Abstract
Ambiguity in real-world problems can be modeled into fuzzy differential equations. The main 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.
1. Introduction
The 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]. This concept of fuzzy set theory was first introduced by Zadeh [2] in 1965. Chang and Zadeh explicated the concept of fuzzy derivatives [3]. The term fuzzy differential equation was formulated by Kandal and Byatt [4] in 1978. These 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. These 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 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. The 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.
2. Preliminaries
Let where is the space of points and is the generic element of .
Definition 1 (see [2]). A fuzzy subset of the set in is a function .
Definition 2 (see [17]). The -level set of the fuzzy set of is a crisp set if .
Definition 3 (see [17]). Let be a triangular fuzzy number (TFN) which is defined as where is the support, is the core, and the membership function iswhere .
Let us denote the set of all fuzzy numbers on as which is a fuzzy number such that .
Definition 4 (see [18]). Let . If there exists such that , then is the Hukuhara difference of and . This can be denoted as . To define the differentiability of a fuzzy function, we can make use of this difference as follows.
Let be differentiable at . If there exists some element such thatthen H is said to be Hukuhara differentiable at .
Suppose H is differential at the point , then all its -level sets, , are Hukuhara differentiable at and , where denotes the Hukuhara derivatives of and as the multivalued mapping.
Theorem 1 (see [19]). Let and be a sequence of partitions on , with ; then, for the interpolate cubic spline , uniformly for ,If satisfies the Holder condition on with , then
Proof. This 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 data points be , , where . Let us define the cubic spline , which is defined in the interval as follows.(i)For and , is a polynomial whose degree is one(ii) is at most a cubic polynomial in each subinterval , where (iii), and are continuous at each point , where (iv), where
If and , and are all continuous in , 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 . The cubic spline formula for an initial value problem in in terms of its first derivatives can be obtained by using Hermite’s interpolation formula as follows [21, 22]:where for all :
Setting and for all 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 and , the above equation becomes
On equating (8) and (11), we obtain
From this, we can compute ’s. Substituting these ’s in (5) gives the required solution. The convergence of this method has been proved by Patricio [21].
3. Fuzzy Initial Value Problem
Consider the first-order fuzzy differential equation aswith the initial condition , where is a fuzzy function of the crisp variable ; that is, , which is unknown. , which is a fuzzy function. is the fuzzy derivative of , and is a fuzzy number. Here, let us assume the fuzzy number to be a triangular fuzzy number.
For , let us denote the -level sets:
Also, where
The mapping is a fuzzy process, and the derivatives , for , are defined aswhere
Equation (13) can be replaced by an equivalent system of equations, and hence,where
The system of equations (20) and (21) will have a unique solution, . Thus, given fuzzy differential equation (13) possesses a unique solution on .
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.
4. Cubic Spline Method for Solving Fuzzy Initial Value Problem
Assume thatas the exact solution of (13):as the approximated solution of (13) at where .
Now let us calculate the solutions by mesh points at , , and , where .
The cubic spline function for a fuzzy initial value problem in in terms of its first derivatives is given aswhere . But, we know thatwhere
By carrying out simple and similar calculations for (26) and (27) as given in “cubic spline function approximation for initial value problems” (especially equations from (6)–(12)), we obtain the following set of equations:where and . From (28), ’s can be computed, and they are substituted in (26) to obtain the solution, . Similarly, ’s can be evaluated from (29) and are substituted in (27) to yield . Each value depends on th value, for .
Both these solutions collectively yield the desired solution of (13) at a fixed , .
4.1. Convergence of Fuzzy Cubic Spline Method
Let us consider the equations:
According to the results given in the work by Ahlberg et al. [19] (p. 34) and Theorem 1, if , we have
If , then the above equation can be written as
At , for , we have
Again, with (31), we obtain explicitly or not according to the linearity or nonlinearity of in . Then, we can writewhere and are constants and or
Hence, the order of the method is sustained, and it is true for .
From (26), we havewhere , .
Similarly, by considering the equations,we get
Thus, from (24), we obtain
5. Numerical Illustration (Exponential Decay Problem with Decay Constant as 1)
Consider the fuzzy differential equationwith as its fuzzy initial condition. Let us find the solution of (41) at and 0.3.
Equation (41) can be modified into a system of ordinary differential equations as follows:
The solution of these two equations collectively gives the solution of (41). Therefore, the exact solution of (41) is
Now let us compute the numerical solution of (41) by using the cubic spline method.
For simplicity, assume h = 0.1.
Consider equation (42), here and so and .
Also, .
Using (28) at i = 1 and , we get
Since , the above equation on simplification gives
Similarly, at i = 2 and , (28) becomes
This is the approximate solution of (42) at .
By using (42) at i = 3, we obtainwhere and . This equation gives the approximate solution of (42) at .
Now consider equation (43), here , and hence, , , and .
By using (29) at i = 1 and , we obtainwhere , for all i.
From (29), taking i = 2 and , we have
Similarly, for i = 3 and in (29), we getwhere which is given by (50) and . This equation on further simplification gives the approximate solution of (43) at .
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, at with . Comparison of exact and cubic spline solutions at is graphically given in Figure 1. Similarly, Figure 2 interprets the compared results of exact and cubic spline at of step length .
In general, the numerical solution of the fuzzy differential equation by using the cubic spline method can be given aswhere , i.e., for a fixed .
6. Conclusion
In this article, a new class of cubic spline function method is introduced for solving fuzzy differential equations subject to fuzzy initial conditions. The desired solution which is obtained is of convergence based on certain conditions on the derivatives. This 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, . 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.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally to this work. And, all the authors have read and approved the final version manuscript.
Acknowledgments
This research was funded by the Deanship of Scientific Research at Princess Nourah Bint Abdulrahman University through the Fast-Track Research Funding Program.