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Generalized Seikkala Differentiability and its Application to Fuzzy initial value problem

This paper proposes a new generalized Seikkala derivative (gS-derivative) of a fuzzy-valued function. We see that, there are many elementary fuzzy-valued functions which occur frequently as solution of fuzzy differential equation, are not Seikkala differentiable but they are generalized Seikkala differentiable. We discuss some of the properties of proposed differentiability. Using gS-differentiability, we find the solution of fuzzy initial value problem.


Introduction
To study the solution of fuzzy initial and boundary value problems, we need the concept of differentiability of fuzzy-valued function. The fuzzy differential calculus is developed by different authors, like Dubois (1982) 1 , Puri and Ralescu (1983) 7 , Kaleva (1987) 5 and Seikkala (1987) 8 . With different types of differentiability concept, theory of fuzzy differential equation is extensively studied. Recently, solutions to fuzzy differential equations under generalized Hukuhara derivative using contractive-like mapping principles is studied by Elder et al. (2015) 2 .
Seikkala differentiability is simpler and stronger than the other differentiability. Unfortunately, there are many elementary fuzzy-valued functions which frequently occur in solution of fuzzy differential equations, are not Seikkala differentiable. The purpose of the paper is to generalize the Seikkala differentiability. We see that a larger class of elementary fuzzy-valued functions belong to a generalized Seikkala differentiability (gS-differentiability ) concept. We find the fuzzy solution of a fuzzy initial value problem using gSdifferentiability of a fuzzy-valued function.
The paper is organized in following manner. Section 2 contains basics of fuzzy numbers. The generalized Seikkala differentiability of a fuzzy-valued function is proposed and compared with Seikkala differentiability with appropriate examples in Section 3. The properties of gSdifferentiability are discussed in the same section. Conclusion is given in the following section.

Fuzzy numbers
Fuzzy numbers have been introduced by Lotfi A. Zadeh to handle nonspecific numerical quantities in a practical way. For instance, profit on some product is approximately known, 2 Rs, say, then we can express this approximate amount 2 by means of fuzzy number. The numbers which are near to 2, can also included in fuzzy number 2 with varying membership grade. That is, 1 is regarded as approximately 2 with degree of membership 0.5. Similarly, 3 can be regarded as approximately 2 with degree of membership 0.6. In general, when numbers are imprecise/inexact/partially known, they can be represented as fuzzy numbers rather than real numbers. The mathematical definition of fuzzy number is given as follows.
Definition 2.1 Let R be the set of real numbers and a: R  [0, 1] be a membership function of a fuzzy set. Then a is said to be a fuzzy number if there is some r 0  R such that a(r 0 ) = 1; function a is quasi-concave and upper semi-continuous on R and clousure of a set{r  R / a(r) > 0} forms a compact set. The set of fuzzy numbers on R is denoted by F(R).
To study the properties of fuzzy numbers, we define corresponding crisp (ordinary) sets.

Definition 2.2
The α-level set a α of any a  F(R) is a crisp set a α = {r  R / a(r) ≥ α}, for α(0,1] and the 0-level set a 0 is defined as the closure of the set {r  R / a(r) > 0}.
Using the theorem of Goetschel and Voxman 4 , we have following the characterization of a fuzzy number. Theorem 2.1 A fuzzy number a is defined by any pair a = (a 1 (α), a 2 (α)) of functions a i : [0,1]  R, defining the end-points of the α-level sets, satisfying the following conditions: (i) a 1 is a bounded nondecreasing left-continuous function for α ∈(0,1] and right-continuous at α = 0; (ii) a 2 is a bounded nonincreasing left-continuous function for α ∈(0,1] and right continuous at α = 0; (iii) a 1 (α) ≤ a 2 (α) for all α.
Addition, subtraction, multiplication of two fuzzy numbers are not the same as defined for real numbers. As fuzzy numbers are defined by membership functions, their arithmetic operations can be defined using continuous operation.

Definition 2.3
Addition and scalar multiplication using their α-level sets are given as follows: , for α [0, 1] and scalar λ R.

Generalized Seikkala differentiability and its properties
Rate of change of function is a derivative of that function. If we consider rate of change of imprecise or fuzzy function, we have fuzzy derivative concept. For example, some object is falling from specific height. The motion of the object after t seconds is known approximately can be expressed by fuzzy function. To find instantaneous rate of change of object with respect to time is given as fuzzy derivative at that time. Many authors have defined fuzzy derivatives mathematically in literature. In this paper, we use concept of Seikkala derivative of fuzzy function to study the fuzzy growth and decay systems.

