Towards granular calculus of single-valued neutrosophic functions under granular computing

Abstract

Neutrosophic theory studies objects whose values vary in the sets of elements and are not true or false, but in between, that can be called by neutral, indeterminate, unclear, vague, ambiguous, incomplete or contradictory quantities. In this paper, we firstly introduce preliminaries on granular calculus and analysis related to single-valued neutrosophic functions. Based on horizontal membership functions approach, we establish some basic arithmetic operations of single-valued neutrosophic numbers, that red allow us to directly introduce the terms of neutrosophic function in usual mathematical formulas. Additionally, we build metrics on the space of single-valued neutrosophic numbers induced from Hamming distance. Then, we define some backgrounds on the limit, derivative and integral of single-valued neutrosophic functions. Finally, in order to demonstrate the usable of our theoretical results, we present some applications to well-known problems arising in engineering such as logistic model, the inverted pendulum system, Mass - Spring - Damper model.

1 Introduction

1.1 Briefly review the calculus of uncertainty functions

Fuzzy sets were introduced by Zadeh [68] to manipulate data and information possessing nonstatistical uncertainties. After that, Zadeh and numerous researchers from the whole world have promoted fuzzy theory reaching to every aspects of engineering science. Nowaday, based on the Mathematics Subject Classification of American Mathematical Society (MSC2010 database), fuzzy theory has formed many different branches such as fuzzy logic, fuzzy graph theory, fuzzy algebraic structures, fuzzy real analysis, fuzzy measure theory, fuzzy differential equations, fuzzy topology, fuzzy control systems, fuzzy probability, etc. Fuzzy theory have a bright future like today, beside many breakthrough researches in algebraic structures of fuzzy numbers space, there has been many researches on fuzzy calculus and fuzzy analysis. In order to model real world systems containing uncertainty by fuzzy differential equations or dinamic systems, the concept of derivative calculus must be introduced. Derivative calculus of fuzzy valued functions were dependened on the type of difference arithmetic correspondently. The first fuzzy derivative seem to be introduced in 1972 [17]. Not long after that, extensive researches on this issue were conducted, namely by Dubois-Prade derivatives [21], Puri-Ralescu derivative based on Hukuhara distance [39], Goetschel-Voxman derivative [23], Seikkala derivative [42] and Friedman-Ming-Kandel derivative [22]. However, when applying these derivatives into engineering problems, there have been appeared some disadvantage and drawback such as the uncertainty of solution of one engineering problem modeling by fuzzy dinamic systems increases when time tends to infinity. It was not until 2005 [8] that Bede and Gal invented strongly generalized Hukuhara derivative. With slightly different notion, Bede and Stefanini [10, 51] introduced generalized Hukuhara derivative. These concepts of fuzzy derivative have been openning up a period of applied researches of fuzzy mathematics in modeling of control system, dynamic scale of economy, etc, see [9] for example.

The fuzzy set of Zadeh is actually characterized by a membership function with the range of [0, 1],i.e., we measure the uncertainty degree of an object belonging to a fuzzy set via single value in interval [0, 1]. However in actual practice, due to the influence of some margin of hesitation, an element may neither belong to fuzzy set nor do not belong to fuzzy set. In the language of fuzzy set the total degree of membership with non-membership of an element in a fuzzy set is generally not equal to. Therefore, Atanassov [5] introduced Intuitionistic fuzzy sets as an extension of fuzzy set of Zadeh. In the view of intuitionistic fuzzy set, an element has degrees of membership and non-membership, relatively independent. A comprihensive study on intuitionistic fuzzy sets can be referenced from [6, 7]. However, as we know, the up to date researchers on intuitionistic fuzzy sets focus on algebraic structure, rarely studies on analysis and topological structures of intuitionistic fuzzy sets space. That has greatly limited the applications of intuitionistic fuzzy logic in engineering, where systems are often modeled by differential equations or control problems.

Neutrosophic set (NS) and neutrosophic logic were invented by Smarandache [43], which are really extension of appeared earlier logic in the the philosophical and mathematical aspects. Neutrosophy logic orients the study of statement that are not true, nor false, but neutral, indeterminate, contradictory or something in between. On the mathematical side, every field posses its own neutrosophic part, namely indeterminacy part. Thus, engineering studies rise to research topics which the underlying are the neutrosophic set and logic, the neutrosophic probability and statistics, the neutrosophic dynamic system and modeling, etc. Smarandache [46] laid the first attempt to study neutrosophic precalculus and neutrosophic calculus based on the existing calculus of interval analysis. Neutrosophic algebraic structures and neutrosophic cognitive maps were investigated in [14, 16]. Neutrosophic measure, neutrosophic probability and statistics were studied in [13, 44, 45, 52]. Neutrosophic systems application in decision making seem to be very successful. Ye [62, 64] proposed a multi-attribute decision making (MADM) method using the correlation coefficient under single-valued neutrosophic environment. Ye [63] further developed clustering method and decision making methods by similarity measures of SVNS. Meanwhile, Ye [65] presented cross entropy measures of SVNS and applied them to decision making (for more details, see [1,2,3,4, 11, 12, 18,19,20, 24,25,26,27,28, 32,33,34,35, 40, 41, 47,48,49,50, 53,54,55,56,57,58,59, 61, 66, 67]).

In some latest publications, based on horizontal membership function approach and granular computing, Mazandarani et al [29,30,31] studied fuzzy differential systems and related problems, which can be considered as a particular scenario of neutrosophic dynamic systems. However, neutrosophic set theory in general and neutrosophic dynamic systems in particular are still in the first stage of development. The main achievements focus on algebraic structures of neutrosophic sets. Recently, there are only some literature that have attained the first step in defining the distance between neutrosophic sets and neutrosophic numbers, see [34, 35, 63, 66] or have introduced some most fundamental concepts in neutrosophic calculus, see [44, 46, 52] for example. However, until now, the studies on analysis structures and topological structures on the space of neutrosophic set and neutrosophic numbers have almost never appeared. The reason for this disadvantages comes from the intrinsic nature of space of neutrosophic sets. For more details, due to opposite number law does not make sense in the space of neutrosophic sets, i.e., if $\mathcal A$ is a NS and $-\mathcal A$ is the opposite element then in general $\mathcal A + (-A)$ is not the zero NS. Thus, the subtraction operation defined by $\mathcal A-\mathcal B=\mathcal A+(-\mathcal B)$ is not the candidate for difference operator in neutrosophic derivative calculus. Hence, it leads to big challenges for researchers if we want to study the analysis properties as well as constructing dynamical models for this object. Furthermore, the multi-coordinate $\mathcal A(T_{\mathcal A},I_{\mathcal A},F_{\mathcal A})$ of neutrosophic set makes more complicate when studying topological structure of the space. As the best of our knowledge, there does not have any suitable derivative concept defined for the neutrosophic-valued functions yet. Hence, it is one of the dynamics that promotes us in this work.

1.2 Contributions and structure of the paper

As the aforesaid in previous section, the difficulty in defining a suitable difference between neutrosophic sets is the limit of research in analysis calculus of neutrosophic-valued functions. Consequently, this leads to the study of many significant engineering problems related to derivative of a neutrosophic-valued functions such as modeling a systems by neutrosophic differential equations, modeling the evolution of a species by neutrosophic dynamic systems, the control problems of a neutrosophic-valued target or approximation of an underlying Input/Output systems by a neutrosophic systems, having no progress. Hence, the aims of this paper are

• 1. Throughout this paper, we introduce three types of single valued triangular neutrosophic numbers with triangular memberships functions for each components. The reason is that, the advantage and simply when presenting the parametric metric form as the classical fuzzy numbers. To this ends, we will define the (α, β, γ) −cuts of neutrosophic fuzzy numbers and through the linearity of triangular membership functions, we can convert neutrosophic numbers into parametric forms as intervals. This parametric forms have advantage that, we can easily define levels-set wise of the derivatives and integral as well as building numerical algorithms.

• 2. The concept of arithmetic operations on the set of neutrosophic numbers is defined via horizontal membership functions. This idea original introduced Piegat et al. (see [36,37,38]) and developed for granular differentiability of fuzzy-valued functions by Mazandarani et al. [29,30,31]. Especially, we can define the granular difference between neutrosophic numbers - one important step to define further differentiability of neutrosophic-valued functions as well as neutrosophic differential equations and other applications. It can be seen that this approach does not necessitate that the decreased diameter of neutrosophic-valued function or multi-case of solution related to so-call switching points as we often face in fuzzy analysis.

• 3. We laid the first step in constructing topological structures on the set of neutrosophic numbers by introducing Hamming metric and building complete metric space $(\mathcal T, D^{gr})$. Due to the fact that the space $\mathcal {T}$ endowed with the metric D^gr ensures the convergent of Cauchy sequence, we can further study the qualitative and quantitative nature of solution to some dynamical systems and processes arising in science and engineering.

• 4. At last, we demonstrate the effectiveness and significance of our theoretical results by applying them in some engineering problems related to logistic model or some mechanical models such inverted pendulum systems or Mass- Spring- Damper model. Our research will open up many potential applications in applied science and engineering that directly employ derivative and integral calculus as the essential tools such as optimal control of wireless networks, modeling a wires in circuits by a dynamic system of neutrosophic objects, etc.

The organizational structure of this paper is as follows: Section 2 recalls some preliminaries on single valued neutrosophic set and neutrosophic numbers, in which we introduce the levels set notion as the bridge between neutrosophic set with granular computing. Next, we introduce some types of single valued triangular neutrosophic numbers along with their respective parametric form. For more clearly, we give some numerical examples for each subsection. Section 3 is used to present granular representation of single valued triangular neutrosophic numbers, that is the foundation to build some calculus properties such as the neutrosophic gr-derivative and neutrosophic gr-integral before some applications to engineering problems are presented in Section 4. Finally, some conclusions and future works are discussed in Section 5.

2 Single valued triangular neutrosophic number

We call a neutrosophic set (NS) $\mathcal {A}$ defined in the universal of discourse X, denote generally by x, if it is represented by the form

$$\begin{array}{@{}rcl@{}} \mathcal{A}=\{\langle x, T_{\mathcal{A}}(x),I_{\mathcal{A}}(x),F_{\mathcal{A}}(x)\rangle :x\in X\} \end{array} $$

where $T_{\mathcal {A}}:X\to ^{-}]0, 1[^{+}$ is denoted for the truth membership function representing the degree of confidence, $I_{\mathcal {A}}:X\to ^{-}]0, 1[^{+}$ is called the indeterminacy membership function which represents the degree of uncertainty and $F_{\mathcal {A}}:X\to ^{-}]0, 1[^{+}$ is called the falsity membership function which represents the degree of scepticism such that the following relation holds

$$\begin{array}{@{}rcl@{}} 0^{-}\leq T_{\mathcal{A}}(x)+I_{\mathcal{A}}(x)+F_{\mathcal{A}}(x)\leq 3^{+}. \end{array} $$

In this paper, we consider single valued NS, which is a NS $\mathcal {A}$ with x is a single valued independent variable (see [15]), whose the truth, indeterminacy and falsity membership functions exhibit the relation

$$\begin{array}{@{}rcl@{}} 0\leq T_{\mathcal{A}}(x)+I_{\mathcal{A}}(x)+F_{\mathcal{A}}(x)\leq 3. \end{array} $$

A single valued NS $\mathcal {A}$ defined on the universal set of real numbers $\mathbb {R}$ is said to be single valued neutrosophic number or neutrosophic number (NN) for short if it has following properties

• (i) $\mathcal {A}$ is neut-normal, i.e., there exist three points a₀, b₀, $c_{0}\in \mathbb {R}$ such that $T_{\mathcal {A}}(a_{0})= 1$, $I_{\mathcal {A}}(b_{0})= 1$ and $F_{\mathcal {A}}(c_{0})= 1$.

• (ii) $\mathcal {A}$ is neut-convex, i.e., the following conditions hold

  $$\begin{array}{@{}rcl@{}} & T_{\mathcal{A}}\left( \lambda x_{1}+(1-\lambda)x_{2}\right)\geq \min\left\{T_{\mathcal{A}}(x_{1}),T_{\mathcal{A}}(x_{2}) \right\},\\ & I_{\mathcal{A}}\left( \lambda x_{1}+(1-\lambda)x_{2}\right)\leq \max\left\{I_{\mathcal{A}}(x_{1}),I_{\mathcal{A}}(x_{2}) \right\},\\ & F_{\mathcal{A}}\left( \lambda x_{1}+(1-\lambda)x_{2}\right)\leq \max\left\{F_{\mathcal{A}}(x_{1}),F_{\mathcal{A}}(x_{2}) \right\}, \end{array} $$

  for each λ ∈ [0, 1] and x₁, $x_{2}\in \mathbb {R}$.

Definition 2.1

The (α, β, γ) - cut (or level set) of a single valued NS $\mathcal {A}$, denoted by $\mathcal {A}_{(\alpha ,\beta ,\gamma )}$, is defined by $\mathcal {A}_{(\alpha ,\beta ,\gamma )}=\{x\in X:T_{\mathcal {A}}(x)\geq \alpha ,I_{\mathcal {A}}(x)\leq \beta , F_{\mathcal {A}}(x)\leq \gamma \}$, where α, β, γ ∈ [0, 1] such that α + β + γ ≤ 3.

Here, we consider a special type of single valued neutrosophic number, namely single valued triangular neutrosophic number.

Definition 2.2

A single valued triangular NN is given by

$$\begin{array}{@{}rcl@{}} \mathcal{A}=\langle[(p_{1},q_{1},r_{1});\alpha],[(p_{2},q_{2},r_{2});\beta],[(p_{3},q_{3},r_{3});\gamma]\rangle, \end{array} $$

where α, β, γ ∈ [0, 1] and the truth membership function $T_{\mathcal {A}}:\mathbb {R}\to [0,\alpha ]$, the indeterminacy membership function $I_{\mathcal {A}}:\mathbb {R}\to [\beta ,1]$ and the falsity membership function $F_{\mathcal {A}}:\mathbb {R}\to [\gamma ,1]$ satisfy following condition

$$0\leq T_{\mathcal{A}}(x)+I_{\mathcal{A}}(x)+F_{\mathcal{A}}(x)\leq 3\text{ for all} x\in \mathcal{A}.$$

We denote by $\mathcal {T}$ the set of all single valued triangular NNs. Then, based on the dependence between quantities: the truth, the indeterminacy and the falsity, we can classify the set $\mathcal {T}$ of single valued triangular NNs into three following types

2.1 Single valued triangular NN of type 1

The quantities of truth, indeterminacy and falsity are not dependent. Then, a single valued triangular NN of type 1 is defined as $\mathcal {A}=\langle p_{1},q_{1},r_{1};p_{2},q_{2},r_{2};p_{3},q_{3},r_{3}\rangle$, with membership functions are defined as follows, respectively

$$\begin{array}{@{}rcl@{}} T_{\mathcal{A}}(x)=\left\{\begin{array}{ll} \frac{x-p_{1}}{q_{1}-p_{1}}&\quad p_{1}\leq x\leq q_{1},\\ \quad 1 &\quad x= q_{1},\\ \frac{r_{1}-x}{r_{1}-q_{1}}&\quad q_{1}\leq x\leq r_{1},\\ \quad 0&\quad\text{otherwise}, \end{array}\right.\\ I_{\mathcal{A}}(x)=\left\{\begin{array}{ll} \frac{x-p_{2}}{q_{2}-p_{2}}&\quad p_{2}\leq x\leq q_{2},\\ \quad 0 &\quad x= q_{2},\\ \frac{r_{2}-x}{r_{2}-q_{2}}&\quad q_{2}\leq x\leq r_{2},\\ \quad 1&\quad\text{otherwise}, \end{array}\right. \end{array} $$

$$\begin{array}{@{}rcl@{}} F_{\mathcal{A}}(x)=\left\{\begin{array}{ll} \frac{x-p_{3}}{q_{3}-p_{3}}&\quad p_{3}\leq x\leq q_{3},\\ \quad 0 &\quad x= q_{3},\\ \frac{r_{3}-x}{r_{3}-q_{3}}&\quad q_{3}\leq x\leq r_{3},\\ \quad 1 &\quad\text{otherwise}, \end{array}\right. \end{array} $$

We can easily find the parametric form of $\mathcal {A}$ as

$$ \mathcal{A}_{(\alpha,\beta,\gamma)}=\left[T_{\mathcal{A}}^{-}(\alpha),T_{\mathcal{A}}^{+}(\alpha);I_{\mathcal{A}}^{-}(\beta), I_{\mathcal{A}}^{+}(\beta);F_{\mathcal{A}}^{-}(\gamma),F_{\mathcal{A}}^{+}(\gamma)\right], $$

(1)

where α, β, γ ∈ [0, 1] such that 0 ≤ α + β + γ ≤ 3 and

$$\begin{array}{@{}rcl@{}} &T_{\mathcal{A}}^{-}(\alpha)=p_{1}+\alpha(q_{1}-p_{1}), &T_{\mathcal{A}}^{+}(\alpha)=r_{1}-\alpha(r_{1}-q_{1}),\\ &I_{\mathcal{A}}^{-}(\beta)=q_{2}-\beta(q_{2}-p_{2}), &I_{\mathcal{A}}^{+}(\beta)=q_{2}+\beta(r_{2}-q_{2}),\\ &F_{\mathcal{A}}^{-}(\gamma)=q_{3}-\gamma(q_{3}-p_{3}), &F_{\mathcal{A}}^{+}(\gamma)=q_{3}+\gamma(r_{3}-q_{3}). \end{array} $$

Example 2.1

Let $\mathcal {A}=(8,14,20;12,16,22;10, 15,24)$ be a single valued triangular NN. Then, from (1) its parametric form is

$$\begin{array}{@{}rcl@{}} \mathcal{A}_{(\alpha,\beta,\gamma)}=\left[8 + 6\alpha,20-6\alpha;16-4\beta,16 + 6\beta;15-5\gamma,15 + 9\gamma\right],\quad\alpha, \beta,\gamma\in[0, 1]. \end{array} $$

In Table 1, we give some values of $T_{\mathcal {A}}^{-}(\alpha )$, $T_{\mathcal {A}}^{+}(\alpha )$, $I_{\mathcal {A}}^{-}(\beta )$, $I_{\mathcal {A}}^{+}(\beta )$, $F_{\mathcal {A}}^{-}(\gamma )$, $F_{\mathcal {A}}^{+}(\gamma )$ of the number $\mathcal {A}$ at some concrete levels whose graphical representation is shown Fig. 1.

Table 1 Values of $T_{\mathcal {A}}^{-}(\alpha )$, $T_{\mathcal {A}}^{+}(\alpha )$, $I_{\mathcal {A}}^{-}(\beta )$, $I_{\mathcal {A}}^{+}(\beta )$, $F_{\mathcal {A}}^{-}(\gamma )$, $F_{\mathcal {A}}^{+}(\gamma )$

Fig. 1

Membership functions of NN of type 1 $\mathcal {A}$

2.2 Single valued triangular NN of type 2

In this type of number, two quantities: indeterminacy membership function and falsity membership function are dependent. Then, a single valued triangular NN of type 2 is defined as $\mathcal {A}=\langle p_{1},q_{1},r_{1};p_{2},q_{2},r_{2};\beta _{neu};\gamma _{neu}\rangle$, whose membership functions are defined in compact form as

$$\begin{array}{@{}rcl@{}} &T_{\mathcal{A}}(x)=\left\{\begin{array}{ll} \frac{x-p_{1}}{q_{1}-p_{1}}&\quad p_{1}\leq x\leq q_{1},\\ \quad 1 &\quad x= q_{1},\\ \frac{r_{1}-x}{r_{1}-q_{1}}&\quad q_{1}\leq x\leq r_{1},\\ \quad 0&\quad\text{otherwise}, \end{array}\right.\\ &I_{\mathcal{A}}(x)=\left\{\begin{array}{ll} \frac{q_{2}-x+\beta_{neu}(x-p_{2})}{q_{2}-p_{2}}&\quad p_{2}\leq x\leq q_{2},\\ \qquad \beta_{neu} &\quad x= q_{2},\\ \frac{x-q_{2}+\beta_{neu}(r_{2}-x)}{r_{2}-q_{2}}&\quad q_{2}\leq x\leq r_{2},\\ \qquad 1&\quad\text{otherwise}, \end{array}\right.\\ &F_{\mathcal{A}}(x)=\left\{\begin{array}{ll} \frac{q_{2}-x+\gamma_{neu}(x-p_{2})}{q_{2}-p_{2}}&\quad p_{2}\leq x\leq q_{2},\\ \qquad \gamma_{neu} &\quad x= q_{2},\\ \frac{x-q_{2}+\gamma_{neu}(r_{2}-x)}{r_{2}-q_{2}}&\quad q_{2}\leq x\leq r_{2},\\ \qquad 1 &\quad\text{otherwise}, \end{array}\right. \end{array} $$

Similarly, the parametric form of $\mathcal {A}$ is

$$\begin{array}{@{}rcl@{}} \mathcal{A}_{(\alpha,\beta,\gamma)}=\left[T_{\mathcal{A}}^{-}(\alpha),T_{\mathcal{A}}^{+}(\alpha);I_{\mathcal{A}}^{-}(\beta), I_{\mathcal{A}}^{+}(\beta);F_{\mathcal{A}}^{-}(\gamma),F_{\mathcal{A}}^{+}(\gamma)\right], \end{array} $$

where

$$\begin{array}{@{}rcl@{}} \begin{array}{ll}T_{\mathcal{A}}^{-}(\alpha)=p_{1}+\alpha(q_{1}-p_{1}), &T_{\mathcal{A}}^{+}(\alpha)=r_{1}-\alpha(r_{1}-q_{1}),\\ I_{\mathcal{A}}^{-}(\beta)=\frac{q_{2}-\beta_{neu}p_{2}-\beta(q_{2}-p_{2})}{1-\beta_{neu}}, & I_{\mathcal{A}}^{+}(\beta)=\frac{q_{2}-\beta_{neu}r_{2}+\beta(r_{2}-q_{2})}{1-\beta_{neu}},\\ F_{\mathcal{A}}^{-}(\gamma)=\frac{q_{2}-\gamma_{neu}p_{2}-\gamma(q_{2}-p_{2})}{1-\gamma_{neu}},& F_{\mathcal{A}}^{+}(\gamma)=\frac{q_{2}-\gamma_{neu}r_{2}+\gamma(r_{2}-q_{2})}{1-\gamma_{neu}}. \end{array} \end{array} $$

Here, α ∈ [0, 1], β ∈ [β_neu,1] and γ ∈ [γ_neu,1] such that 0 ≤ α + β + γ ≤ 3.

Example 2.2

Let $\mathcal {A}=(8,14,20;12,16,22;0.5;0.6)$ be a single valued NN. Then, its parametric form is

$$\begin{array}{@{}rcl@{}} \mathcal{A}_{(\alpha,\beta,\gamma)}=\left[8 + 6\alpha,20-6\alpha;20-8\beta,10 + 12\beta;22-10\gamma,7 + 15\gamma\right], \end{array} $$

for α ∈ [0, 1], β ∈ [0.5,1], γ ∈ [0.6,1].

In Table 2, we give some values of $T_{\mathcal {A}}^{-}(\alpha )$, $T_{\mathcal {A}}^{+}(\alpha )$, $I_{\mathcal {A}}^{-}(\beta )$, $I_{\mathcal {A}}^{+}(\beta )$, $F_{\mathcal {A}}^{-}(\gamma )$, $F_{\mathcal {A}}^{+}(\gamma )$ of the number $\mathcal {A}$ whose graphical representation is shown Fig. 2.

Table 2 Values of $T_{\mathcal {A}}^{-}(\alpha )$, $T_{\mathcal {A}}^{+}(\alpha )$, $I_{\mathcal {A}}^{-}(\beta )$, $I_{\mathcal {A}}^{+}(\beta )$, $F_{\mathcal {A}}^{-}(\gamma )$, $F_{\mathcal {A}}^{+}(\gamma )$

Fig. 2

Membership functions of NN of type 2 $\mathcal {A}$

2.3 Single valued triangular NN of type 3

Here, the quantities: the truth, indeterminacy and falsity membership function are dependent. Then, a single valued triangular NN of type 3 is defined as $\mathcal {A}=\langle p_{1},q_{1},r_{1};\alpha _{neu};\beta _{neu};\gamma _{neu}\rangle$, whose membership functions are defined as follows

$$\begin{array}{@{}rcl@{}} &T_{\mathcal{A}}(x)=\left\{\begin{array}{ll} \alpha_{neu}\frac{x-p_{1}}{q_{1}-p_{1}}&\quad p_{1}\leq x\leq q_{1},\\ \quad \alpha_{neu} &\quad x= q_{1},\\ \alpha_{neu}\frac{r_{1}-x}{r_{1}-q_{1}}&\quad q_{1}\leq x\leq r_{1},\\ \quad 0&\quad\text{otherwise}, \end{array}\right.\\ &I_{\mathcal{A}}(x)=\left\{\begin{array}{ll} \frac{q_{1}-x+\beta_{neu}(x-p_{1})}{q_{1}-p_{1}}&\quad p_{1}\leq x\leq q_{1},\\ \qquad \beta_{neu} &\quad x= q_{1},\\ \frac{x-q_{1}+\beta_{neu}(r_{1}-x)}{r_{1}-q_{1}}&\quad q_{1}\leq x\leq r_{1},\\ \qquad 1&\quad\text{otherwise}, \end{array}\right.\\ &F_{\mathcal{A}}(x)=\left\{\begin{array}{ll} \frac{q_{1}-x+\gamma_{neu}(x-p_{1})}{q_{1}-p_{1}}&\quad p_{1}\leq x\leq q_{1},\\ \qquad \gamma_{neu} &\quad x= q_{1},\\ \frac{x-q_{1}+\gamma_{neu}(r_{1}-x)}{r_{1}-q_{1}}&\quad q_{1}\leq x\leq r_{1},\\ \qquad 1 &\quad\text{otherwise}, \end{array}\right. \end{array} $$

The parametric form of the number $\mathcal {A}$ is

$$\begin{array}{@{}rcl@{}} \mathcal{A}_{(\alpha,\beta,\gamma)}=\left[T_{\mathcal{A}}^{-}(\alpha),T_{\mathcal{A}}^{+}(\alpha);I_{\mathcal{A}}^{-}(\beta), I_{\mathcal{A}}^{+}(\beta);F_{\mathcal{A}}^{-}(\gamma),F_{\mathcal{A}}^{+}(\gamma)\right], \end{array} $$

where

$$\begin{array}{@{}rcl@{}} \begin{array}{ll}T_{\mathcal{A}}^{-}(\alpha)=p_{1}+\frac{\alpha}{\alpha_{neu}}(q_{1}-p_{1}),& T_{\mathcal{A}}^{+}(\alpha)=r_{1}-\frac{\alpha}{\alpha_{neu}}(r_{1}-q_{1}),\\ I_{\mathcal{A}}^{-}(\beta)=\frac{q_{1}-\beta_{neu}p_{1}-\beta(q_{1}-p_{1})}{1-\beta_{neu}},& I_{\mathcal{A}}^{+}(\beta)=\frac{q_{1}-\beta_{neu}r_{1}+\beta(r_{1}-q_{1})}{1-\beta_{neu}},\\ F_{\mathcal{A}}^{-}(\gamma)=\frac{q_{1}-\gamma_{neu}p_{1}-\gamma(q_{1}-p_{1})}{1-\gamma_{neu}},& F_{\mathcal{A}}^{+}(\gamma)=\frac{q_{1}-\gamma_{neu}r_{1}+\gamma(r_{1}-q_{1})}{1-\gamma_{neu}}. \end{array} \end{array} $$

Here, α ∈ [0,α_neu], β ∈ [β_neu,1] and γ ∈ [γ_neu,1] such that 0 ≤ α + β + γ ≤ 3.

Example 2.3

Let $\mathcal {A}=(14,16,22;0.5;0.8;0.7)$ be a single valued NN. Then, its parametric form is

$$\begin{array}{@{}rcl@{}} \mathcal{A}_{(\alpha,\beta,\gamma)}=\left[14 + 4\alpha,22-12\alpha;24-10\beta,-8 + 30\beta;\frac{62}{3}-\frac{20}{3}\gamma,2 + 20\gamma\right], \end{array} $$

for α ∈ [0,0.5], β ∈ [0.8,1], γ ∈ [0.7,1].

In Table 3, we give some values of $T_{\mathcal {A}}^{-}(\alpha )$, $T_{\mathcal {A}}^{+}(\alpha )$, $I_{\mathcal {A}}^{-}(\beta )$, $I_{\mathcal {A}}^{+}(\beta )$, $F_{\mathcal {A}}^{-}(\gamma )$, $F_{\mathcal {A}}^{+}(\gamma )$ of the number $\mathcal {A}$ whose graphical representation is shown Fig. 3.

Table 3 Values of $T_{\mathcal {A}}^{-}(\alpha )$, $T_{\mathcal {A}}^{+}(\alpha )$, $I_{\mathcal {A}}^{-}(\beta )$, $I_{\mathcal {A}}^{+}(\beta )$, $F_{\mathcal {A}}^{-}(\gamma )$, $F_{\mathcal {A}}^{+}(\gamma )$

Fig. 3

Membership functions of NN of type 3 $\mathcal {A}$

3 Granular presentation of single valued triangular neutrosophic number

3.1 Horizontal membership function

Definition 3.1

Let $\mathcal {A}=(a_{1},b_{1},c_{1};a_{2},b_{2},c_{2};a_{3},b_{3},c_{3})$ be a single valued triangular neutrosophic number whose parametric form is

$$\begin{array}{@{}rcl@{}} \mathcal{A}_{(\alpha,\beta,\gamma)}=\left[T_{\mathcal{A}}^{-}(\alpha),T_{\mathcal{A}}^{+}(\alpha);I_{\mathcal{A}}^{-}(\beta), I_{\mathcal{A}}^{+}(\beta);F_{\mathcal{A}}^{-}(\gamma),F_{\mathcal{A}}^{+}(\gamma)\right]. \end{array} $$

Then, we can represent the horizontal membership function (HMF) of $\mathcal {A}$ as an element A^gr(α, β, γ, μ), which is given by

$$\begin{array}{@{}rcl@{}} A^{gr}: [0,\alpha_{neu}]\times[\beta_{neu},1]\times[\gamma_{neu},1]\times[0, 1]^{3} &\to [a_{1},c_{1}]\times[a_{2},c_{2}]\times[a_{3},c_{3}], \end{array} $$

and maps (α, β, γ, μ) into $\left (x_{\alpha }(\mu _{1}),x_{\beta }(\mu _{2}),x_{\gamma }(\mu _{3})\right )\in \mathbb {R}^{3}$, where the notion ”gr” represents for the granular information that are contained in (x_α,x_β,x_γ) ∈ [a₁,c₁] × [a₂,c₂] × [a₃,c₃] and μ ∈ [0, 1]³ standing for μ₁, μ₂, μ₃ is called relative-distance-measure (RDM for short) variables. In particular, we have A^gr(α, β, γ, μ) = (x_α(μ₁),x_β(μ₂),x_γ(μ₃)), where

$$\begin{array}{@{}rcl@{}} &&x_{\alpha}(\mu_{1}):=T_{\mathcal{A}}^{gr}(\alpha,\mu_{1})= T_{\mathcal{A}}^{-}(\alpha)+\left( T_{\mathcal{A}}^{+}(\alpha)-T_{\mathcal{A}}^{-}(\alpha)\right)\mu_{1},\\ &&x_{\beta}(\mu_{2}):=I_{\mathcal{A}}^{gr}(\beta,\mu_{2})=I_{\mathcal{A}}^{-}(\beta)+\left( I_{\mathcal{A}}^{+}(\beta)-I_{\mathcal{A}}^{-}(\beta)\right)\mu_{2},\\ &&x_{\gamma}(\mu_{3}):=F_{\mathcal{A}}^{gr}(\gamma,\mu_{3})=F_{\mathcal{A}}^{-}(\gamma)+\left( F_{\mathcal{A}}^{+}(\gamma)-F_{\mathcal{A}}^{-}(\gamma)\right)\mu_{3}. \end{array} $$

Proposition 3.1

The HMF of a number$\mathcal {A}\in \mathcal {T}$isdenoted by$\mathcal {H}(\mathcal {A})\triangleq A^{gr}(\alpha ,\beta ,\gamma ,\mu )$.Moreover, the (α, β, γ) − cuts of$\mathcal {A}$canbe obtained by using following inverse transformation

$$\begin{array}{@{}rcl@{}} \mathcal{A}_{(\alpha,\beta,\gamma)}&:=&\mathcal{H}^{-1}(A^{gr}(\alpha,\beta,\gamma,\mu))\\& =& \left\{\left[\inf\limits_{\xi\geq\alpha} \min\limits_{\mu_{1}} T_{\mathcal{A}}^{gr}(\xi,\mu_{1}), \sup\limits_{\xi\geq\alpha} \max\limits_{\mu_{1}} T_{\mathcal{A}}^{gr}(\xi,\mu_{1})\right],\right.\\ &&\qquad \left.\left[\inf\limits_{\xi\geq\beta} \min\limits_{\mu_{2}} I_{\mathcal{A}}^{gr}(\xi,\mu_{2}), \sup\limits_{\xi\geq\beta} \max\limits_{\mu_{2}} I_{\mathcal{A}}^{gr}(\xi,\mu_{2})\right],\right.\\ &&\qquad \left.\left[\inf_{\xi\geq\gamma} \min\limits_{\mu_{3}} F_{\mathcal{A}}^{gr}(\xi,\mu_{3}), \sup\limits_{\xi\geq\gamma} \max\limits_{\mu_{3}} F_{\mathcal{A}}^{gr}(\xi,\mu_{3})\right]\right\}. \end{array} $$

Definition 3.2

Two elements $\mathcal {A}$ and $\mathcal {\tilde {A}}\in \mathcal {T}$ are said to be equal and written by $\mathcal {A}=\mathcal {\tilde {A}}$ if and only if $\mathcal {H}(\mathcal {A})=\mathcal {H}(\mathcal {\tilde {A}})$ for all triplet $(\mu _{1},\mu _{2},\mu _{3})=(\tilde {\mu }_{1},\tilde {\mu }_{2},\tilde {\mu }_{3})\in [0, 1]\times [0, 1]\times [0, 1]$, i.e.,

$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{ll} T_{\mathcal{A}}^{gr}(\alpha,\mu_{1})=T_{\mathcal{\tilde{A}}}^{gr}(\alpha,\mu_{1})\\ I_{\mathcal{A}}^{gr}(\beta,\mu_{2})=I_{\mathcal{\tilde{A}}}^{gr}(\beta,\mu_{2})\\ F_{\mathcal{A}}^{gr}(\gamma,\mu_{3})=F_{\mathcal{\tilde{A}}}^{gr}(\gamma,\mu_{3}) \end{array}\right. \end{array} $$

for each (α, β, γ) ∈ [0,α_neu] × [β_neu,1] × [γ_neu,1] and for all (μ₁,μ₂,μ₃) ∈ [0, 1] × [0, 1] × [0, 1].

3.2 Arithmetic operations

Definition 3.3

Denote ”⊗” by one of three arithmetic operations in $\mathcal {T}$: addition, subtraction or multiplication operation. Then $\mathcal {H}(\mathcal {A}_{1}\otimes \mathcal {A}_{2})\triangleq \mathcal {H}(\mathcal {A}_{1}) \otimes \mathcal {H}(\mathcal {A}_{2})$. It should be noted that the difference in this sense is called granular difference (gr-difference), denoted by ⊖^gr.

Example 3.1

Let $\mathcal {A}=(5,10, 15;3,6,9;10, 16,22)$ and $\mathcal {\tilde {A}}=(4,6,8;1,3,5;9,11,13)$ be two triangular neutrosophic numbers of type 1 whose parametric representations are given by

$$\begin{array}{@{}rcl@{}} &&\mathcal{A}_{(\alpha,\beta,\gamma)}=\left\{[5 + 5\alpha,15-5\alpha],[6-3\beta,6 + 3\beta],[16-6\gamma,16 + 6\gamma]\right\},\\ &&\mathcal{\tilde{A}}_{(\alpha,\beta,\gamma)}=\left\{[4 + 2\alpha,8-2\alpha],[3-2\beta,3 + 2\beta],[11-2\gamma,11 + 2\gamma]\right\}. \end{array} $$

From Definition 3.1, it implies that

$$\begin{array}{@{}rcl@{}} A^{gr}(\alpha,\beta,\gamma,\mu)&=\left( 5 + 5\alpha+(10-10\alpha)\mu_{1}; 6-3\beta+ 6\beta\mu_{2};16-6\gamma+ 12\gamma\mu_{3}\right),\\ \tilde{A}^{gr}(\alpha,\beta,\gamma,\tilde{\mu})&=\left( 4 + 2\alpha+(4-4\alpha)\tilde{\mu}_{1};3-2\beta+ 4\beta\tilde{\mu}_{2};11-2\gamma+ 4\gamma\tilde{\mu}_{3}\right). \end{array} $$

In addition, employing Definition 3.3, we immediately obtain that

$$\begin{array}{@{}rcl@{}} \bullet &&A^{gr}(\alpha,\beta,\gamma,\mu)+\tilde{A}^{gr}(\alpha,\beta,\gamma,\mu)=\left( 9 + 7\alpha+(14-14\alpha)\mu_{1};9-5\beta+ 10\beta\mu_{2};27-8\gamma+ 16\gamma\mu_{3}\right),\\ \bullet &&A^{gr}(\alpha,\beta,\gamma,\mu)-\tilde{A}^{gr}(\alpha,\beta,\gamma,\mu)=\left( 1 + 2\alpha+(6-6\alpha)\mu_{1};3-\beta+ 2\beta\mu_{2};5-4\gamma+ 8\gamma\mu_{3}\right),\\ \bullet &&0.5\cdot A^{gr}(\alpha,\beta,\gamma,\mu)=\left( 2.5 + 2.5\alpha+(5-5\alpha)\mu_{1}; 3-1.5\beta+ 3\beta\mu_{2};8-3\gamma+ 6\gamma\mu_{3}\right),\\ \bullet &&A^{gr}(\alpha,\beta,\gamma,\mu)\times\tilde{A}^{gr}(\alpha,\beta,\gamma,\mu)=\left( 20 + 30\alpha+ 10\alpha^{2}+ 20(1-\alpha)(2 + 3\alpha)\mu_{1}+ 40(1-\alpha)^{2}{\mu_{1}^{2}};\right.\\ &&\hspace{5cm}\left.18-21\beta+ 6\beta^{2}+ 6\beta(7-4\beta)\mu_{2}+ 24\beta^{2}{\mu_{2}^{2}};\right.\\ &&\hspace{5cm}\left.176-98\gamma+ 12\gamma^{2}+ 4\gamma(49-12\gamma)\mu_{3}+ 48\gamma^{2}{\mu_{3}^{2}}\right). \end{array} $$

Then, by using Proposition 3.1 and (α, β, γ) −cuts representation theorem, we obtain

• The addition

  $$\begin{array}{@{}rcl@{}} \mathcal{A}+\mathcal{\tilde{A}}=(9,17,24;4,9,14;19,27,35). \end{array} $$

• The subtraction

  $$\begin{array}{@{}rcl@{}} \mathcal{A}\ominus^{gr}\mathcal{\tilde{A}}=(-3,4,11;-2,3,8;-3,5,13). \end{array} $$

• The multiplication by the scalar λ = 0.5

  $$\begin{array}{@{}rcl@{}} \lambda\mathcal{A}=(2.5,5,7.5;1.5,3,4.5;5,8,11). \end{array} $$

• The multiplication

  $$\begin{array}{@{}rcl@{}} \mathcal{A}\cdot\mathcal{\tilde{A}}=(20,60, 120;3,18,45;90, 176,286). \end{array} $$

  Especially, we have the spare of $\mathcal {A}$ is given by

  $$\begin{array}{@{}rcl@{}} \mathcal{A}^{2}:=\mathcal{A}\cdot \mathcal{A}=(25,100,225;9,36,81;100,256,484). \end{array} $$

Definition 3.4

Let $f: [a,b]\subset \mathbb {R} \to \mathcal {T}$ be a $\mathcal {T}$-valued function including n distinct single valued triangular neutrosophic numbers $\mathcal {A}_{1},, \mathcal {A}_{2},\ldots ,\mathcal {A}_{n}$. Then, the HMF of f at t ∈ [a, b], denoted by $\mathcal {H}(f(t))\triangleq f^{gr}(t,\alpha ,\beta ,\gamma ,\mu _{f})$, is as

$$\begin{array}{@{}rcl@{}} f^{gr}:[a,b]\times[0,\alpha_{neu}]\times[\beta_{neu},1]\times[\gamma_{neu},1]\times [0, 1]^{3n} \rightarrow \mathbb{R}^{3}, \end{array} $$

where $\mu _{f} \triangleq (\mu _{1,\mathcal {A}_{1}},\ldots ,\mu _{1,\mathcal {A}_{n}},\mu _{2,\mathcal {A}_{1}},\ldots ,\mu _{2,\mathcal {A}_{n}},\mu _{3,\mathcal {A}_{1}},\ldots ,\mu _{3,\mathcal {A}_{n}})$.

Remark 3.1

Let f be a $\mathcal {T} -$ valued function defined in closed interval $[a,b]\subset \mathbb {R}$. Then, since the value $f(t)\in \mathcal {T}$, we can write f(t) as $f(t)=\left \{\langle x,T_{f(t)}(x),I_{f(t)}(x),F_{f(t)}(x)\rangle :x\in \mathbb {R}\right \}$ for each t ∈ [a, b], whose HMF can be written as

$$\begin{array}{@{}rcl@{}} f^{gr}(t,\alpha,\beta,\gamma,\mu_{f})=\left( T_{f(t)}^{gr}(\alpha,\mu_{1,f}),I_{f(t)}^{gr}(\beta,\mu_{2,f}),F_{f(t)}^{gr}(\gamma,\mu_{3,f})\right). \end{array} $$

Example 3.2

Let $\mathcal {A}=(5,10, 15;3,6,9;10, 16,22)$ and $\mathcal {\tilde {A}}=(4,6,8;1,3,5;9,11,13)$ be two triangular neutrosophic numbers of type 1 with respective HMF is as follows

$$\begin{array}{@{}rcl@{}} A^{gr}(\alpha,\beta,\gamma,\mu)&=\left( 5 + 5\alpha+(10-10\alpha)\mu_{1}; 6-3\beta+ 6\beta\mu_{2};16-6\gamma+ 12\gamma\mu_{3}\right),\\ \tilde{A}^{gr}(\alpha,\beta,\gamma,\tilde{\mu})&=\left( 4 + 2\alpha+(4-4\alpha)\tilde{\mu}_{1};3-2\beta+ 4\beta\tilde{\mu}_{2};11-2\gamma+ 4\gamma\tilde{\mu}_{3}\right), \end{array} $$

where α, β, γ ∈ [0, 1] and $\mu =(\mu _{1},\mu _{2},\mu _{3}), \tilde {\mu }=(\tilde {\mu }_{1},\tilde {\mu }_{2},\tilde {\mu }_{3})\in [0, 1]\times [0, 1]\times [0, 1]$. Consider a $\mathcal {T}$-valued function $f(t)=\mathcal {A}t+\mathcal {\tilde {A}}\sin {2t}$ on the interval [0,7]. The horizontal membership function of f is given by

$$\begin{array}{@{}rcl@{}} \mathcal{H}(f(t)) &=&tA^{gr}(\alpha,\beta,\gamma,\mu)+\tilde{A}^{gr}(\alpha,\beta,\gamma,\tilde{\mu})\sin{2t}\\ &=&\left( \left[5 + 5\alpha+(10-10\alpha)\mu_{1}\right]t+\left[4 + 2\alpha+(4-4\alpha)\tilde{\mu}_{1}\right]\sin{2t};\right.\\ &&\quad \left.\left[6-3\beta+ 6\beta\mu_{2}\right]t+\left[3-2\beta+ 4\beta\tilde{\mu}_{2}\right]\sin{2t};\right.\\&&\quad \left.\left[16-6\gamma+ 12\gamma\mu_{3}\right]t+\left[11-2\gamma+ 4\gamma\tilde{\mu}_{3}\right]\sin{2t}\right). \end{array} $$

and the graphical representation of $\mathcal {T} -$ valued function f(t) is shown in Fig. 4

Fig. 4

The (α, β, γ)- cuts of $\mathcal {T}$-valued function f in Example 3.2, where the black curve corresponds to the certain values, the blue curves show the left end-points, while the red show the right end-points

3.3 Neutrosophic metric space

Definition 3.5

Let $\mathcal {A}$, $\mathcal {\tilde {A}}\in \mathcal {T}$. The function $D^{gr}:\mathcal {T}\times \mathcal {T}\to \mathbb {R}^{+}\cup \{0\}$ defined by

$$ \begin{array}{cc}D^{gr}(\mathcal{A},\mathcal{\tilde{A}})&=\sup_{\alpha,\beta,\gamma}\max_{\mu_{i},\tilde{\mu}_{i}}\frac{1}{3}\left\{\left|T_{\mathcal{A}}^{gr}(\alpha,\mu_{1})-T_{\mathcal{\tilde{A}}}^{gr}(\alpha,\tilde{\mu}_{1})\right|+\left|I_{\mathcal{A}}^{gr}(\beta,\mu_{2})-I_{\mathcal{\tilde{A}}}^{gr}(\beta,\tilde{\mu}_{2})\right|\right.\\ &\left.+\left|F_{\mathcal{A}}^{gr}(\gamma,\mu_{3})-F_{\mathcal{\tilde{A}}}^{gr}(\gamma,\tilde{\mu}_{3})\right|\right\}, \end{array} $$

(2)

is a distance between two type 1 single valued triangular neutrosophic numbers $\mathcal {A}$ and $\mathcal {\tilde {A}}$.

Proposition 3.2

Such real-valued function D ^gr is said to be a metric on $\mathcal {T}$ .

Proof

Let $\mathcal {A}$ and $\mathcal {\tilde {A}}$ be two numbers in $\mathcal {T}$ with respective horizontal membership functions

$$\begin{array}{@{}rcl@{}} &&A^{gr}(\alpha,\beta,\gamma,\mu)=(T_{\mathcal{A}}^{gr}(\alpha,\mu_{1}),I_{\mathcal{A}}^{gr}(\beta,\mu_{2}),F_{\mathcal{A}}^{gr}(\gamma,\mu_{3})),\\ &&\tilde{A}^{gr}(\alpha,\beta,\gamma,\tilde{\mu})=(T_{\mathcal{\tilde{A}}}^{gr}(\alpha,\tilde{\mu}_{1}),I_{\mathcal{\tilde{A}}}^{gr}(\beta,\tilde{\mu}_{2}),F_{\mathcal{\tilde{A}}}^{gr}(\gamma,\tilde{\mu}_{3})). \end{array} $$

From the formula (2), it implies that $D^{gr}(\mathcal {A},\mathcal {\tilde {A}})\geq 0$ for all $\mathcal {A}, \mathcal {\tilde {A}}\in \mathcal {T}$ and if $D^{gr}(\mathcal {A},\mathcal {\tilde {A}})= 0$ then we deduce that

$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{ll} \max\limits_{\mu_{1},\tilde{\mu}_{1}}\left|T_{\mathcal{A}}^{gr}(\alpha,\mu_{1})-T_{\mathcal{\tilde{A}}}^{gr}(\alpha,\tilde{\mu}_{1})\right|= 0\\ \max\limits_{\mu_{2},\tilde{\mu}_{2}}\left|I_{\mathcal{A}}^{gr}(\beta,\mu_{2})-I_{\mathcal{\tilde{A}}}^{gr}(\beta,\tilde{\mu}_{2})\right|= 0\\ \max\limits_{\mu_{3},\tilde{\mu}_{3}}\left|F_{\mathcal{A}}^{gr}(\gamma,\mu_{3})-F_{\mathcal{\tilde{A}}}^{gr}(\gamma,\tilde{\mu}_{3})\right|= 0. \end{array}\right. \end{array} $$

Equivalently, it implies

$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{ll} T_{\mathcal{A}}^{gr}(\alpha,\mu_{1})=T_{\mathcal{\tilde{A}}}^{gr}(\alpha,\tilde{\mu}_{1})\\ I_{\mathcal{A}}^{gr}(\beta,\mu_{2})=I_{\mathcal{\tilde{A}}}^{gr}(\beta,\tilde{\mu}_{2})\\ F_{\mathcal{A}}^{gr}(\gamma,\mu_{3})=F_{\mathcal{\tilde{A}}}^{gr}(\gamma,\tilde{\mu}_{3}), \end{array}\right. \end{array} $$

for all $\mu _{i}=\tilde {\mu }_{i}\in [0, 1]$ (i = 1,2,3) which follows that $\mathcal {A}=\mathcal {\tilde {A}}$.

Since symmetry of D^gr is obvious, the rest of proof is to show that

$$ D^{gr}(\mathcal{A}_{1},\mathcal{A}_{2})\leq D^{gr}(\mathcal{A}_{1},\mathcal{A}_{3})+ D^{gr}(\mathcal{A}_{3},\mathcal{A}_{2})\quad \text{for all} \mathcal{A}_{1},\mathcal{A}_{2},\mathcal{A}_{3}\in\mathcal{T}. $$

(3)

Indeed, since the following inequality

$$\begin{array}{@{}rcl@{}} \left|T_{\mathcal{A}_{1}}^{gr}(\alpha,\mu_{1})-T_{\mathcal{A}_{2}}^{gr}(\alpha,\tilde{\mu}_{1})\right|\leq \left|T_{\mathcal{A}_{1}}^{gr}(\alpha,\mu_{1})-T_{\mathcal{A}_{3}}^{gr}(\alpha,\overline{\mu}_{1})\right|+\left|T_{\mathcal{A}_{3}}^{gr}(\alpha,\overline{\mu}_{1})-T_{\mathcal{A}_{2}}^{gr}(\alpha,\tilde{\mu}_{1})\right| \end{array} $$

holds for each α ∈ [0,α_neu], $\mu _{1},\tilde {\mu }_{1},\overline {\mu }_{1}\in [0, 1]$, we deduce that

$$\begin{array}{@{}rcl@{}} \left|T_{\mathcal{A}_{1}}^{gr}(\alpha,\mu_{1})-T_{\mathcal{A}_{2}}^{gr}(\alpha,\tilde{\mu}_{1})\right|\leq \max\limits_{\mu_{1},\tilde{\mu}_{1},\overline{\mu}_{1}}\left\{\left|T_{\mathcal{A}_{1}}^{gr}(\alpha,\mu_{1})-T_{\mathcal{A}_{3}}^{gr}(\alpha,\overline{\mu}_{1})\right|+\left|T_{\mathcal{A}_{3}}^{gr}(\alpha,\overline{\mu}_{1})-T_{\mathcal{A}_{2}}^{gr}(\alpha,\tilde{\mu}_{1})\right|\right\}. \end{array} $$

Therefore,

$$\begin{array}{@{}rcl@{}} \sup_{\alpha}\max\limits_{\mu_{1},\tilde{\mu}_{1}} \left|T_{\mathcal{A}_{1}}^{gr}(\alpha,\mu_{1})-T_{\mathcal{A}_{2}}^{gr}(\alpha,\tilde{\mu}_{1})\right|&\leq \sup_{\alpha}\max\limits_{\mu_{1},\overline{\mu}_{1}}\left|T_{\mathcal{A}_{1}}^{gr}(\alpha,\mu_{1})-T_{\mathcal{A}_{3}}^{gr}(\alpha,\overline{\mu}_{1})\right|\\&\quad+\sup_{\alpha}\max\limits_{\overline{\mu}_{1},\tilde{\mu}_{1}}\left|T_{\mathcal{A}_{3}}^{gr}(\alpha,\overline{\mu}_{1})-T_{\mathcal{A}_{2}}^{gr}(\alpha,\tilde{\mu}_{1})\right|. \end{array} $$

By similar arguments, we also obtain

$$\begin{array}{@{}rcl@{}} \sup_{\beta}\max\limits_{\mu_{2},\tilde{\mu}_{2}} \left|I_{\mathcal{A}_{1}}^{gr}(\beta,\mu_{2})-I_{\mathcal{A}_{2}}^{gr}(\beta,\tilde{\mu}_{2})\right|&\leq \sup_{\beta}\max\limits_{\mu_{2},\overline{\mu}_{2}}\left|I_{\mathcal{A}_{1}}^{gr}(\beta,\mu_{2})-I_{\mathcal{A}_{3}}^{gr}(\beta,\overline{\mu}_{2})\right|\\&\quad+\sup_{\beta}\max\limits_{\overline{\mu}_{2},\tilde{\mu}_{2}}\left|I_{\mathcal{A}_{3}}^{gr}(\beta,\overline{\mu}_{2})-I_{\mathcal{A}_{2}}^{gr}(\beta,\tilde{\mu}_{2})\right|,\\ \sup_{\gamma}\max\limits_{\mu_{3},\tilde{\mu}_{3}} \left|F_{\mathcal{A}_{1}}^{gr}(\gamma,\mu_{3})-F_{\mathcal{A}_{2}}^{gr}(\gamma,\tilde{\mu}_{3})\right|&\leq \sup_{\gamma}\max\limits_{\mu_{3},\overline{\mu}_{3}}\left|F_{\mathcal{A}_{1}}^{gr}(\gamma,\mu_{3})-F_{\mathcal{A}_{3}}^{gr}(\gamma,\overline{\mu}_{3})\right|\\&\quad+\sup_{\gamma}\max\limits_{\overline{\mu}_{3},\tilde{\mu}_{3}}\left|F_{\mathcal{A}_{3}}^{gr}(\gamma,\overline{\mu}_{3})-F_{\mathcal{A}_{2}}^{gr}(\gamma,\tilde{\mu}_{3})\right|. \end{array} $$

Finally, by adding both sides of three above inequalities, we obtain the inequality (3). □

Remark 3.2

Such metric D^gr is said to be the granular metric on the space $\mathcal {T}$ of all single valued triangular neutrosophic numbers. Hence, the space $\mathcal {T}$ equipped with the metric D^gr is a metric space.

Theorem 3.1

The metric space $(\mathcal {T},D^{gr})$ is complete space.

Proof

Let $\{\mathcal {A}_{n}\}_{n\geq 1}\subset \mathcal {T}$ be a Cauchy sequence in $\mathcal {T}$, that means

$$\begin{array}{@{}rcl@{}} \forall \epsilon>0, \exists N\in\mathbb{N}^{\ast}\text{ such that} \forall n\geq N, p\geq 1,\text{ we have} D^{gr}(\mathcal{A}_{n+p},\mathcal{A}_{n})<\epsilon, \end{array} $$

or equivalently,

$$\begin{array}{@{}rcl@{}} &&\sup\limits_{\alpha,\beta,\gamma}\max\limits_{\mu_{1},\tilde{\mu}_{1}} \left|T_{\mathcal{A}_{n+p}}^{gr}(\alpha,\mu_{1})-T_{\mathcal{A}_{n}}^{gr}(\alpha,\tilde{\mu}_{1})\right|< \epsilon,\\ &&\sup\limits_{\alpha,\beta,\gamma}\max\limits_{\mu_{2},\tilde{\mu}_{2}} \left|I_{\mathcal{A}_{n+p}}^{gr}(\beta,\mu_{2})-I_{\mathcal{A}_{n}}^{gr}(\beta,\tilde{\mu}_{2})\right|<\epsilon,\\ &&\sup\limits_{\alpha,\beta,\gamma}\max\limits_{\mu_{3},\tilde{\mu}_{3}} \left|F_{\mathcal{A}_{n+p}}^{gr}(\gamma,\mu_{3})-F_{\mathcal{A}_{n}}^{gr}(\gamma,\tilde{\mu}_{3})\right|<\epsilon. \end{array} $$

As a result, we directly obtain that

$$\begin{array}{@{}rcl@{}} && \left|T_{\mathcal{A}_{n+p}}^{gr}(\alpha,\mu_{1})-T_{\mathcal{A}_{n}}^{gr}(\alpha,\tilde{\mu}_{1})\right|< \epsilon,\\ && \left|I_{\mathcal{A}_{n+p}}^{gr}(\beta,\mu_{2})-I_{\mathcal{A}_{n}}^{gr}(\beta,\tilde{\mu}_{2})\right|<\epsilon,\\ && \left|F_{\mathcal{A}_{n+p}}^{gr}(\gamma,\mu_{3})-F_{\mathcal{A}_{n}}^{gr}(\gamma,\tilde{\mu}_{3})\right|<\epsilon. \end{array} $$

Thus, we deduce that $\left \{T_{\mathcal {A}_{n}}^{gr}(\alpha ,\mu _{1})\right \}_{n\geq 1}$, $\left \{I_{\mathcal {A}_{n}}^{gr}(\beta ,\mu _{2}) \right \}_{n\geq 1}$, $\left \{F_{\mathcal {A}_{n}}^{gr}(\gamma ,\mu _{3})\right \}_{n\geq 1}$ are Cauchy sequences in the space of real numbers $\mathbb {R}$, and then, these sequences are convergent in $\mathbb {R}$.

Particularly, let us consider the sequence $\left \{T_{\mathcal {A}_{n}}^{gr}(\alpha ,\mu _{1})\right \}_{n\geq 1}$. By Definition 3.1, we can rewrite $T_{\mathcal {A}_{n}}^{gr}(\alpha ,\mu _{1})=T_{\mathcal {A}_{n}}^{-}(\alpha )+\left (T_{\mathcal {A}_{n}}^{+}(\alpha )-T_{\mathcal {A}_{n}}^{-}(\alpha )\right )\mu _{1}$.

Since $\left \{T_{\mathcal {A}_{n}}^{gr}(\alpha ,\mu _{1})\right \}_{n\geq 1}$ is a convergent sequence and 0 ≤ μ₁ ≤ 1, it follows that the sequences $\left \{T_{\mathcal {A}_{n}}^{-}(\alpha )\right \}$ and $\left \{T_{\mathcal {A}_{n}}^{+}(\alpha )\right \}$ are also convergent. No loss generality, we assume that $\lim \limits _{n\to \infty }T_{\mathcal {A}_{n}}^{-}(\alpha )=T_{\mathcal {A}}^{-}(\alpha )$, $\lim \limits _{n\to \infty }T_{\mathcal {A}_{n}}^{+}(\alpha )=T_{\mathcal {A}}^{+}(\alpha )$ and due to the fact that $T_{\mathcal {A}_{n}}^{-}(\alpha )\leq T_{\mathcal {A}_{n}}^{+}(\alpha ), \forall n\geq 1$, we obtain that $T_{\mathcal {A}}^{-}(\alpha )\leq T_{\mathcal {A}}^{+}(\alpha )$. At last, by employing analogous method as in proof of Theorem 8.5 in [9], we deduce that the interval $[T_{\mathcal {A}}^{-}(\alpha ), T_{\mathcal {A}}^{+}(\alpha )]$ is the α − cuts of a fuzzy number. As a consequence, the similar results are also obtained for the sequences $\left \{I_{\mathcal {A}_{n}}^{gr}(\beta ,\mu _{2}) \right \}_{n\geq 1}$ and $\left \{F_{\mathcal {A}_{n}}^{gr}(\gamma ,\mu _{3})\right \}_{n\geq 1}$.

Therefore, we can see that if $\{\mathcal {A}_{n}\}_{n\geq 1}$ is a Cauchy sequence in $\mathcal {T}$ then $\mathcal {A}_{n}$ converges to an element $\mathcal {A}\in \mathcal {T}$. Hence, this achieves the proof. □

3.4 The continuity

Definition 3.6

Let $f: [a,b]\subset \mathbb {R} \to \mathcal {T}$ be a $\mathcal {T}$-valued function and t₀ ∈ [a, b]. An element $\ell \in \mathcal {T}$ is called the limit of f(t) as t tends to t₀ and written by $\lim \limits _{t\to t_{0}}f(t)=\ell$ iff for all 𝜖 > 0, there exists δ > 0 such that ∀t ∈ [a, b] satisfying 0 < |t − t₀| < δ then D^gr(f(t),ℓ) < 𝜖.

Especially, we have

• If t₀ = a then $\lim \limits _{t\to a^{+}}f(t)=\ell$ means that for all 𝜖 > 0, there exists δ > 0 such that ∀t ∈ [a, b] satisfying 0 < t − a < δ then D^gr(f(t),ℓ) < 𝜖.

• If t₀ = b then $\lim \limits _{t\to b^{-}}f(t)=\ell$ means that for all 𝜖 > 0, there exists δ > 0 such that ∀t ∈ [a, b] satisfying 0 < b − t < δ then D^gr(f(t),ℓ) < 𝜖.

Definition 3.7

$\mathcal {T} -$ valued function $f: (a,b)\subset \mathbb {R} \to \mathcal {T}$ is said to be continuous on (a, b) if for all t₀ ∈ (a, b), for all 𝜖 > 0, there exists δ > 0 such that ∀t ∈ (a, b) satisfying |t − t₀| < δ then D^gr(f(t),f(t₀) < 𝜖.

3.5 The neutrosophic derivatives

Definition 3.8

Let $f:U\subset \mathbb {R}\to \mathcal {T}$ be a $\mathcal {T} -$ valued function. Then, f is called granular differentiable (gr-differentiable for short) at a point t₀ ∈ U if there exists an element $f^{\prime }_{gr}(t_{0})\in \mathcal {T}$ such that the following limit

$$\begin{array}{@{}rcl@{}} \lim\limits_{h\rightarrow 0}\frac{f(t_{0}+h)\ominus^{gr}f(t_{0})}{h} = f^{\prime}_{gr}(t_{0}), \end{array} $$

exists for h sufficiently near 0 and then, the value $f^{\prime }_{gr}(t_{0})$ is called the granular derivative (gr-derivative) of $\mathcal {T}$-valued function f at the point t₀. The function f is called gr-differentiable on U if the gr-derivative of f exists for all points t₀ ∈ U and mapping $t\in U\mapsto f^{\prime }_{gr}(t)$ is then called gr-derivative of f and denoted by $f^{\prime }_{gr}$ or $\dot {f}_{gr}$.

Proposition 3.3

A necessary and sufficient condition for afunction$f:U\subset \mathbb {R} \to \mathcal {T}$isgr-differentiable at a pointt₀ ∈ Uis the differentiability of its horizontal membership function at that point.Moreover, we have$\mathcal {H}\left (f^{\prime }_{gr}(t_{0}) \right ) =\frac {\partial f^{gr}(t_{0},\alpha ,\beta ,\gamma , \mu _{f})}{\partial t}$.

Proof

Since the assumption $\mathcal {T} -$ valued function f is gr-differentiable at t₀ ∈ U, we have

$$\begin{array}{@{}rcl@{}} \forall\epsilon>0, \exists\delta>0\text{ such that} \forall h: 0<h<\delta \Rightarrow D^{gr}\left( \frac{f(t_{0}+h)\ominus^{gr}f(t_{0})}{h},f^{\prime}_{gr}(t_{0})\right)<\epsilon. \end{array} $$

For simplicity in presentation, let us denote $\frac {f(t_{0}+h)\ominus ^{gr}f(t_{0})}{h}$ and $f^{\prime }_{gr}(t_{0})$ by $\mathcal {A}$ and $\mathcal {A}^{\prime }$, respectively. Then, by employing the concept of granular metric, the above statement can be rewritten as follows

$$\begin{array}{@{}rcl@{}} &&\forall\epsilon>0, \exists\delta>0\text{ such that} \forall h: 0<h<\delta \Rightarrow\\ &&\sup\limits_{\alpha,\beta,\gamma}\max\limits_{\mu_{i},\tilde{\mu}_{i}}\frac{1}{3}\left\{\left|T_{\mathcal{A}}^{gr}(\alpha,\mu_{1})-T_{\mathcal{A}^{\prime}}^{gr}(\alpha,\tilde{\mu}_{1})\right|+\left|I_{\mathcal{A}}^{gr}(\beta,\mu_{2})-I_{\mathcal{A}^{\prime}}^{gr}(\beta,\tilde{\mu}_{2})\right|\right.\\ &&\left.+\left|F_{\mathcal{A}}^{gr}(\gamma,\mu_{3})-F_{\mathcal{A}^{\prime}}^{gr}(\gamma,\tilde{\mu}_{3})\right|\right\}<\epsilon, \end{array} $$

that is equivalent to

$$\begin{array}{@{}rcl@{}} \left\|\frac{1}{h}\left[f^{gr}(t_{0}+h,\alpha,\beta,\gamma, \mu_{f})-f^{gr}(t_{0},\alpha,\beta,\gamma, \mu_{f})\right]-\left( f^{\prime}_{gr}\right)^{gr}(t_{0},\alpha,\beta,\gamma,\mu_{f^{\prime}})\right\| \end{array} $$

is getting as small as h tends to 0. Here, we denote

$$\begin{array}{@{}rcl@{}} &\frac{1}{h}\left[f^{gr}(t_{0}+h,\alpha,\beta,\gamma, \mu_{f})-f^{gr}(t_{0},\alpha,\beta,\gamma, \mu_{f})\right]=\left( T_{\mathcal{A}}^{gr}(\alpha,\mu_{1}),I_{\mathcal{A}}^{gr}(\beta,\mu_{2}),F_{\mathcal{A}}^{gr}(\gamma,\mu_{3})\right),\\ & \left( f^{\prime}_{gr}\right)^{gr}(t_{0},\alpha,\beta,\gamma,\mu_{f^{\prime}})=\left( T_{\mathcal{A}^{\prime}}^{gr}(\alpha,\tilde{\mu}_{1}),I_{\mathcal{A}^{\prime}}^{gr}(\beta,\tilde{\mu}_{2}),F_{\mathcal{A}^{\prime}}^{gr}(\gamma,\tilde{\mu}_{3})\right). \end{array} $$

Therefore, this follows that the gr-differentiability of f implies the differentiability of its horizontal membership function. By analogous arguments, we also prove the converse statement. The proof is complete. □

Proposition 3.4

Let $f, g:[a,b]\to \mathcal {T}$ be differentiable $\mathcal {T} -$ valued functions. Then, based on the horizontal membership function approach, the following statements are fulfilled:

• (i) $(\mathcal {A})'_{gr}=\tilde {0}$ , where $\mathcal {A}\in \mathcal {T}$ and $\tilde {0}$ is zero neutrosophic number.

• (ii)$\left (\alpha f(t)\pm \beta g(t)\right )'_{gr}=\alpha f^{\prime }_{gr}(t)+\beta g^{\prime }_{gr}(t)$,wheret ∈ [a, b] andα,$\beta \in \mathbb {R}$.

• (iii)$(fg)'_{gr}(t)=f^{\prime }_{gr}(t)g(t)+f(t)g^{\prime }_{gr}(t)$,wheret ∈ [a, b].

