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

K Loganathan,S Karpagappriya,P Jayaraman,Nazek Alessa,José Francisco Gómez Aguilar

doi:10.1155/2021/5553732

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.

Highlights

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 in the field of economics [11] where the information are not given in the crisp set [12]
The authors concluded that a fuzzy differential equation can be modified into a system of ordinary differential equations (ODEs)
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]

Summary

Preliminaries

Let X′ {x} where X′ is the space of points and x is the generic element of X′. Definition 1 (see [2]). A fuzzy subset μA′ of the set A′ in X′ is a function μA′ : A′ ⟶ [0, 1]. 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. Let us denote the set of all fuzzy numbers on R as F which is a fuzzy number such that μ: R ⟶ [0, 1]. To define the differentiability of a fuzzy function, we can make use of this difference as follows. Suppose H is differential at the point t0 ∈ (u, v), all its α-level sets, Hα(t) [H(t)]α, are Hukuhara differentiable at t0 and [H′(t0)]α DHα(t0), where DHα denotes the Hukuhara derivatives of Hα and Hα as the multivalued mapping. Proof. is theorem has been proved in the work by Ahlberg et al [19] (p. 29)

Cubic Spline Function Approximation for Initial Value

Fuzzy Initial Value Problem

Cubic Spline Method for Solving Fuzzy Initial Value Problem

Conclusion