A Novel Approach for Solving Fuzzy Differential Equations Using Cubic Spline Method

Ambiguity in real-world problems can bemodeled into fuzzy differential equations.-emain objective of this work is to introduce a new class of cubic spline function approach to solve fuzzy initial value problems efficiently. Further, the convergence of this method is shown. As it is a single-step method that converges faster, the complexity of the proposed method is too low. Finally, a numerical example is illustrated in order to validate the effectiveness and feasibility of the proposed method, and the results are compared with the exact as well as Taylor’s method of order two.


Introduction
e entire real world is complex; it is found that the complexity arises from uncertainty in the form of ambiguity. Uncertainties in the real-world problem can be modeled easily with the help of fuzzy set theory when one lacks complete information about the variables and parameters [1].
is concept of fuzzy set theory was first introduced by Zadeh [2] in 1965. Chang and Zadeh explicated the concept of fuzzy derivatives [3]. e term fuzzy differential equation was formulated by Kandal and Byatt [4] in 1978. ese equations help in modeling the propagation of epistemic uncertainty in a dynamical environment [5]. Kaleva [6], Seikkala [7], and Song and Wu [8] have extensively studied the existence and uniqueness of solutions of these equations. A general formulation of the first-order fuzzy initial value problem was given by Buckley and Feuring [9]. Later, the fuzzy initial and boundary value differential equation was given by O'Regan et al. [10].
First-order linear fuzzy differential equations have inspired several authors to focus on solving them numerically since they appear in many real-world applications. ese applications include different fields of science such as medical diagnosis, biology, and civil engineering and also in the field of economics [11] where the information are not given in the crisp set [12]. Based on Zadeh's extension principle, a new fuzzy version of Euler's method was developed by Ahamed and Hasan [13]. Solving of these equations by the Taylor method of order p has been studied by Abbasbandy and Viranloo [14], and the same was discussed by Allahviranloo et al. [15] by using the predictor-corrector algorithm. Finally, the authors concluded that a fuzzy differential equation can be modified into a system of ordinary differential equations (ODEs). Also, they found out that there are two solutions for a fuzzy differential equation by solving the associated ODEs. e convergence, consistency, and stability for approximating the solution of fuzzy differential equations with initial value conditions have been studied by Ezzati et al. [16]. All the numerical results of these equations and their applications were summarized by Chakraverty et al. [12].
In this paper, the fuzzy initial value problem is solved numerically by using a new class of function approximation called cubic spline, for better accuracy of the solution.

Preliminaries
Let X ′ � x { } where X ′ is the space of points and x is the generic element of X ′ .
Definition 2 (see [17]). e α-level set of the fuzzy set Definition 3 (see [17]). Let A ′ be a triangular fuzzy number (TFN) which is defined as 〈l, m, n〉 where [l, n] is the support, m { } is the core, and the membership function is where l < m < n.
Let us denote the set of all fuzzy numbers on R as F which is a fuzzy number such that μ: R ⟶ [0, 1].
Definition 4 (see [18]). Let l and m ∈ F . If there exists n ∈ F such that l � m + n, then n is the Hukuhara difference of l and m. is can be denoted as n � l⊖m. To define the differentiability of a fuzzy function, we can make use of this difference as follows.
Let H: then H is said to be Hukuhara differentiable at t 0 . Suppose H is differential at the point t 0 ∈ (u, v), then all its α-level sets, H α (t) � [H(t)] α , are Hukuhara differentiable at t 0 and [H ′ (t 0 )] α � DH α (t 0 ), where DH α denotes the Hukuhara derivatives of H α and H α as the multivalued mapping.

(3)
If q ‴ (x) satisfies the Holder condition on [u, v] with 0 < α < 1, then Proof. is theorem has been proved in the work by Ahlberg et al. [19] (p. 29). □ 2.1. Cubic Spline Function Approximation for Initial Value Problems. Let the given (n + 1) data points be (u i , v i ), , and P i ″ (u) are all continuous in (u 0 , u n ), then this cubic spline is called as natural spline [20].
Many applications make use of slopes. So let us denote the cubic spline function that is obtained in terms of first derivatives to be m i . e cubic spline P(u) formula for an initial value problem in u i− 1 ≤ u ≤ u i in terms of its first derivatives P ′ (u i ) � m i can be obtained by using Hermite's interpolation formula as follows [21,22]: where h � u i − u i− 1 for all i: 2

Mathematical Problems in Engineering
Setting u � u i and P(u i ) � v i for all i in (7), we have Now consider a differential equation of first order with the initial condition as follows: On differentiating (9) twice with respect to u, Taking u � u i and P(u i ) � v i , the above equation becomes On equating (8) and (11), we obtain From this, we can compute P i 's. Substituting these P i 's in (5) gives the required solution.
e convergence of this method has been proved by Patricio [21].

Fuzzy Initial Value Problem
Consider the first-order fuzzy differential equation as with the initial condition u( Here, let us assume the fuzzy number to be a triangular fuzzy number.
e system of equations (20) and (21) will have a unique solution, . us, given fuzzy differential equation (13) possesses a unique solution on J.
Usually, equations (20) and (21) can be solved analytically. Yet, in most of the cases, this becomes tedious, and hence, a numerical approach to these systems of equations has to be considered.

and 3. (32)
If p � 2, then the above equation can be written as Again, with (31), we obtain [P l ] i explicitly or not according to the linearity or nonlinearity of G(ξ, u(ξ); α) in u(ξ). en, we can write where c 1 and c 2 are constants and α ∈ [0, 1] or Hence, the order of the method is sustained, and it is true From (26), we have where ξ j− 1 ≤ ξ ≤ ξ j , ∀j � 0, 1, . . . , n.
Similarly, by considering the equations, Mathematical Problems in Engineering we get us, from (24), we obtain

Numerical Illustration (Exponential Decay Problem with Decay Constant as 1)
Consider the fuzzy differential equation with y(0) � (0.5, 1, 1.5) as its fuzzy initial condition. Let us find the solution of (41) at ξ � 0.2 and 0.3. Equation (41) can be modified into a system of ordinary differential equations as follows: e solution of these two equations collectively gives the solution of (41). erefore, the exact solution of (41) is

(44)
Now let us compute the numerical solution of (41) by using the cubic spline method.
Tables 1 and 2 represent the comparison of the solutions for equation (41) that are obtained by exact, cubic spline method and Taylor's method of order, p � 2 at ξ � 0.2 with h � 0.1. Comparison of exact and cubic spline solutions at ξ � 0.2 is graphically given in Figure 1. Similarly, Figure 2 interprets the compared results of exact and cubic spline at ξ � 0.3 of step length h � 0.1.
In general, the numerical solution of the fuzzy differential equation by using the cubic spline method can be given as where i � 1, 2, . . .

Conclusion
In this article, a new class of cubic spline function method is introduced for solving fuzzy differential equations subject to fuzzy initial conditions. e desired solution which is obtained is of O(h 4 ) convergence based on certain conditions on the derivatives. is numerical method is verified with an example, and the results are compared with the exact as well as with the solution obtained by Taylor's method of order, p � 2. From the comparison of results, one can conclude that the proposed method is a single-step method that converges faster and has greater accuracy than the Taylor method of order two. In future, one can extend this method to solve higher-order linear and nonlinear fuzzy initial value problems.

Data Availability
No data were used to support the findings of the study.