A New Technique for solving Picture Fuzzy Differential Equation

Picture Fuzzy set (PFS) is an extension of fuzzy set (FS) and intuitionistic fuzzy set (IFS) that can model the uncertainty by integrating the concept of positive, negative and neutral membership degree of an element. In this paper, the solution of Picture Fuzzy ordinary differential equation of first order by means of picture fuzzy number is exemplified and intend to define the picture fuzzy number for (∝, δ, β)-cut. Finally, we illustrate the numerical example for drug distribution in human body for different drug levels is discussed for determining its effectiveness and practicality of the first order differential equation involving picture fuzzy numbers.


Introduction
Ambiguity is an inevitable constituent of daily life and to deal with the uncertainty Fuzzy set theory (FST) is explored. FST established by Zadeh [1], plays a vital part in decision making (DM) under ambiguous situation. Though, in some case FST is not much skilled to play decisive role. Various extensions of fuzzy set (FS) have been made successfully in most of the real-world uncertain problems. A significant generalization of FST is the intuitionistic fuzzy set (IFS) theory developed by Atanassov [2] describing two functions stating the degree of membership and the degree of non-membership distinctly the sum of two degrees must not exceed one. Although IFS has been useful in various problems of real-life situations, it has some lacking to deal with neutrality. Degree of neutrality concept can be found when we come across with human opinions which involves more answers of type: yes, no, abstain, refusal.
In the Literature, Cuong and Kreinovich [3], presented the concept PFS which is a direct extension of FS and IFS by integrating the concepts of positive, negative and neutral membership degree of an element. Cuong [4] proposed distance measure between PFS and discussed its properties. Compositions of picture fuzzy relations was studied by Phong and co-authors [5]. Cuong and Hai [6] constructed fuzzy inference processes for some operations in picture fuzzy systems. Cuong [7] presented De Morgan fuzzy triples on PFS, Viet [8] presented picture fuzzy inference system based on membership graph, Singh [9] presented  [10] defined (α, δ, β)-cut of PFS and studied some of its properties. There are some imprecise or uncertain parameters arises, by modeling science and engineering problems. For an uncertainty model with differential equations, the imprecise differential equation concept emerges. Differentiation plays a vital role in science and engineering with imprecise or uncertain parameters. Many researchers [11][12][13][14][15][16][17][18][19] introduces Fuzzy differential equation to model this uncertainty. Later, intuitionistic fuzzy differential equation [20][21][22][23][24][25] was emerged. But these three logics have not the refusal term. To change such refusal situation, Picture fuzzy set were developed. The different factors of PFS have been applied in the differential equations (DE).
The structure of the article is as follows. The pre-requisite concepts and definitions are given in the preliminary section. Followed by preliminaries, the solution of the first-order differential equation with triangular picture fuzzy numbers as initial conditions are derived. Finally, the numerical example for drug distribution in human body for different drug levels are illustrated and its graphical interpretations are also shown. The future research scope is discussed in the conclusion part.

Preliminaries
Some of the basic definitions are discussed in this section.

Picture Fuzzy Set (PFS) [3]
A PFS ‫ܣ‬ on a universe X is an object of the form When , ∀ ‫,ܺ߳ݔ‬ ߟ ‫)ݔ(‬ = 0, then the PFS reduces into IFS.

(∝, , ࢼ)-cut of PFS [10]
Let (∝, ߜ, ߚ)-cut of PFS A of a universe set X is given as crisp subset defined by

Triangular Picture Fuzzy Sets [10]
A triangular picture fuzzy set is denoted by Here the solution of PFS is

Definition 3.2
The solution of the picture fuzzy differential equation be ‫)ݔ(ݕ‬ with its (∝, ߜ, ߚ)-cut is ‫, The solution is a strong if Otherwise the solution is weak.  (1) is With the initial condition That is, ----------(4) Applying the initial conditions, we get,  (5) and (6), Similarly, the solution of the first order DE is given by Then the solution is strong.

Case 4.2
When ܿ is positive constant, i.e., ܿ > 0, then (∝, ߜ, ߚ)-cut of the equation (1) we get With the initial condition With the initial condition Then the solution is strong.

Application
Consider a drug distribution (Concentration) in human body. In such an appropriate dose of medicine is crucial to fight the infection to human body. In the human body the amount of blood decreases in time, then the multiple doses of medicine are essential. The drug in a patient's blood decays at the rate y can be modeled by the decay equation. This rule can be expressed to a differential equation: