Interval-valued fuzzy derivatives and solution to interval-valued fuzzy differential equations

Abstract

In this paper, we define the concept of fuzzy derivatives for perfect and semi-perfect interval-valued fuzzy mappings. Based on this definition, we then present a method for solving interval-valued fuzzy differential equations using the alpha-cut extension principle. Finally, several examples are provided to demonstrate the effectiveness of the proposed method.

Keywords

Generalized differentiability perfect and semi-perfect interval-valued fuzzy sets fuzzy differential equations

1 Introduction

Differential equations are powerful tools in modeling and simulating dynamical systems. However, in most of the real world systems, it is not possible to exactly determine the model structure and parameters and there is always a level of uncertainty in the model. In these cases, we often use some uncertainty handling methods such as those in probabilistic models as well as in rough [22, 56] and fuzzy set theories [26]. Recently, solving fuzzy differential equations has become an important part of describing the actual behavior of the real world systems, potentially also leading to their improved control. Such a theoretical paradigm has a wide array of applications, mostly when there is an abundance of imprecision, such as in engineering [1, 34], advertising [38], medical sciences [20], population models [33] and quantum optics and gravity [36].

In many of the above cases, however, the amount of uncertainties may be so high that the precisiation of information by the ordinary fuzzy sets is not sufficient. In such occasions, we employ higher order fuzzy sets such as interval-valued fuzzy sets and type-2 fuzzy sets to handle/manage the uncertainty. The model of such systems will then be in the form of higher order fuzzy differential equations. Hence, to describe these behaviors, we need to solve higher order fuzzy differential equations. In this paper, we consider the solution of fuzzy differential equations defined on the space of perfect, i.e. with normal memberships, and semi-perfect, i.e. with sub-normal memberships, interval-valued fuzzy numbers as a special case of higher order fuzzy sets.

Interval-valued fuzzy sets (IVFSs) are extensions of fuzzy sets (FSs). The development phases of the theory of the IVFSs took different pathways [8–11 , 40]. Hamrawi in his thesis [17] provided a comprehensive study about definitions and operations on the interval-valued fuzzy sets. He presented a methodology to allow functions and operations to be extended to the type-2 fuzzy sets. Additionally, he presented arithmetic operations for the type-2 fuzzy sets using a novel alpha-cut extension principle. More information about the type-2 fuzzy sets and the IVFS is provided in [5 , 27–29]. Up to now, the type-2 fuzzy sets have been used for applications such as: forecasting of time-series [39], controlling of mobile [19] and parallel robots [32 , 54], the truck backing-up control problem [3], clustering [7], pattern recognition [18], machine learning [4], design controls [30, 37] and signal processing [24]. Therefore, it seems that the application of IVFS in differential equations will remain a promising area of research.

Chang and Zadeh [44] proposed the concept of a fuzzy derivative in 1972. In 1983, this concept was followed by Puri and Ralescu [35]. They suggested a generalized and extended concept of Hukuhara differentiability (H-derivative) for a class of fuzzy mappings. Many works have been completed by several authors in the area of solving the Fuzzy Differential Equations (FDE) using numerical methods, like: Euler method [31], Adomain method [6, 50], Taylor methods [43], predictor-corrector method [49], improved predictor-corrector method [48] and by using concept of α-path [25]. Additionally, analytical methods for solving FDEs are suggested in [41, 55]. Khastan et al. [2] presented a general form of solution for the first order linear fuzzy differential equations by using the generalized Hukuhara differentiability concept. The gH-difference has been used extensively to propose some theories in the area of fuzzy differential equations (FDEs) [43] and fuzzy partial differential equations (FPDEs) [45 , 50] and also to devise some methods of solving such equations. For example, Allahviranloo et al. [46] used gH-difference to solve nonlinear fuzzy differential equation (NFDE), second-order fuzzy differential equations [51] and fuzzy Caputo fractional differential equation [53]. Moreover, Allahviranloo et al. [52] used gH-difference to solve FPDE of heat equation and also to show the existence and uniqueness of the answer.

In this paper, by extending the work of Cano and Flores [55], we introduce a generalized differentiability concept for the perfect and semi-perfect interval-valued fuzzy numbers. Then, we utilize this concept to solve the first order perfect and semi-perfect interval-valued fuzzy differential equations.

The rest of this paper is organized as follows. First in Section 2, some preliminaries are provided. A brief introduction to the PIVFNs and SPIVFNs is then provided in Section 3. Section 4 proposes the concept of generalized differentiability for the PIVFNs and SPIVFNs. In Section 5, the method of solving PIVFDEs and SPIVFDEs are introduced and the solutions of several different first order PIVFDEs and SPIVFDEs are provided. Finally, Section 6 concludes the paper.

2 Preliminaries

Let A and B be two non-empty bounded subsets of $ℝ^{n}$ . The Hausdorff distance between sets A and B is defined as $d_{2} (A, B) = sup_{x \in A} d_{1} (x, B)$ where $d_{1} (x, B) = inf_{y \in B} ∥ x - y ∥$ . Here ∥ .∥ stands for Euclidean norm. The Hausdorff metric between these two sets is obtained as follows $d (A, B) = max {d_{2} (A, B), d_{2} (B, A)} .$

A fuzzy set u in the universe W is defined as a mapping which assigns a membership value 0 ≤ μ_u (w)≤1 to each element w ∈ W. Each normal, convex, compact fuzzy set is called a fuzzy number (FN). The α-cut set of the fuzzy set u is defined as the set of all points whose membership value is greater than or equal to α. In other words, $[u]^{α} = {w \in W | μ_{u} (w) \geq α} .$

Using the α-cut definition, the arithmetic operations in fuzzy space is as follows $\begin{matrix} [u + v]^{α} = [u]^{α} + [v]^{α} . \\ [λ u]^{α} = λ [u]^{α} . \end{matrix}$

Let f(t) be a fuzzy function defined on the space of fuzzy numbers, then for all α ∈ [0, 1] the α-cut set of f(t) is a close interval on $ℝ$ that can be represented as $[f (t)]^{α} = [{\underline{f}}^{α} (t), {\bar{f}}^{α} (t)],$ where $\underline{f}$ and $\bar{f}$ are the two end points of the interval, which are functions of α and t. Knowing all the α-cut sets, the original function f(t) can be reconstructed using the decomposition theorem.

3 Basic concepts

In this section, we provide basic concepts necessary to define the perfect interval-valued fuzzy sets which are the main subjects of this paper. Since the main subject of this article is fuzzy numbers, it is assumed that all of the fuzzy sets and numbers are defined on the space of n dimensional real numbers, i.e. $X \subseteq ℝ^{n}$ . However, in some cases it may be possible to extend the definition to more general universes.