Example 3.3

Let $\mathcal {A}=(5,10, 15;3,6,9;10, 16,22)$ and $\mathcal {\tilde {A}}=(4,6,8;1,3,5;9,11,13)$ be two triangular neutrosophic numbers of type 1. Consider the $\mathcal {T}$-valued function $f(t)=\mathcal {A}t+\mathcal {\tilde {A}}\sin {2t}$ with t ∈ [0,7]. Then, the horizontal membership function of f is given in Example 3.2 and its derivative is as

$$\begin{array}{@{}rcl@{}} \frac{\partial f^{gr}(t,\alpha,\beta,\gamma,\mu,\tilde{\mu})}{\partial t}&=&\left( \left[5 + 5\alpha+(10-10\alpha)\mu_{1}\right]+\left[8 + 4\alpha+(8-8\alpha)\tilde{\mu}_{1}\right]\cos{2t};\right.\\ &&\quad \left.\left[6-3\beta+ 6\beta\mu_{2}\right]+\left[6-4\beta+ 8\beta\tilde{\mu}_{2}\right]\cos{2t};\right.\\ &&\quad \left.\left[16-6\gamma+ 12\gamma\mu_{3}\right]+\left[22-4\gamma+ 8\gamma\tilde{\mu}_{3}\right]\cos{2t}\right), \end{array} $$

where α, β, γ ∈ [0, 1] and $\mu =(\mu _{1},\mu _{2},\mu _{3}), \tilde {\mu }=(\tilde {\mu }_{1},\tilde {\mu }_{2},\tilde {\mu }_{3})\in [0, 1]^{3}$. Thus, it follows that the function f is gr-differentiable and the (α, β, γ) - cuts of its derivative is given by

$$\begin{array}{@{}rcl@{}} [f^{\prime}_{gr}(t)]_{(\alpha,\beta,\gamma)}&=&\mathcal{H}^{-1}\left( \frac{\partial f^{gr}(t,\alpha , \alpha_{u},\alpha_{v})}{\partial t}\right)\\&=&\left( \left[5 + 5\alpha,15-5\alpha\right]+\left[8 + 4\alpha,16-4\alpha\right]\cos{2t};\right.\\ &&\quad \left.\left[6-3\beta+ 6 + 3\beta\right]+\left[6-4\beta,6 + 4\beta\right]\cos{2t};\right.\\ &&\quad \left.\left[16-6\gamma,16 + 6\gamma\right]+\left[22-4\gamma,22 + 4\gamma\right]\cos{2t}\right). \end{array} $$

By using (α, β, γ) −cuts representation theorem, we have

$$\begin{array}{@{}rcl@{}} f^{\prime}_{gr}(t)&=&\left( \bigcup\limits_{\alpha}\left\{\left[5 + 5\alpha,15-5\alpha\right]+\left[8 + 4\alpha,16-4\alpha\right]\cos{2t}\right\};\right.\\ &&\quad \left.\bigcup\limits_{\beta}\left\{\left[6-3\beta, 6 + 3\beta\right]+\left[6-4\beta,6 + 4\beta\right]\cos{2t}\right\};\right.\\ &&\quad \left.\bigcup\limits_{\gamma}\left\{\left[16-6\gamma,16 + 6\gamma\right]+\left[22-4\gamma,22 + 4\gamma\right]\cos{2t}\right\}\right)\\ &=&(5,10, 15;3,6,9;10, 16,22)+(8,12,16;2,6,10;18,22,26)\cos{2t} \end{array} $$

Therefore, we obtain that the gr-derivative $f^{\prime }_{gr}(t)$ is $\mathcal {A}+ 2\mathcal {\tilde {A}}\cos {2t}$ which graphical representation is shown in Fig. 5

Fig. 5

The (α, β, γ)- cuts of the gr-derivative of $\mathcal {T}$-valued function f in Example 3.3, where the black curve corresponds to the certain values, while the blue and red curves represent for the left and right end-points

3.6 The neutrosophic integral

Definition 3.9

Let $f:[a,b] \rightarrow \mathcal {T}$ be a continuous $\mathcal {T} -$ function whose HMF $\mathcal {H}\left (f(t)\right )$ is integrable on [a, b]. If there exists a number $\upsilon \in \mathcal {T}$ such that $\mathcal {H}(\upsilon )={\displaystyle {\int }_{a}^{b}} \mathcal {H}\left (f(t)\right )dt$ then such number m is said to be the granular integral (gr-integral for short) of f on [a, b] and denoted by $\upsilon ={\displaystyle {\int }_{a}^{b}} f(t) dt$.

Remark 3.3

Similar to Proposition 3.3, we can easily prove that the integrability of the function f is equivalent to the integrability of its horizontal membership function.

Remark 3.4

Let a function $f:[a,b]\to \mathcal {T}$ and a point x ∈ [a, b]. Then, if f is integrable on [a, b], f is also integrable on the sub-interval [a, x].

Lemma 3.1

If function$f:[a,b] \rightarrow \mathcal {T}$iscontinuous on [a, b] then for eacht ∈ [a, b],the function${\Psi }(t)={\displaystyle {\int }_{a}^{t}}f(s)ds$isan anti-derivative of the function f.

Proof

Let t₀ ∈ [a, b] be arbitrary. Due to the continuity of f at point t₀, we have that for all 𝜖 > 0, there exists δ > 0 such that ∀t ∈ [a, b] satisfying |t − t₀| < δ then D^gr(f(t),f(t₀)) < 𝜖, i.e., $\lim \limits _{t\to t_{0}}f(t)=f(t_{0})$.

For h sufficiently near 0, let us consider the following quotient

$$\begin{array}{@{}rcl@{}} \frac{{\Delta} {\Psi}}{{\Delta} t_{0}}&=\frac{1}{h}\left[{\Psi}(t_{0}+h)\ominus^{gr}{\Psi}(t_{0})\right]=\frac{1}{h}\left[\displaystyle{\int}_{a}^{t_{0}+h}f(s)ds\ominus^{gr}\displaystyle{\int}_{a}^{t_{0}}f(s)ds\right], \end{array} $$

whose horizontal membership function is given by

$$\begin{array}{@{}rcl@{}} \mathcal{H}\left( \frac{{\Delta} {\Psi}}{{\Delta} t_{0}}\right)&=&\frac{1}{h}\left[\displaystyle{\int}_{a}^{t_{0}+h}\mathcal{H}(f(s))ds\ominus^{gr}\displaystyle{\int}_{a}^{t_{0}}\mathcal{H}(f(s))ds\right]\\ &=&\frac{1}{h}\left[\displaystyle{\int}_{a}^{t_{0}+h}f^{gr}(s_{0}+h,\alpha,\beta,\gamma,\mu_{f})ds-\displaystyle{\int}_{a}^{t_{0}}f^{gr}(s_{0},\alpha,\beta,\gamma,\mu_{f})ds\right]\\ &=&\frac{1}{h}\displaystyle{\int}_{t_{0}}^{t_{0}+h}f^{gr}(s_{0}+h,\alpha,\beta,\gamma,\mu_{f})ds. \end{array} $$

Next, by applying mean value theorem for integrals, we obtain that

$$\begin{array}{@{}rcl@{}} \mathcal{H}\left( \frac{{\Delta} {\Psi}}{{\Delta} t_{0}}\right)&=&\frac{1}{h}\displaystyle{\int}_{t_{0}}^{t_{0}+h}f^{gr}(s_{0}+h,\alpha,\beta,\gamma,\mu_{f})ds\\ &=&f^{gr}(t_{0}+\theta h,\alpha,\beta,\gamma,\mu_{f}), \end{array} $$

in which 𝜃 ∈ (0, 1). Since the fact that t₀ + 𝜃h tends to t₀ as h → 0 then we have

$$\begin{array}{@{}rcl@{}} \mathcal{H}\left( {\Psi}^{\prime}_{gr}(t_{0})\right)=\lim\limits_{h\to 0}\mathcal{H}\left( \frac{{\Delta} {\Psi}}{{\Delta} t_{0}}\right)=\lim\limits_{h\to 0}f^{gr}(t_{0}+\theta h,\alpha,\beta,\gamma,\mu_{f})=f^{gr}(t_{0},\alpha,\beta,\gamma,\mu_{f}), \end{array} $$

which implies that $[{\Psi }^{\prime }_{gr}(t_{0})]_{(\alpha ,\beta ,\gamma )}=\mathcal {H}^{-1}(f^{gr}(t_{0},\alpha ,\beta ,\gamma ,\mu _{f}))=[f(t_{0})]_{(\alpha ,\beta ,\gamma )}$ for each α, β, γ ∈ [0, 1]. Since the point t₀ ∈ [a, b] is chosen arbitrarily, this achieves the proof. □

Theorem 3.2 (Newton-Leibniz Formula)

Assume that${\Phi }:[a,b]\subseteq \mathbb {R}\to \mathcal {T}$bea gr-differentiable$\mathcal {T}$-valuedfunction and function$f(t):={\Phi }^{\prime }_{gr}(t)$iscontinuous on [a, b].Then f is gr-integrable and

$$\begin{array}{@{}rcl@{}} {\displaystyle{\int}_{a}^{b}} f(t) dt={\Phi}(b)\ominus^{gr}{\Phi}(a). \end{array} $$

Proof

By Lemma 3.1, we obtain that function ${\Psi }(t)={\displaystyle {\int }_{a}^{t}}f(s)ds$ is an anti-derivative of function f on [a, b] whose horizontal membership function is given as follows

$$\begin{array}{@{}rcl@{}} {\Psi}^{gr}(t,\alpha,\beta,\gamma,\mu_{\Psi})={\displaystyle{\int}_{a}^{t}}f^{gr}(s,\alpha,\beta,\gamma,\mu_{f})ds, \end{array} $$

that means Ψ^gr(t, α, β, γ, μ_Ψ) is also an anti-derivative of f^gr(t, α, β, γ, μ_f) on [a, b]. Hence, if Φ^gr(t, α, β, γ, μ_Φ) is another anti-derivative of f^gr(t, α, β, γ, μ_f) on [a, b] then

$$\begin{array}{@{}rcl@{}} {\Phi}^{gr}(t,\alpha,\beta,\gamma,\mu_{\Phi})&=&{\Psi}^{gr}(t,\alpha,\beta,\gamma,\mu_{\Psi})+\mathcal{C}\\ &=&{\displaystyle{\int}_{a}^{t}}f^{gr}(s,\alpha,\beta,\gamma,\mu_{f})ds+\mathcal{C}, \end{array} $$

where $\mathcal {C}$ is a constant. In addition, by substituting t = a, we have ${\Phi }^{gr}(a,\alpha ,\beta ,\gamma ,\mu _{\Phi })={\displaystyle {\int }_{a}^{a}}f^{gr}(s,\alpha ,\beta ,\gamma ,\mu _{f})$$ds+\mathcal {C}$, or equivalently, ${\Phi }^{gr}(a,\alpha ,\beta ,\gamma ,\mu _{\Phi })=\mathcal {C}$. Thus, it implies that

$$\begin{array}{@{}rcl@{}} &&{\Phi}^{gr}(t,\alpha,\beta,\gamma,\mu_{\Phi})={\displaystyle{\int}_{a}^{t}}f^{gr}(s,\alpha,\beta,\gamma,\mu_{f})ds+{\Phi}^{gr}(a,\alpha,\beta,\gamma,\mu_{\Phi})\\ &\Longleftrightarrow & {\displaystyle{\int}_{a}^{t}}f^{gr}(s,\alpha,\beta,\gamma,\mu_{f})ds={\Phi}^{gr}(t,\alpha,\beta,\gamma,\mu_{\Phi})-{\Phi}^{gr}(a,\alpha,\beta,\gamma,\mu_{\Phi}). \end{array} $$

Letting t = b then we obtain the following formula

$$\begin{array}{@{}rcl@{}} {\displaystyle{\int}_{a}^{b}}f^{gr}(s,\alpha,\beta,\gamma,\mu_{f})ds={\Phi}^{gr}(b,\alpha,\beta,\gamma,\mu_{\Phi})-{\Phi}^{gr}(a,\alpha,\beta,\gamma,\mu_{\Phi}), \end{array} $$

whose (α, β, γ) − cuts can be given as follows

$$\begin{array}{@{}rcl@{}} {\displaystyle{\int}_{a}^{b}}\mathcal{H}^{-1}\left( f^{gr}(s,\alpha,\beta,\gamma,\mu_{f})\right)ds=\mathcal{H}^{-1}\left( {\Phi}^{gr}(b,\alpha,\beta,\gamma,\mu_{\Phi})-{\Phi}^{gr}(a,\alpha,\beta,\gamma,\mu_{\Phi})\right), \end{array} $$

that means the following integral equality holds

$$\begin{array}{@{}rcl@{}} {\displaystyle{\int}_{a}^{b}} f(t) dt={\Phi}(b)\ominus^{gr}{\Phi}(a). \end{array} $$

