For some given positive constants and . Then Therefore we can write, where, | | = | | + | |.
6. Algorithm for finding the numerical solution:

Step 7 Next
Step 8 End

Next
Step 8 End    Error with respect to MRK method when = .

Numerical Example
Error with respect to MRK method when = .
Error with respect to RKM method when = .
Error with respect to RKM method when = .
Error with respect to RKM method when = .  Error with respect to MRK method when = .
Error with respect to MRK method when = .
Error with respect to RKM method when = .
Error with respect to RKM method when = .
Error with respect to RKM method when = .

Application in Industry:
A tank initially contains 300 gals of brine which has dissolved in it lbs of salt. Coming into the tank at 3 gals/min is brine with concentration lbs salt/gals and the well stirred mixture leaves at the rate 3 gals/min. Let ( ) lbs be the salt in the tank at any time ≥ 0. Then     Error with respect to MRK method when = .
Error with respect to MRK method when = .
Error with respect to RKM method when = .
Error with respect to RKM method when = .
Error with respect to RKM method when = .  Error with respect to MRK method when = .
Error with respect to MRK method when = .
Error with respect to RKM method when = .
Error with respect to RKM method when = .
Error with respect to RKM method when = .

Remarks from above table on the solution
Our main aim is to compare the results between exact solution ((i)-gH and (ii)-gH) and two numerical methods namely Modified Runge Kutta method and Runge Kutta Mersion method on fuzzy differential equation. We show the comparison results between them with error estimation also. It is very difficult to consider the betterment of a method by solving a problem. In fuzzy case sometimes for different numerical example give different closure to exact solution. In crisp sense when we decrease the step length the numerical solution is tending to close to exact solution. But for fuzzy case it is happen not always.

Conclusion:
In this paper, Modified Runge Kutta Method and Runge Kutta Mersion method was taken into account to estimate the solution of first order linear ordinary differential equation with fuzzy initial condition, which is considered to be an important area of research with fuzzy differential equation. The approach generalized Hukuhara derivative concept, were applied to elucidate the exact fuzzy solutions of the given differential equation. Comprehensively, the whole deliberation reaches its conclusion with the following remarks: • Demonstrating numerical methods for finding the solution of first order linear ordinary differential equation with fuzzy initial condition • Compare the results with the exact solution which was found using generalized Hukuhara derivative approach.
• Numerical examples and application are taken to show for the importance of the mentioned numerical techniques.
• The accuracy for the result are discussed.
Thus in future we seek to apply these concepts to different types of differential equation and applications in fuzzy environments.