Definition 3.1.(IVFS) [12 –17] An interval-valued fuzzy set (IVFS), $\hat{A}$ , over X is a function defined as follows $\begin{matrix} \hat{A} : X \to I (U) \\ x \mapsto [{\underline{u}}_{x}, {\bar{u}}_{x}] \subset [0, 1] . \end{matrix}$ where U is the universe of all real numbers between 0 and 1 and I (U) is an interval defined on U. Thereby, $\hat{A} (x) = [{\underline{u}}_{x}, {\bar{u}}_{x}] \in I (U)$ represents interval grade of membership of x in $\hat{A}$ . IVFSs can be described as follows by classical set representation. $\begin{matrix} \hat{A} & = & {(x, \hat{A} (x)) | x \in X, \hat{A} (x) \subseteq U} \\ = & {(x, [{\underline{u}}_{x}, {\bar{u}}_{x}]) | x \in X, [{\underline{u}}_{x}, {\bar{u}}_{x}] \subseteq U} . \end{matrix}$

Any IVFS can be represented by $\hat{A} = (\underline{A}, \bar{A})$ , where $\bar{A}$ is the upper membership function (UMF) and $\underline{A}$ is the lower membership function (LMF) of $\hat{A}$ . Figure 1 shows a typical normal and convex IVFS.

Definition 3.2.(α-cut) [12 –17] The α-cut of an IVFS, $\hat{A} \in X$ , is defined by ${\hat{A}}_{α} = ({\underline{A}}_{α}, {\bar{A}}_{α})$ where ${\underline{A}}_{α} = [^{L} {\underline{a}}_{α},^{R} {\underline{a}}_{α}]$ and ${\bar{A}}_{α} = [^{L} {\bar{a}}_{α},^{R} {\bar{a}}_{α}]$ are the α-cuts of its LMF and UMF at the same level α, respectively (see Fig. 1).

Definition 3.3.(IVFN) [12 –17] Generally, an IVFS is said to be an interval-valued fuzzy number (IVFN) if both of its LMF and UMF are convex, and either,

its LMF and UMF are both normal or,

its UMF is normal.

Definition 3.4.(PIVFN) [17] When the first condition in Definition 3.3 occurs, then the IVFNs are called perfect IVFN (PIVFN). In other words, in this case, both of its LMF and UMF are FNs. Then, from representation Theorem we have $\begin{matrix} \underline{A} & = & ⋃_{\forall α} α [^{L} {\underline{a}}_{α},^{R} {\underline{a}}_{α}] \\ \bar{A} & = & ⋃_{\forall α} α [^{L} {\bar{a}}_{α},^{R} {\bar{a}}_{α}] \end{matrix}$ (1) $\hat{A} = ⋃_{\forall α} α ([^{L} {\underline{a}}_{α},^{R}], [^{L} {\bar{a}}_{α},^{R} {\bar{a}}_{α}])$ (2)

Definition 3.5.(SPIVFN) [17] When the second condition in Definition 3.3 occurs, then the IVFNs is called semi-perfect IVFN (SPIVFN). Therefore, from representation Theorem, we have, $\begin{matrix} \underline{A} & = & ⋃_{\forall α \in [0, h (a)]} α [^{L} {\underline{a}}_{α},^{R} {\underline{a}}_{α}] \\ \bar{A} & = & ⋃_{\forall α \in [0, 1]} α [^{L} {\bar{a}}_{α},^{R} {\bar{a}}_{α}] \end{matrix}$ (3) $\hat{A} = ⋃_{\forall α} α (\begin{matrix} [^{L} {\underline{a}}_{α},^{R} {\underline{a}}_{α}], [^{L} {\bar{a}}_{α},^{R} {\bar{a}}_{α}] α \leq H (\underline{A}) \\ 0, [^{L} {\bar{a}}_{α},^{R} {\bar{a}}_{α}] α > H (\underline{A}) \end{matrix}$ (4) where $H (\underline{A})$ is the height of LMF which is the largest membership grade attained by any element in its domain.

Definition 3.6.(PIVFN Arithmetic) [17] Let $\hat{A} = (\underline{A}, \bar{A})$ and $\hat{B} = (, \bar{B})$ be two PIVFNs and let * be any of the four basic arithmetic operations. Then, $\hat{A} * \hat{B}$ can be computed as given in Equation (5). $\begin{matrix} \hat{A} * \hat{B} & = & ⋃_{\forall α} α ([^{L} {\underline{a}}_{α},^{R} {\underline{a}}_{α}], [^{L} {\bar{a}}_{α},^{R} {\bar{a}}_{α}]) \\ * ([^{L} {\underline{b}}_{α},^{R} {\underline{b}}_{α}], [^{L} {\bar{b}}_{α},^{R} {\bar{b}}_{α}]) \\ = & ⋃_{\forall α} α (([^{L} {\underline{a}}_{α},^{R}] * [^{L} {\underline{b}}_{α},^{R}]), \\ ([^{L} {\bar{a}}_{α},^{R} {\bar{a}}_{α}] * [^{L} {\bar{b}}_{α},^{R} {\bar{b}}_{α}])) \end{matrix}$ (5)Definition 3.7.(SPIVFN Arithmetic) [17] Let $\hat{A} = (\underline{A}, \bar{A})$ and $\hat{B} = (\underline{B}, \bar{B})$ be two SPIVFNs represented as, $\begin{matrix} \hat{A} = ⋃_{\forall α} α (\begin{matrix} [^{L} {\underline{a}}_{α},^{R} {\underline{a}}_{α}], [^{L} {\bar{a}}_{α},^{R} {\bar{a}}_{α}] & α \leq H (\underline{A}) \\ 0, [^{L} {\bar{a}}_{α},^{R}] & α > H (\underline{A}) \end{matrix}) \\ \hat{B} = ⋃_{\forall α} α (\begin{matrix} [^{L} {\underline{b}}_{α},^{R}], [^{L} {\bar{b}}_{α},^{R}] & α \leq H (\underline{B}) \\ 0, [^{L} {\bar{b}}_{α},^{R} {\bar{b}}_{α}] & α > H (B) \end{matrix}) \end{matrix}$

In this case, either one or both of the LMFs may be sub-normal.

Next, assume that $H_{min} = min (H (\underline{A}), H (\underline{B}))$ and let * be any of the four basic arithmetic operations, then the basic arithmetic operation between $\hat{A}$ and $\hat{B}$ can be obtained as (6),

$\begin{matrix} \hat{A} * \hat{B} \\ = ⋃_{\forall α} α {\begin{matrix} ([^{L} {\underline{a}}_{α},^{R} {\underline{a}}_{α}] * [^{L} {\underline{b}}_{α},^{R} {\underline{b}}_{α}]), \\ ([^{L} {\underline{a}}_{α},^{R} {\bar{a}}_{α}] * [^{L} {\bar{b}}_{α},^{R}]) α \leq H_{min} \\ 0, ([^{L} {\underline{a}}_{α},^{R} {\bar{a}}_{α}] * [^{L} {\bar{b}}_{α},^{R} {\bar{b}}_{α}]) α > H_{min} \end{matrix} \end{matrix}$ (6)

Note that the operator *∈ { + , - , × , div } and 0 ∉ B if * = div. For more information about how the operations work on IVFSs refer to [17].

