Fuzzy linear singular differential equations under granular differentiability concept

Abstract

In this paper, fuzzy linear singular differential equations under so-called granular differentiability are investigated in which the coefficients and initial conditions are as fuzzy numbers. Some new notions such as fuzzy nilpotent matrix, fuzzy linearly independent vectors, fuzzy eigenvectors, rank, index and fuzzy Jordan canonical form of a fuzzy matrix are introduced. Moreover, we compare the proposed method with other ones based on other derivatives, using some examples.

Introduction

Mathematical models based on linear and nonlinear differential equations are often used for modeling the behavior of phenomena. This paper concentrates on a general form of autonomous linear differential equations as:

$$E\dot{x}(t)=Ax(t)+B$$

where $E={[e_{ij}]}_{m\times n}$, $x(t)={[x_i(t)]}_{n\times 1}$, $A={[a_{ij}]}_{m\times n}$, and $B={[b_{ij}]}_{m\times 1}$. Two following cases can be considered for studying the differential equation (1):

• If the matrix E is nonsingular and $n=m$, then the differential equation (1) reduces to the following ordinary differential equation:

  $$\dot{x}(t)=E^{-1}Ax(t)+E^{-1}B$$

  where $E^{-1}$ is the inverse matrix of E.

• If the matrix E is singular then we have a singular differential equation. The basic theory of singular differential equations has been established in the nineteenth century; see the papers and books [1], [2], [3], [4], [5]. Since singular differential equations are more general than ordinary differential equations, they capture much attention in many fields of science such as in mathematics, computer science, engineering, control theory, fluid dynamics, and many other areas. The potential of the theory of singular differential equations in modeling of dynamical systems is explained in [6], [7], [8], [9], [10].

Because the existence of uncertainty in determining of mathematical model parameters of dynamical system is inevitable, then studying uncertain differential equations is very important. If there exists uncertainty in equation (2) and this uncertainty is modeled by fuzzy sets, then we have a fuzzy ordinary differential equation that has been studied in many papers such as [11], [12], [13], [14], [15]. Moreover, considering the uncertainty in equation (1), whenever E is a singular matrix, leads to studying of Fuzzy Linear Singular Differential Equations (FLSDEs) which is investigated in this paper with more details. In [16], using Strongly Generalized Hukuhara derivative (SGH-derivative), the fuzzy singular differential equation has been studied where the coefficients matrices are crisp and the initial conditions are fuzzy numbers. Here we consider FLSDEs in which not only the coefficients, but also initial conditions are uncertain, a more general form than what has been considered in [16]. In addition, the approach presented in [16] is on the basis of the Fuzzy Standard Interval Arithmetic (FSIA) and suffers from a number of limitations that are outlined in the sequel.

• Unnatural behavior in modeling phenomenon (UBM phenomenon); Based on the mentioned approach in [16], the solutions of the following FLSDEs are not the same:

  $$\{\begin{array}{c}E\dot{\tilde{x}}(t)=A\tilde{x}(t)+B\hfill \\ \tilde{x}(t_0)={\tilde{x}}_0\hfill \end{array}\{\begin{array}{c}E\dot{\tilde{x}}(t)-A\tilde{x}(t)=B\hfill \\ \tilde{x}(t_0)={\tilde{x}}_0\hfill \end{array}$$

  $$\{\begin{array}{c}E\dot{\tilde{x}}(t)-B=A\tilde{x}(t)\hfill \\ \tilde{x}(t_0)={\tilde{x}}_0\hfill \end{array}\{\begin{array}{c}E\dot{\tilde{x}}(t)-A\tilde{x}(t)-B=0\hfill \\ \tilde{x}(t_0)={\tilde{x}}_0\hfill \end{array}$$

• Doubling property; Based on the mentioned approach, an FLSDE might have two solutions - the solution corresponding to the first form of SGH-differentiability and the solution corresponding to the second form of SGH-differentiability - see Lemma 2.4 in [16].

Due to the mentioned restrictions imposed by the approach based on FSIA, a new approach dealing with FLSDEs is presented in this paper. The proposed approach is based on the Relative-Distance-Measure Fuzzy Interval Arithmetic (RDM-FIA) studied in [17], [18], [19], [20], [21], [22].

Briefly, in this paper the fuzzy linear singular differential equation under granular differentiability based on uncertain information is investigated. In these systems, all the coefficients and initial conditions can be uncertain. Therefore, based on the aforementioned explanations, a list of novel contributions of the paper is as follows:

• Considering the fuzzy linear singular differential equation structure as a fully fuzzy singular differential equation.

• Defining the fuzzy nilpotent matrix, fuzzy linearly independent vectors, fuzzy eigenvectors.

• Introducing the rank, index and fuzzy Jordan canonical form of a fuzzy matrix.

• Presenting the advantages and efficiency of the proposed approach in comparison with the previous approaches.

• Finally, the solution of fully fuzzy linear singular differential equations is obtained.

The rest of paper is organized as follows: Section 2 presents some preliminaries. Section 3 comes with fuzzy nilpotent matrix, fuzzy linearly independent vectors, fuzzy eigenvectors, rank, index and fuzzy Jordan canonical form of a fuzzy matrix. Section 4 will discuss how to solve a FLSDEs under granular differentiability. Finally, by presenting two examples in Section 5, the efficiency and applicability of the proposed approach are illustrated.