Seikkala differentiability
Seikkala differentiability of fuzzy-valued function y: I F(R) is defined as follows. The definition is adopted from Seikkala 3 . Definition 3.1 Let I be subset of R and y be a fuzzy-valued function defined on I. The α-level sets y α (t) = [y 1 (t, α), y 2 (t, α)] for α [0,1] and t I. We assume that derivatives of y i (t, α), i = 1, 2 exist for all t  I and for each α. We define (y'(t)) α = [y 1 '(t, α), y 2 '(t, α)] for all t I, all α.
If, for each fixed t  I, (y'(t)) α defines the α-level set of a fuzzy number, then we say that Seikkala derivative of y(t) exists at t and it is denoted by fuzzy-valued function y'(t).
The Seikkala derivative involves two steps: (1) Check both level functions are differentiable or not (2) Check level sets of derivatives define fuzzy numbers or not.
Exponential functions are used to represent real-world applications, such as bacterial growth/decay, population growth/decline. If the initial population is imprecise or inexact then to represent the system following fuzzy functions are used. And to find rate of change of such function, we use fuzzy derivative. In particular, Seikkala derivative. So we consider some illustrations of how to find Seikkala derivatives of functions those are useful in real life situations.
The following fuzzy function is exists in fuzzy decay problem. We study the rate of change of the function using uncertain (fuzzy) derivative.
Therefore, Seikkala derivative of g does not exist.
We consider another example of derivative of fuzzy function which occur in uncertain periodic motion of an object.
We see that the uncertain functions defined in Example 3.1 and 3.2 exist in real life, are gSdifferentiable.

Properties
We prove some properties of gS-differentiability. Proof. Let f be a S-differentiable function then f α '(t) = [f 1 '(t, α), f 2 '(t, α)] defines a fuzzy number for each t in I. We can write which defines fuzzy number for each t in I. Hence the proof.
In all these four cases, f+g is gS-differentiable as f and g are gS-differentiable.

Theorem 3.4 If fuzzy-valued function f is gS-differentiable and λ in R then (λf) is gSdifferentiable.
Proof. Similar arguments as in Theorem 3.3, the result can be proved.

Application
Now we consider an application of gS-derivative to solve a fuzzy initial value problem. The first example is real life example of decay of radioactive isotope where the initial amount is uncertain. Thus the system is studied as fuzzy system because uncertainty exists in the system. The mathematical modeling of such problem gives us fuzzy initial value problem. To find rate of change of approximate amount of radioactive isotope at any time t, we use gSdifferentiability concept.
Example 4.1 A radioactive isotope has initial mass is around 200 mg, which is decreasing at a rate of present mass. Find the expression for the approximate amount of isotope remaining at any time.

Solution: Fuzzy Model
In this case, we have a fuzzy initial value problem
If y is Seikkala differentiable and [y 1 , y 2 ] defines fuzzy number, for each t  I, then y will be fuzzy solution of the problem. But as we have discussed in Pirzada(2017) 6 , y is not Seikkala differentiable and therefore, it cannot be a solution of problem (1). Now we consider the given problem under gS-differentiability, we see that y(t) is gSdifferentiable with y′ α (x) = [y 2 ′(t, α), y 1 ′(t, α)], as sufficient conditions for existence of a fuzzy number y' for each x: y 2 '(t, α) / α > 0 and y 1 '(t, α) / α < 0, for given fuzzy initial condition y(0) = c = (around 200) = (195,200,205). Hence equations (4) and (5) define fuzzy solution of the problem.

Interpretation
The amount of isotope at any time t is represent approximately using functions y 1 (t, α) and y 2 (t, α), α[0,1] given in equations (4) and (5) respectively. That is, we have infinite family of solution (y 1 , y 2 ). For α = 0.5, solution can be represented by two curves given following figure.
Example 4.2 Suppose biochemical oxygen demand (BOD) in water is of amount approximately D 0 . The rate of decay is proportional to the amount of dissolved oxygen in water at present. Determine the approximate amount of dissolved oxygen in water at time t.

Conclusions
The generalization of Seikkala derivative is defined in the proposed work. Different properties are discussed. We observed that the gS-differentiability is useful in finding the solution of fuzzy initial value problem. We have seen that solution of fuzzy decay model exists under gS-differentiability only. The solution of the problem does not exist under Seikkala differentiability of fuzzy-valued function. Further, we can extend its application to solve fuzzy boundary value problems and fuzzy partial differential equations.