4 Interval-valued fuzzy derivatives

In this section we extend the concept of fuzzy derivative to interval-valued fuzzy mappings. As it is mentioned, we introduce a differentiability concept for the perfect interval-valued fuzzy numbers, PIVFN, and semi-perfect interval-valued fuzzy numbers, SPIVFN, by extending the work of Cano and Flores [55].

4.1 The perfect interval-valued fuzzy derivative

Definition 4.1.1. Let $\hat{F} : I \to \hat{ℱ}$ be a PIVFN as a function of t. Fix t₀ ∈ I. We say $\hat{F}$ is differentiable at t₀, if there exists an element ${\hat{F}}^{'} (t_{0}) = (\underline{F^{'}} (t_{0}), {\bar{f}}^{'} (T_{0})) \in \hat{ℱ}$ such that either

(1) For all h>0 sufficiently close to 0 we have,

$\begin{matrix} lim_{h \to 0^{+}} \frac{(\underline{F} (t_{0} + h), \bar{F} (t_{0} + h)) - (\underline{F} (t_{0}), \bar{F} (t_{0}))}{h} \\ = lim_{h \to 0^{+}} \frac{(\underline{F} (t_{0}), \bar{F} (t_{0})) - (\underline{F} (t_{0} - h), \bar{F} (t_{0} - h))}{h} \\ = ({\underline{F}}^{'} (t_{0}), {\bar{F}}^{'} (t_{0})) \end{matrix}$ (7)

(2) For all h<0 sufficiently close to 0 we have,

$\begin{matrix} lim_{h \to 0^{-}} \frac{(\underline{F} (t_{0} + h), \bar{F} (t_{0} + h)) - (\underline{F} (t_{0}), \bar{F} (t_{0}))}{h} \\ = lim_{h \to 0^{-}} \frac{(\underline{F} (t_{0}), \bar{F} (t_{0})) - (\underline{F} (t_{0} - h), \bar{F} (t_{0} - h))}{h} \\ = ({\underline{F}}^{'} (t_{0}), {\bar{F}}^{'} (t_{0})) \end{matrix}$ (8)

Definition 4.1.2. Let $\hat{F} : I \to \hat{ℱ}$ . We say $\hat{F}$ is (1)-differentiable on I if $\hat{F}$ is differentiable in the sense (1) of Definition 4.1.1 and its derivative is denoted as D₁F. Similarly we say $\hat{F}$ is (2)-differentiable on I if $\hat{F}$ is differentiable in the sense (2) of Definition 4.1.1 and its derivative is denoted as D₂F.

Theorem 4.1.1.Suppose that $\hat{F}$ is a PIVFN and a function of time with its LMF and UMF denoted as f and g, respectively. For eachα ∈ [0, 1],

$\begin{matrix} [{\overset{α}{F}}^{'} (t)]^{α} & = & {[^{L} f_{α} (t),^{R} f_{α} (t)], \\ [^{L} g_{α} (t),^{R} g_{α} (t)]} . \end{matrix}$ (9)

i) If $\hat{F}$ is (1)-differentiable then f_α and g_α are (1)-differentiable and we have,

$\begin{matrix} [{\hat{F}}^{'} (t)]^{α} & = & {[^{L} f_{α}^{'} (t),^{R} f_{α}^{'} (t)], \\ [^{L} g_{α}^{'} (t),^{R} g_{α}^{'} (t)]} . \end{matrix}$ (10)

ii) If $\hat{F}$ is (2)-differentiable then f_α and g_α are (2)-differentiable and we have,

$\begin{matrix} [{\hat{F}}^{'} (t)]^{α} & = & {[^{R} f_{α}^{'} (t),^{L} f_{α}^{'} (t)], \\ [^{R} g_{α}^{'} (t),^{L} g_{α}^{'} (t)]} . \end{matrix}$ (11)Proof. i) If h>0, we have, $\begin{matrix} [\hat{F} (t + h) - \hat{F} (t)]^{α} \\ = {[^{L} f_{α} (t + h) -^{L} f_{α} (t),^{R} f_{α} (t + h) -^{R} f_{α} (t)], \\ [^{L} g_{α} (t + h) -^{L} g_{α} (t),^{R} g_{α} (t + h) -^{R} g_{α} (t)]}, \end{matrix}$

By multiplying 1/h, we obtain, $\begin{matrix} \frac{[\hat{F} (t + h) - \hat{F} (t)]^{α}}{h} \\ = {[^{L} f_{α} (t + h) -^{L} f_{α} (t),^{R} f_{α} (t + h) -^{R} f_{α} (t)], \\ [^{L} g_{α} (t + h) -^{L} g_{α} (t),^{R} g_{α} (t + h) -^{R} g_{α} (t)]} / h \\ = {[^{L} f_{α} (t + h) -^{L} f_{α} (t),^{R} f_{α} (t + h) -^{R} f_{α} (t)] / h, \\ [^{L} g_{α} (t + h) -^{L} g_{α} (t),^{R} g_{α} (t + h) -^{R} g_{α} (t)] / h} . \end{matrix}$

Also we have, $\begin{matrix} \frac{[\hat{F} (t) - \hat{F} (t - h)]^{α}}{h} \\ = {[^{L} f_{α} (t) -^{L} f_{α} (t - h),^{R} f_{α} (t) -^{R} f_{α} (t - h)] / h, \\ [^{L} g_{α} (t) -^{L} g_{α} (t - h),^{R} g_{α} (t) -^{R} g_{α} (t - h)] / h} . \end{matrix}$

Finally, by letting h approach zero we have, $[{\hat{F}}^{'} (t)]^{α} = {[^{L} f_{α}^{'} (t),^{R} f_{α}^{'} (t)], [^{L} g_{α}^{'} (t),^{R} g_{α}^{'} (t)]} .$

This completes proof (i) of Theorem 4.1.1.

ii) If h<0, we have, $\begin{matrix} [\hat{F} (t + h) - \hat{F} (t)]^{α} \\ = {[^{L} f_{α} (t + h) -^{L} f_{α} (t),^{R} f_{α} (t + h) -^{R} f_{α} (t)], \\ [^{L} g_{α} (t + h) -^{L} g_{α} (t),^{R} g_{α} (t + h) -^{R} g_{α} (t)]} . \end{matrix}$ and multiplying both sides by 1/h we have, $\begin{matrix} \frac{[\hat{F} (t + h) - \hat{F} (t)]^{α}}{h} \\ = \frac{1}{h} {[^{L} f_{α} (t + h) -^{L} f_{α} (t),^{R} f_{α} (t + h) -^{R} f_{α} (t)], \\ [^{L} g_{α} (t + h) -^{L} g_{α} (t),^{R} g_{α} (t + h) -^{R} g_{α} (t)]} \\ = {[\frac{^{R} f_{α} (t + h) -^{R} f_{α} (t)}{h}, \frac{^{L} f_{α} (t + h) -^{L} f_{α} (t)}{h}], \\ [\frac{^{R} g_{α} (t + h) -^{R} g_{α} (t)}{h}, \frac{^{L} g_{α} (t + h) -^{L} g_{α} (t)}{h}]} . \end{matrix}$

Also we obtain, $\begin{matrix} \frac{[\hat{F} (t) - \hat{F} (t - h)]^{α}}{h} \\ = {[\frac{^{R} f_{α} (t) -^{R} f_{α} (t - h)}{h}, \frac{^{L} f_{α} (t) -^{L} f_{α} (t - h)}{h}], \\ [\frac{^{R} g_{α} (t) -^{R} g_{α} (t - h)}{h}, \frac{^{L} g_{α} (t) -^{L} g_{α} (t - h)}{h},]} . \end{matrix}$

Finally, by letting h approach zero we have, $[{\hat{F}}^{'} (t)]^{α} = {[^{R} f_{α}^{'} (t),^{L} f_{α}^{'} (t)], [^{R} g_{α}^{'} (t),^{L} g_{α}^{'} (t)]} .$

This completes proof (ii) of Theorem 4.1.1.

4.2 The semi-perfect interval-valued fuzzy derivative

In this section, we extend the concept of fuzzy derivative to semi-perfect interval-valued fuzzy mappings.

Definition 4.2.1. Let $\hat{F} : I \to \hat{ℱ}$ be a SPIVFN and function of t. Fix t₀ ∈ I. We say $\hat{F}$ is differentiable at t₀, if there exists an element ${\hat{F}}^{'} (t_{0}) = (\underline{F^{'}} (t_{0}), {\bar{F}}^{'} (t_{0})) \in \hat{ℱ}$ such that either

(1) For all h>0 sufficiently close to 0 we have,

(2) For all h < 0 sufficiently close to 0 we have,

Definition 4.2.2. We say $\hat{F}$ is (1)-differentiable on I if $\hat{F}$ is differentiable in the sense (1) of Definition 4.2.1 and its derivative is denoted as D₁F. Similarly we say $\hat{F}$ is (2)-differentiable on I if $\hat{F}$ is differentiable in the sense (2) of Definition 4.2.1 and its derivative is denoted as D₂F.

Theorem 4.2.1.Let $\hat{F}$ be a SPIVFN and a function of time. For eachα ∈ [0, 1] denote,

$\begin{matrix} [\hat{F} (t)]^{α} \\ = {\begin{matrix} [^{L} f_{α} (t),^{R} f_{α} (t)], α \leq H (f) \\ [^{L} g_{α} (t),^{R} g_{α} (t)] \\ 0, [^{L} g_{α} (t),^{R} g_{α} (t)] α > H (f) \end{matrix} \end{matrix}$ (14)

(i) If $\hat{F}$ is (1)-differentiable then f_α and g_α are (1)-differentiable and we have,

$\begin{matrix} [{\hat{F}}^{'} (t)]^{α} \\ = {\begin{matrix} [^{L} f_{α}^{'} (t),^{R} f_{α}^{'} (t)], α \leq H (f) \\ [^{L} g_{α}^{'} (t),^{R} g_{α}^{'} (t)] \\ 0, [^{L} g_{α}^{'} (t),^{R} g_{α}^{'} (t)] α > H (f) \end{matrix} \end{matrix}$ (15)

(ii) If $\hat{F}$ is (2)-differentiable then f_α and g_α are (2)-differentiable and we have,

$\begin{matrix} [{\hat{F}}^{'} (t)]^{α} \\ {\begin{matrix} [^{R} f_{α}^{'} (t),^{L} f_{α}^{'} (t)], α \leq H (f) \\ [^{R} g_{α}^{'} (t),^{L} g_{α}^{'} (t)] \\ 0, [^{R} g_{α}^{'} (t),^{L} g_{α}^{'} (t)] α > H (f) \end{matrix} \end{matrix}$ (16)

Proof. (i) If h > 0, we have, $\begin{matrix} [\hat{F} (t + h) - \hat{F} (t)]^{α} \\ = {\begin{matrix} \begin{matrix} {[^{L} f_{α} (t + h) -^{L} f_{α} (t),^{R} f_{α} (t + h) -^{R} f_{α} (t)], \\ [^{L} g_{α} (t + h) -^{L} g_{α} (t),^{R} g_{α} (t + h) -^{R} g_{α} (t)]} \end{matrix} & α \leq H (f) \\ 0, [^{L} g_{α} (t + h) -^{L} g_{α} (t),^{R} g_{α} (t + h) -^{R} g_{α} (t)] & α > H (f) \end{matrix} \end{matrix}$ By multiplying 1/h, we obtain, $\begin{matrix} [\hat{F} (t + h) - \hat{F} (t)]^{α} \\ = {\begin{matrix} \frac{1}{h} {[^{L} f_{α} (t + h) -^{L} f_{α} (t),^{R} f_{α} (t + h) -^{R} f_{α} (t)], α \leq H (f) \\ [^{L} g_{α} (t + h) -^{L} g_{α} (t),^{R} g_{α} (t + h) -^{R} g_{α} (t)]} \\ {0, [^{L} g_{α} (t + h) -^{L} g_{α} (t),^{R} g_{α} (t + h) -^{R} g_{α} (t)] / h} α > H (f) \end{matrix} \\ = {\begin{matrix} {[^{L} f_{α} (t + h) -^{L} f_{α} (t),^{R} f_{α} (t + h) -^{R} f_{α} (t)] / h, α \leq H (f) \\ [^{L} g_{α} (t + h) -^{L} g_{α} (t),^{R} g_{α} (t + h) -^{R} g_{α} (t)] / h} \\ {0, [^{L} g_{α} (t + h) -^{L} g_{α} (t),^{R} g_{α} (t + h) -^{R} g_{α} (t)] / h} α > H (f) \end{matrix} \end{matrix}$

Also, we obtain, $\begin{matrix} [\hat{F} (t) - \hat{F} (t - h)]^{α} \\ = {\begin{matrix} \begin{matrix} {[^{R} f_{α} (t) -^{R} f_{α} (t - h),^{L} f_{α} (t) -^{L} f_{α} (t - h)] / h, \\ [^{R} g_{α} (t) -^{R} g_{α} (t - h),^{R} g_{α} (t) -^{R} g_{α} (t - h)] / h} \end{matrix} & α \leq H (f) \\ {0, [^{R} g_{α} (t) -^{R} g_{α} (t - h),^{R} g_{α} (t) -^{R} g_{α} (t - h)] / h} & α > H (f) \end{matrix} \end{matrix}$

Finally, we have, $[{\hat{F}}^{'} (t)]^{α} = {\begin{matrix} \begin{matrix} [^{L} f_{α}^{'} (t),^{R} f_{α}^{'} (t)], [^{L} g_{α}^{'} (t),^{R} g_{α}^{'} (t)] & α \leq H (f) \end{matrix} \\ \begin{matrix} 0, [^{L} g_{α}^{'} (t),^{R} g_{α}^{'} (t)] & α > H (f) \end{matrix} \end{matrix}$

This completes proof (i) of Theorem 4.2.1.

(ii) If h<0, we have, $\begin{matrix} [\hat{F} (t + h) - \hat{F} (t)]^{α} \\ = {\begin{matrix} \begin{matrix} {[^{L} f_{α} (t + h) -^{L} f_{α} (t),^{R} f_{α} (t + h) -^{R} f_{α} (t)], \\ [^{L} g_{α} (t + h) -^{L} g_{α} (t),^{R} g_{α} (t + h) -^{R} g_{α} (t)]} \end{matrix} & α \leq H (f) \\ {0, [^{L} g_{α} (t + h) -^{L} g_{α} (t),^{R} g_{α} (t + h) -^{R} g_{α} (t)]} & α > H (f) \end{matrix} \end{matrix}$

By multiplying 1/h we obtain, $\begin{matrix} [\hat{F} (t + h) - \hat{F} (t)]^{α} \\ = {\begin{matrix} \begin{matrix} \frac{1}{h} {[^{L} f_{α} (t + h) -^{L} f_{α} (t),^{R} f_{α} (t + h) -^{R} f_{α} (t)], \\ [^{L} g_{α} (t + h) -^{L} g_{α} (t),^{R} g_{α} (t + h) -^{R} g_{α} (t)]} \end{matrix} & α \leq H (f) \\ {0, \frac{1}{h} [^{L} g_{α} (t + h) -^{L} g_{α} (t),^{R} g_{α} (t + h) -^{R} g_{α} (t)]} & α > H (f) \end{matrix} \\ = {\begin{matrix} \begin{matrix} {[\frac{^{R} f_{α} (t + h) -^{R} f_{α} (t)}{h}, \frac{^{L} f_{α} (t + h) -^{L} f_{α} (t)}{h}], \\ [\frac{^{R} g_{α} (t + h) -^{R} g_{α} (t)}{h}, \frac{^{L} g_{α} (t + h) -^{L} g_{α} (t)}{h}]} \end{matrix} & α \leq H (f) \\ {0, [\frac{^{R} g_{α} (t + h) -^{R} g_{α} (t)}{h}, \frac{^{L} g_{α} (t + h) -^{L} g_{α} (t)}{h}]} & α > H (f) \end{matrix} \end{matrix}$

Also we have, $\begin{matrix} [\hat{F} (t) - \hat{F} (t - h)]^{α} \\ = {\begin{matrix} \begin{matrix} {[^{R} f_{α} (t) -^{R} f_{α} (t - h),^{L} f_{α} (t) -^{L} f_{α} (t - h)] / h, \\ [^{R} g_{α} (t) -^{R} g_{α} (t - h),^{R} g_{α} (t) -^{R} g_{α} (t - h)] / h} \end{matrix} & α \leq H (f) \\ {0, [^{R} g_{α} (t) -^{R} g_{α} (t - h),^{R} g_{α} (t) -^{R} g_{α} (t - h)] / h} & α > H (f) \end{matrix} \end{matrix}$

Finally, we have, $\begin{matrix} [{\hat{F}}^{'} (t)]^{α} \\ = {\begin{matrix} {[^{R} f_{α}^{'} (t),^{L} f_{α}^{'} (t)], [^{R} g_{α}^{'} (t),^{L} g_{α}^{'} (t)]} & α \leq H (f) \\ {0, [^{R} g_{α}^{'} (t),^{L} g_{α}^{'} (t)]} & α > H (f) \end{matrix} \end{matrix}$

This completes proof (ii) of Theorem 4.2.1.

5 Solving perfect interval-valued fuzzy differential equations

The method for solving ordinary differential equations defined on PIVFNs and SPIVFNs is discussed in this section.

5.1 Solving perfect interval-valued fuzzy differential equations

Let us consider the following initial value fuzzy differential equation, ${\hat{x}}^{'} (t) = F (t, \hat{x} (t)), \hat{x} (0) = {\hat{x}}^{0}$ (17)

where F : [0, a] × ℱ → ℱ is a continuous fuzzy mapping and ${\hat{x}}^{0}$ is a PIVFN. The relations (10) and (11) give an almost straightforward method to find solution of the fuzzy differential Equation (17). From α-cut method, we have,

$\begin{matrix} [\hat{x} (t)]^{α} & = & {[^{L} u_{α} (t),^{R} u_{α} (t)], [^{L} v_{α} (t),^{R} v_{α} (t)]}, \\ {[{\hat{x}}^{0} (t)]}^{α} & = & {[^{L} u_{α}^{0} (t),^{R} u_{α}^{0} (t)], [^{L} v_{α}^{0} (t),^{R} v_{α}^{0} (t)]} \end{matrix}$ (18)and,

$\begin{matrix} [F (t, \hat{x} (t))]^{α} \\ = {f_{α} ({t, u}_{α} (t), v_{α} (t)), g_{α} ({t, u}_{α} (t), v_{α} (t))} \end{matrix}$ (19)

where u_α (t) = [^Lu_α (t) , ^Ru_α (t)] and v_α (t) = [^Lv_α (t) , ^Rv_α (t)].

Then, the following procedures should be followed to solve the fuzzy initial value problem.

Case 1. Supposing that ${\hat{x}}^{'} (t)$ is (1)-differentiable, from (10) we have, $[{\hat{x}}^{'} (t)]^{α} = {[^{L} u_{α}^{'} (t), R^{} u_{'}^{α} (t)], [^{L} v_{α} (t),^{R} v_{α} (t)]}$ and, ${\begin{matrix} u_{α}^{'} (t) = f_{α} ({t, u}_{α} (t), v_{α} (t)), u_{α} (0) = u_{α}^{0} \\ v_{α}^{'} (t) = g_{α} ({t, u}_{α} (t), v_{α} (t)), v_{α} (0) = v_{α}^{0} \end{matrix},$ (20)

Therefore, we have ${\begin{matrix} ^{L} u_{α}^{'} (t) =^{L} f_{α} ({t, u}_{α} (t), v_{α} (t)),^{L} u_{α} (0) =^{L} u_{α}^{0} \\ ^{R} u_{α}^{'} (t) =^{R} f_{α} ({t, u}_{α} (t), v_{α} (t)),^{R} u_{α} (0) =^{R} u_{α}^{0} \\ ^{L} v_{α}^{'} (t) =^{L} g_{α} ({t, u}_{α} (t), v_{α} (t)),^{L} v_{α} (0) =^{L} v_{α}^{0} \\ ^{R} v_{α}^{'} (t) =^{R} g_{α} ({t, u}_{α} (t), v_{α} (t)),^{R} v_{α} (0) =^{R} v_{α}^{0} \end{matrix}$ (21)

Case 2. Supposing that ${\hat{x}}^{'} (t)$ is (2)-differentiable, from (11) we have, $[{\hat{x}}^{'} (t)]^{α} = {[^{R} u_{α}^{'} (t),^{L} u_{α}^{'} (t)], [^{R} v_{α} (t),^{L} v_{α} (t)]}$ and, ${\begin{matrix} - u_{α}^{'} (t) = f_{α} ({t, u}_{α} (t), v_{α} (t)), u_{α} (0) = u_{α}^{0} \\ - v_{α}^{'} (t) = g_{α} ({t, u}_{α} (t), v_{α} (t)), v_{α} (0) = v_{α}^{0} \end{matrix}$ (22)

Therefore, we have, ${\begin{matrix} ^{L} u_{α}^{'} (t) =^{R} f_{α} ({t, u}_{α} (t), v_{α} (t)),^{L} u_{α} (0) =^{L} u_{α}^{0} \\ ^{R} u_{α}^{'} (t) =^{L} f_{α} ({t, u}_{α} (t), v_{α} (t)),^{R} u_{α} (0) =^{R} u_{α}^{0} \\ ^{L} v_{α}^{'} (t) =^{R} g_{α} ({t, u}_{α} (t), v_{α} (t)),^{L} v_{α} (0) =^{L} v_{α}^{0} \\ ^{R} v_{α}^{'} (t) =^{L} g_{α} ({t, u}_{α} (t), v_{α} (t)),^{R} v_{α} (0) =^{R} v_{α}^{0} \end{matrix}$ (23)

Example 5.1.1. Consider the fuzzy Malthusian problem, ${\begin{matrix} {\hat{x}}^{'} (t) = - λ \hat{x} (t) \\ \hat{x} (0) = {\hat{X}}^{0} \end{matrix}$ where λ > 0 and ${\hat{X}}^{0}$ is a symmetric triangular PIVFNwith support { [- a, a] , [- b, b] }. That is $[{\hat{X}}^{0}]^{α} = {[- a (1 - α), a (1 - α)], [- b (1 - α), b (1 - α)]} .$ If ${\hat{x}}^{'} (t)$ is (1)-differentiable, then we need to solve the following differential systems, $\begin{matrix} {\begin{matrix} ^{L} u_{α}^{'} (t) = - λ^{R} u_{α} (t),^{L} u_{α} (0) = - a (1 - α) \\ ^{R} u_{α}^{'} (t) = - λ^{L} u_{α} (t),^{R} u_{α} (0) = a (1 - α) \end{matrix} \\ {\begin{matrix} ^{L} v_{α}^{'} (t) = - λ^{R} v_{α} (t),^{L} v_{α} (0) = - b (1 - α) \\ ^{R} v_{α}^{'} (t) = - λ^{L} v_{α} (t),^{R} v_{α} (0) = b (1 - α) \end{matrix} \end{matrix}$

By solving the above differential systems of equations, we have, $\begin{matrix} [\hat{X}]^{α} = {[- a (1 - α) e^{λ t}, a (1 - α) e^{λ t}], \\ [- b (1 - α) e^{λ t}, b (1 - α) e^{λ t}]} for all t \geq 0 . \end{matrix}$

Next, if ${\hat{x}}^{'} (t)$ is (2)-differentiable, then we need to solve the following system of differential equations ${\begin{matrix} ^{L} u_{α}^{'} (t) = - λ^{L} u_{α} (t),^{L} u_{α} (0) = - a (1 - α) \\ ^{R} u_{α}^{'} (t) = - λ^{R} u_{α} (t),^{R} u_{α} (0) = a (1 - α) \\ ^{L} v_{α}^{'} (t) = - λ^{L} v_{α} (t),^{L} v_{α} (0) = - b (1 - α) \\ ^{R} v_{α}^{'} (t) = - λ^{R} v_{α} (t),^{R} v_{α} (0) = b (1 - α) \end{matrix}$ which yields, $\begin{matrix} [\hat{X}]^{α} & = & {[- a (1 - α) e^{- λ t}, a (1 - α) e^{- λ t}], \\ [- b (1 - α) e^{- λ t}, b (1 - α) e^{- λ t}]} \end{matrix}$

Choosing the value of parameters as: a = 1, b = 2 and λ= 1 will yield solutions shown in Fig. 2.

Example 5.1.2. As another example, consider the linear first order fuzzy differential equation, ${\begin{matrix} {\hat{x}}^{'} (t) = 2 t \hat{x} (t) + t {\hat{X}}^{0} \\ \hat{x} (0) = {\hat{X}}^{0} \end{matrix}$

where ${\hat{X}}^{0}$ is a symmetric triangular perfect IVFN with support { [- a, a] , [- b, b]}. If we study ${\hat{x}}^{'} (t)$ in the (1)-differentiable form, we yields, $\begin{matrix} {[\hat{X}]}^{α} & = & {[- a (1 - α) q, a (1 - α) q], \\ [- b (1 - α) q, b (1 - α) q]} \end{matrix}$ where q = 0.5 (3e^t² - 1) for all t ≥ 0. Next, if we study ${\hat{x}}^{'} (t)$ in the (2)-differentiable form, we have, $\begin{matrix} [\hat{X}]^{α} & = & {[- a (1 - α) q, a (1 - α) q], \\ [- b (1 - α) q, b (1 - α) q]} \end{matrix}$ where q = 0.5 (3e^-t² - 1). These solutions are depicted in Fig. 3.

Example 5.1.3. Consider the linear first order fuzzy differential equation, ${\begin{matrix} {\hat{x}}^{'} (t) = - 2 t \hat{x} (t) + t {\hat{X}}^{0} \\ \hat{x} (0) = {\hat{X}}^{0} \end{matrix}$ where ${\hat{X}}^{0}$ is a symmetric triangular perfect IVFN with support { [- a, a] , [- b, b] }.

By solving in in the (1)-differentiable form, we have, $\begin{matrix} [\hat{X}]^{α} & = & {[- a (1 - α) q, a (1 - α) q], \\ [- b (1 - α) q, b (1 - α) q]} \end{matrix}$ where q = 0.5 (3e^t² - 1) for all t ≥ 0. And next, if we study ${\hat{x}}^{'} (t)$ in the (2)-differentiable form, it yields to, $\begin{matrix} [\hat{X}]^{α} & = & {[- a (1 - α) q, a (1 - α) q], \\ [- b (1 - α) q, b (1 - α) q]} \end{matrix}$ where q = 0.5 (3e^-t² - 1). The (1) and (2) solutions of this example is similar to the solution of Fig. 5.1.2,as shown in Fig. 3.

5.2 Solving semi-perfect interval-valued fuzzy differential equations

Let us consider the following initial value fuzzy differential equation, ${\hat{x}}^{'} (t) = F (t, \hat{x} (t)), \hat{x} (0) = {\hat{x}}^{0}$ (24)

where F : [0, a] × ℱ → ℱ is a continuous fuzzy mapping and ${\hat{x}}^{0}$ is a fuzzy number. The relation (15) and (16) provide a method to find solution of the fuzzy differential Equation (24). By using α-cut method we have, $\begin{matrix} [\hat{x} (t)]^{α} \\ = {\begin{matrix} [^{L} u_{α} (t),^{R} u_{α} (t)], α \leq H (u) \\ [^{L} v_{α} (t),^{R} v_{α} (t)] \\ 0, [^{L} v_{α} (t),^{R} v_{α} (t)], α > H (u) \end{matrix} \\ [{\hat{x}}^{0} (t)]^{α} \end{matrix}$ (25) $= {\begin{matrix} [^{L} u_{α}^{0} (t),^{R} u_{α}^{0} (t)], α \leq H (u^{0}) \\ [^{L} v_{α}^{0} (t),^{R} v_{α}^{0} (t)] \\ 0, [^{L} v_{α}^{0} (t),^{R} v_{α}^{0} (t)] α > H (u^{0}) \end{matrix}$

and, $\begin{matrix} [F (t, \hat{x} (t))]^{α} \\ = {\begin{matrix} {f_{α} ({t, u}_{α} (t), v_{α} (t)), α \leq H (u) \\ g_{α} ({t, u}_{α} (t), v_{α} (t))} \\ 0, g_{α} ({t, v}_{α} (t)), α > H (u) \end{matrix} \end{matrix}$ (26)Then, the following method may be applied for solving the fuzzy initial value problem.

Case 1. Supposing that ${\hat{x}}^{'} (t)$ is (1)-differentiable, then from (15) we have, $\begin{matrix} [{\hat{x}}^{'} (t)]^{α} \\ = {\begin{matrix} [^{L} u_{α}^{'} (t),^{R} u_{α}^{'} (t)], [^{L} v_{α} (t),^{R} v_{α} (t)] α \leq H (u) \\ 0, [^{L} v_{α} (t),^{R} v_{α} (t)] α > H (u) \end{matrix} \end{matrix}$

If α ≤ h (u), ${\begin{matrix} u_{α}^{'} (t) = f_{α} ({t, u}_{α} (t), v_{α} (t)), u_{α} (0) = u_{α}^{0} \\ v_{α}^{'} (t) = g_{α} ({t, u}_{α} (t), v_{α} (t)), v_{α} (0) = v_{α}^{0} \end{matrix}$ (27)

and if α > h (u), $v_{α}^{'} (t) = g_{α} (t, v_{α} (t)) v_{α} (0) = v_{α}^{0} .$ (28)

Therefore, if α ≤ h (u) we have, $\begin{matrix} if α \leq h (u) then \to \\ {\begin{matrix} ^{L} u_{α}^{'} (t) =^{L} f_{α} ({t, u}_{α} (t), v_{α} (t)),^{L} u_{α} (0) =^{L} u_{α}^{0} \\ ^{R} u_{α}^{'} (t) =^{R} f_{α} ({t, u}_{α} (t), v_{α} (t)),^{R} u_{α} (0) =^{R} u_{α}^{0} \\ ^{L} v_{α}^{'} (t) =^{L} g_{α} ({t, u}_{α} (t), v_{α} (t)),^{L} v_{α} (0) =^{L} v_{α}^{0} \\ ^{R} v_{α}^{'} (t) =^{R} g_{α} ({t, u}_{α} (t), v_{α} (t)),^{R} v_{α} (0) =^{R} v_{α}^{0} \end{matrix} \\ if α > h (u) then \to \\ {\begin{matrix} ^{L} v_{α}^{'} (t) =^{L} g_{α} ({t, v}_{α} (t)),^{L} v_{α} (0) =^{L} v_{α}^{0} \\ ^{R} v_{α}^{'} (t) =^{R} g_{α} ({t, v}_{α} (t)),^{R} v_{α} (0) =^{R} v_{α}^{0} \end{matrix} \end{matrix}$ (29)

Case 2. If ${\hat{x}}^{'} (t)$ is (2)-differentiable, then from (16) we have, $\begin{matrix} [{\hat{x}}^{'} (t)]^{α} \\ = {\begin{matrix} [^{L} u_{α}^{'} (t),^{R} u_{α}^{'} (t)], [^{L} v_{α} (t),^{R} v_{α} (t)] α \leq H (u) \\ 0, [^{L} v_{α} (t),^{R} v_{α} (t)] α > H (u) \end{matrix} \end{matrix}$

If α ≤ h (u), ${\begin{matrix} - u_{α}^{'} (t) = f_{α} (t, u_{α} (t), v_{α} (t)), u_{α} (0) = u_{α}^{0} \\ - v_{α}^{'} (t) = g_{α} (t, u_{α} (t), v_{α} (t)), v_{α} (0) = v_{α}^{0} \end{matrix} .$ (30)

and if α > h (u), $- v_{α}^{'} (t) = g_{α} ({t, v}_{α} (t)), v_{α} (0) = v_{α}^{0} .$ (31)

Therefore, if α ≤ h (u) we have, $\begin{matrix} {\begin{matrix} ^{L} u_{α}^{'} (t) =^{R} f_{α} ({t, u}_{α} (t), v_{α} (t)),^{L} u_{α} (0) =^{L} u_{α}^{0} \\ ^{R} u_{α}^{'} (t) =^{L} f_{α} ({t, u}_{α} (t), v_{α} (t)),^{R} u_{α} (0) =^{R} u_{α}^{0} \\ ^{L} v_{α}^{'} (t) =^{R} g_{α} ({t, u}_{α} (t), v_{α} (t)),^{L} v_{α} (0) =^{L} v_{α}^{0} \\ ^{R} v_{α}^{'} (t) =^{L} g_{α} ({t, u}_{α} (t), v_{α} (t)),^{R} v_{α} (0) =^{R} v_{α}^{0} \end{matrix} \\ and if α > h (u) then, \\ {\begin{matrix} ^{L} v_{α}^{'} (t) =^{R} g_{α} ({t, v}_{α} (t)),^{L} v_{α} (0) =^{L} v_{α}^{0} \\ ^{R} v_{α}^{'} (t) =^{L} g_{α} ({t, v}_{α} (t)), 7^{R} v_{α} (0) =^{R} v_{α}^{0} \end{matrix} \end{matrix}$ (32)

Example 5.2.1. Consider the fuzzy Malthusian problem [55], ${\begin{matrix} {\hat{x}}^{'} (t) = - λ \hat{x} (t) \\ \hat{x} (0) = {\hat{X}}^{0} \end{matrix}$

where λ > 0 and ${\hat{X}}^{0}$ is a symmetric triangular SPIVFN with, $\begin{matrix} [{\hat{X}}^{0}]^{α} \\ = {\begin{matrix} [- a (H (a) - α), a (H (a) - α)] / H (a), & α \leq H (a) \\ \begin{matrix} [- b (1 - α), b (1 - α)] \\ 0, [- b (1 - α), b (1 - α)] \end{matrix} & α > H (a) \end{matrix} \end{matrix}$

If we study ${\hat{x}}^{'} (t)$ in the (1)-differentiable form, we need to solve the following differential systems, $\begin{matrix} if α \leq H (a) then \to \\ {\begin{matrix} ^{L} u_{α}^{'} (t) = - λ^{R} u_{α} (t),^{L} u_{α} (0) = - \frac{a}{H (a)} (H (a) - α) \\ ^{R} u_{α}^{'} (t) = - λ^{L} u_{α} (t),^{R} u_{α} (0) = \frac{a}{H (a)} (H (a) - α) \\ ^{L} v_{α}^{'} (t) = - λ^{R} v_{α} (t),^{L} v_{α} (0) = - b (1 - α) \\ ^{R} v_{α}^{'} (t) = - λ^{L} v_{α} (t),^{R} v_{α} (0) = b (1 - α) \end{matrix} \end{matrix}$

If α > H (a) then ${\begin{matrix} ^{L} v_{α}^{'} (t) = - λ^{R} v_{α} (t),^{L} v_{α} (0) = - b (1 - α) \\ ^{R} v_{α}^{'} (t) = - λ^{L} v_{α} (t),^{R} v_{α} (0) = b (1 - α) \end{matrix}$

By solving the above system of equations, we have, $\begin{matrix} [\hat{X}]^{α} \\ = {\begin{matrix} [- a (H (a) - α) e^{λ t}, a (H (a) - α) e^{λ t}] / H (a), α \leq H (a) \\ [- b (1 - α) e^{λ t}, b (1 - α) e^{λ t}] \\ 0, [- b (1 - α) e^{λ t}, b (1 - α) e^{λ t}] α > H (a) \end{matrix} \end{matrix}$

for all t ≥ 0. Next, if we study ${\hat{x}}^{'} (t)$ in the (2)-differentiable form, we need to solve the following differential systems, $\begin{matrix} if α \leq H (a) then \to \\ {\begin{matrix} ^{L} u_{α}^{'} (t) = - λ^{L} u_{α} (t),^{L} u_{α} (0) = - a (H (a) - α) / H (a) \\ ^{R} u_{α}^{'} (t) = - λ^{R} u_{α} (t),^{R} u_{α} (0) = a (H (a) - α) / H (a) \\ ^{L} v_{α}^{'} (t) = - λ^{L} v_{α} (t),^{L} v_{α} (0) = - b (1 - α) \\ ^{R} v_{α}^{'} (t) = - λ^{R} v_{α} (t),^{R} v_{α} (0) = b (1 - α) \end{matrix} \\ if α > H (a) then \to \\ {\begin{matrix} ^{L} v_{α}^{'} (t) = - λ^{L} v_{α} (t),^{L} v_{α} (0) = - b (1 - α) \\ ^{R} v_{α}^{'} (t) = - λ^{R} v_{α} (t),^{R} v_{α} (0) = b (1 - α) \end{matrix} \end{matrix}$

By solving the above system of equations, we have, $\begin{matrix} [\hat{X}]^{α} \\ = {\begin{matrix} \begin{matrix} [- a (H (a) - α) e^{- λ t}, a (H (a) - α) e^{- λ t}] / H (a), \\ [- b (1 - α) e^{- λ t}, b (1 - α) e^{- λ t}] \end{matrix} & α \leq H (a) \\ 0, [- b (1 - α) e^{- λ t}, b (1 - α) e^{- λ t}] & α > H (a) \end{matrix} \end{matrix}$

Let’s choose parameter values as: a = 1, b = 2, h(a) = 0.75 and λ= 1. These solutions are represented in Fig. 4.

Example 5.2.2. Consider the linear first order fuzzy differential equation, ${\begin{matrix} {\hat{x}}^{'} (t) = 2 t \hat{x} (t) + t {\hat{X}}^{0} \\ \hat{x} (0) = {\hat{X}}^{0} \end{matrix}$

where q = 0.5 (3e^t² - 1), for all t ≥ 0. Next, if we study ${\hat{x}}^{'} (t)$ in the (2)-differentiable form, we find, $\begin{matrix} [\hat{X}]^{α} \\ = {\begin{matrix} [- a (H (a) - α) q, a (H (a) - α) q] / H (a), α \leq H (a) \\ [- b (1 - α) q, b (1 - α) q] \\ 0, [- b (1 - α) q, b (1 - α) q] α > H (a) \end{matrix} \end{matrix}$

where q = 0.5 (3e^-t² - 1). These solutions are depicted in Fig. 5.

Example 5.2.3. As another example, consider the linear first order fuzzy differential equation, ${\begin{matrix} {\hat{x}}^{'} (t) = - 2 t \hat{x} (t) + t {\hat{X}}^{0} \\ \hat{x} (0) = {\hat{X}}^{0} \end{matrix}$

where ${\hat{X}}^{0}$ is a symmetric triangular SPIVFN. If we study ${\hat{x}}^{'} (t)$ in the (1)-differentiable form, we have, $\begin{matrix} [\hat{X}]^{α} \\ = {\begin{matrix} [- a (h (a) - α), a (h (a) - α)] / h (a), α \leq H (a) \\ [- b (1 - α), b (1 - α)] \\ 0, [- b (1 - α), b (1 - α)] α > H (a) \end{matrix} \end{matrix}$ where q = 0.5 (3e^t² - 1), for all t ≥ 0. Next, if we study ${\hat{x}}^{'} (t)$ in the (2)-differentiable form, then we find, $\begin{matrix} [\hat{X}]^{α} \\ = {\begin{matrix} [- a (H (a) - α), a (H (a) - α)] / H (a) α \leq H (a) \\ [- b (1 - α), b (1 - α)] \\ 0, [- b (1 - α), b (1 - α)] α > H (a) \end{matrix} \end{matrix}$

where q = 0.5 (3e^-t² - 1). These solutions are similar to the solution of 5.2.2, as shown in Fig. 5.

It is noteworthy that we have assumed the co-intensive property as defined by Zadeh, where co-intensiveness implies the fuzzy solution being close fitting to its prototype. This can be verified by removing the fuzzy quality from the analysis of this paper. In other words, once we have crisp information, the solution follows that of the fuzzy solution, with the fuzziness removed. It is clear that, if the fuzziness is removed from definition of interval-valued fuzzy derivative, Definitions 4.1.1 and 4.2.1, the classical definition of derivative is obtained. For further justification of this assumption, consider the provided examples in this section. If the fuzziness is removed, these examples reduce to classical initial valued ordinary differential equations with zero initial values, which clearly result in X = 0. Now, if the fuzziness is removed from the fuzzy solutions of these examples by setting α = 1, the crisp solutions of X = 0 are obtained again.

6 Conclusion

In this paper, we extend the concept of fuzzy differentiation to perfect and semi-perfect interval-valued fuzzy numbers using α-cut sets. This study presents a novel methodology to allow differential functions and operations to be extended directly from crisp sets to interval fuzzy sets. Utilizing the alpha-cut extension principle, we propose a novel approach to solve first order differential equations for perfect interval valued fuzzy numbers (PIVFNs) and semi-perfect interval valued fuzzy numbers (SPIVFNs). Several examples confirm the usefulness of the presented method. We also demonstrate that the principle of co-intensiveness applies here, i.e. by removing fuzziness, trivial crisp solutions are obtained.

The methodology outlined in this paper can be applied to all fuzzy ordinary differential equation that may appear in complex engineering applications of dynamical systems in general. Specifically, as future work, we hope to focus on studying the existence and uniqueness of the solutions of interval valued fuzzy differential equations and also on applying these proposed approaches in modeling and control of robotic test beds that are made by authors.

We also hope that the present work may form the basis for further research on fuzzy differentiation and their applications. Such approaches can serve towards more realistic modeling frameworks and more appropriate handling of uncertainty and complexity of dynamical systems, and subsequently overcoming the challenges of reaching higher efficiency and performance.

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