□

Example 3.4

Consider a function ${\Phi }:[0,5]\to \mathcal {T}$ given by ${\Phi }(t)=\mathcal {A}_{1}e^{-t}+\mathcal {A}_{2}t^{2}$ in which $\mathcal {A}_{1}=\left (4,7,10;0, 1,2;3,5,7\right )$ and $\mathcal {A}_{2}=\left (2,4,6;1,2,3;1,3,5\right )\in \mathcal {T}$. Then, the horizontal membership function of Φ(t) is given by

$$\begin{array}{@{}rcl@{}} \mathcal{H}({\Phi}(t))&=&{\Phi}^{gr}(t,\alpha,\beta,\gamma,\mu,\tilde{\mu})\\ &=&A_{1}^{gr}(\alpha,\beta,\gamma,\mu)e^{-t}+A_{2}^{gr}(\alpha,\beta,\gamma,\tilde{\mu})t^{2}\\ &=&\left( \left[4 + 3\alpha+(6-6\alpha)\mu_{1}\right]e^{-t}+\left[2 + 2\alpha+(4-4\alpha)\tilde{\mu}_{1}\right]t^{2};\right.\\ &&\quad \left.\left[1-\beta+ 2\beta\mu_{2}\right]e^{-t}+\left[2-\beta+ 2\beta\tilde{\mu}_{2}\right]t^{2};\left[5-2\gamma+ 4\gamma\mu_{3}\right]e^{-t}\right.\\ &&\left.+\left[3-2\gamma+ 4\gamma\tilde{\mu}_{3}\right]t^{2}\right). \end{array} $$

By using similar arguments as in Example 3.2, we have that the $\mathcal {T} -$ valued function Φ(t) is gr-differentiable on [0,5] and its derivative is ${\Phi }^{\prime }_{gr}(t)=f(t)$ whose horizontal membership function is given by

$$\begin{array}{@{}rcl@{}} f^{gr}(t,\alpha,\beta,\gamma,\mu,\tilde{\mu})&=&\frac{\partial {\Phi}^{gr}(t,\alpha,\beta,\gamma,\mu,\tilde{\mu})}{\partial t}\\ &=&-A_{1}^{gr}(\alpha,\beta,\gamma,\mu)e^{-t}+ 2A_{2}^{gr}(\alpha,\beta,\gamma,\tilde{\mu})t\\ &=&\left( \left[-4-3\alpha-(6-6\alpha)\mu_{1}\right]e^{-t}+\left[4 + 4\alpha+(8-8\alpha)\tilde{\mu}_{1}\right]t;\right.\\ &&\quad \left.\left[-1+\beta-2\beta\mu_{2}\right]e^{-t}+\left[4-2\beta+ 4\beta\tilde{\mu}_{2}\right]t;\right.\\ &&\quad \left.\left[-5 + 2\gamma- 4\gamma\mu_{3}\right]e^{-t}+\left[6-4\gamma+ 8\gamma\tilde{\mu}_{3}\right]t\right), \end{array} $$

where α, β, γ ∈ [0, 1] and $\mu =(\mu _{1},\mu _{2},\mu _{3}), \tilde {\mu }=(\tilde {\mu }_{1},\tilde {\mu }_{2},\tilde {\mu }_{3})\in [0, 1]^{3}$. Then, employing (α, β, γ) − level sets representation theorem, we obtain that

$$\begin{array}{@{}rcl@{}} f(t)=\mathcal{A}_{3}e^{-t}+\mathcal{A}_{4}t, \end{array} $$

in which $\mathcal {A}_{3}=(-10,-7,-4;-2,-1,0;-7,-5,-3)$, $\mathcal {A}_{4}=(4,8,12;2,4,6;2,6,10)$ are single valued triangular neutrosophic numbers. In addition, we can see that f is continuous function on [0, 10]. Then, by applying Theorem 3.2, it follows that

$$\begin{array}{@{}rcl@{}} {\displaystyle{\int}_{0}^{5}}f(t)dt&={\Phi}(5)\ominus^{gr}{\Phi}(0)=\left( \mathcal{A}_{3}e^{-5}+ 5\mathcal{A}_{4}\right)\ominus^{gr}\mathcal{A}_{3}=\left( 1-e^{-5}\right)\mathcal{A}_{1}+ 5\mathcal{A}_{4}. \end{array} $$

4 Applications to $\mathcal {T} -$ valued differential equations

4.1 $\mathcal {T} -$ valued differential equations

In the following, based on the HMF approach, we will investigate some classes of $\mathcal {T} -$ valued differential equations. Indeed, let us consider following initial problem to $\mathcal {T} -$ valued differential equations

$$ \left\{\begin{array}{ll} x^{\prime}_{gr}(t)&=f(t,x(t))\\ x(t_{0})&=x_{0} \end{array}\right.\qquad t\in [t_{0},T], $$

(4)

where $f:[t_{0},T]\to \mathcal {T}$ is a $\mathcal {T} -$ valued function including n distinct neutrosophic numbers $\mathcal {A}_{1}$, $\mathcal {A}_{2}$, …, $\mathcal {A}_{n}$, $x^{\prime }_{gr}(\cdot )$ represents for the gr-derivative of x(⋅) w.r.t t and $x_{0}\in \mathcal {T}$ is initial condition. According to Definition 3.2, the initial problem (4) can be transformed into following form

$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{ll} \mathcal{H}(x^{\prime}_{gr}(t))&=\mathcal{H}(f(t,x(t)))\\ \mathcal{H}(x(t_{0}))&=\mathcal{H}(x_{0}) \end{array}\right.\qquad t\in [t_{0},T], \end{array} $$

Then, by using Proposition 3.3, we obtain that

$$ \left\{\begin{array}{ll} \frac{\partial x^{gr}(t,\alpha,\beta,\gamma, \mu_{x})}{\partial t}&=f^{gr}(t, x^{gr}(t,\alpha,\beta,\gamma, \mu_{x}),\alpha,\beta,\gamma,\mu_{f})\\ x^{gr}(t_{0},\alpha,\beta,\gamma,\mu_{x})&=x_{0}^{gr}(\alpha,\beta,\gamma,\mu_{x}) \end{array}\right.\qquad t\in [t_{0},T], $$

(5)

where α, β, γ ∈ [0, 1], μ_x ∈ [0, 1]³ and $\mu _{f} \triangleq (\mu _{1,\mathcal {A}_{1}},\ldots ,\mu _{1,\mathcal {A}_{n}},\mu _{2,\mathcal {A}_{1}},\ldots ,$$\mu _{2,\mathcal {A}_{n}},\mu _{3,\mathcal {A}_{1}},\ldots ,\mu _{3,\mathcal {A}_{n}})$.

Thus, under the HMF approach, we can see that the use of gr-differentiability help us only need to solve just one differential equation that is equivalent to the given equation and we call this equivalent equation is granular differential equation. Moreover, we can see that if $\mathcal {T} -$ valued differential equation (4) doesn’t have solution then the corresponding granular differential equation also does not. Conversely, if $\tilde {x}^{gr}(t,\alpha ,\beta ,\gamma ,\mu _{x})$ is a solution of problem (5) then it is also the solution of problem (4).

Remark 4.1

Some important results such the well-posedness or the existence and uniqueness of solution to Cauchy problems (4) for $\mathcal {T} -$ valued differential equations correspond to those of Cauchy problem (5) for granular differential equations.

Example 4.1

Consider following $\mathcal {T} -$ valued differential equations

$$ \left\{\begin{array}{ll} \dot{x}_{gr}(t)&=y(t)-\tilde{u}\\ \dot{y}_{gr}(t)&=-4x(t) \end{array}\right. $$

(6)

subject to the initial condition

$$\begin{array}{@{}rcl@{}} x(0)=y(0)=\tilde{0}, \end{array} $$

where $\tilde {u}=(-1,0, 1;0, 1,2;-2,0,2)$, $\tilde {0}=(0,0,0;0,0,0;0,0,0)$ are single valued triangular neutrosophic numbers and t ∈ [0, 10]. By using the similar method to obtain the system (5), it follows that the corresponding granular system of differential equations of (6) is

$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{ll} \frac{\partial x^{gr}(t,\alpha,\beta,\gamma,\mu_{x})}{\partial t}&=y^{gr}(t,\alpha,\beta,\gamma,\mu_{y})-\tilde{u}^{gr}(\alpha,\beta,\gamma,\mu_{\tilde{u}})\\ \frac{\partial y^{gr}(t,\alpha,\beta,\gamma,\mu_{y})}{\partial t}&=-x^{gr}(t,\alpha,\beta,\gamma,\mu_{x})\\ x^{gr}(0,\alpha,\beta,\gamma,\mu_{x})&=y^{gr}(0,\alpha,\beta,\gamma,\mu_{y})= 0, \end{array}\right. \end{array} $$

in which $\tilde {u}^{gr}(\alpha ,\beta ,\gamma ,\mu _{\tilde {u}})=\left (\alpha -1+(2-2\alpha )\mu _{\tilde {u},1};1-\beta + 2\beta \mu _{\tilde {u},2};4\gamma \mu _{\tilde {u},3} \right )$ and the triplets $\mu _{x}=\mu _{y}=\mu _{\tilde {u}}=\overline {\mu }$. Then, the solution of the above system is given as

$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{ll} x^{gr}(t,\alpha,\beta,\gamma,\overline{\mu})&=-\tilde{u}^{gr}(\alpha,\beta,\gamma,\overline{\mu})\sin 2t\\ y^{gr}(t,\alpha,\beta,\gamma,\overline{\mu})&=\frac{1}{2} \tilde{u}^{gr}(\alpha,\beta,\gamma,\overline{\mu})\left( 1-\cos 2t\right), \end{array}\right. \end{array} $$

whose (α, β, γ) − cuts can be given as

$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{llll} [x(t)]_{(\alpha,\beta,\gamma)}=\left( [-1+\alpha,1-\alpha];[-1-\beta,-1+\beta];[-2\gamma,2\gamma]\right)\sin 2t\\ \left[y(t)\right]_{(\alpha,\beta,\gamma)}=\left( \left[-\frac{1}{2}+\frac{\alpha}{2},\frac{1}{2}-\frac{\alpha}{2}\right];\left[\frac{1}{2}-\frac{\beta}{2},\frac{1}{2}+\frac{\beta}{2}\right];\left[-\gamma,\gamma\right]\right)(1-\cos 2t) \end{array}\right. \end{array} $$

Figures 6 and 7 show the (α, β, γ) − level sets of solution of the system (6).

Fig. 6

The (α, β, γ) − cuts of function x(t) that corresponds to the solution of system (6)

Fig. 7

The (α, β, γ) − cuts of function y(t) that corresponds to the solution of system (6)

4.2 Some real-life models

Example 4.2 (Logistic equations)

In this example, we consider dynamics of a single population model. We denote by x = Φ(t) the number of individuals of a given species at the time t and r by the percent change of the population. If r is not impacted by the limitation of space and food then we can assume it as a constant. However, in real world, this assumption is unrealistic. Thus, in modeling models of population by dynamic system, we often modify the unrestricted growth rate r to ensure that the environment can only support a certain number of the species, denoted by $\mathcal {K}$, namely the carrying capacity of the environment with populations living in. If $x>\mathcal {K}$ then it cause consequences the lack of food and space available to support x, more species will be die than will be born, which leads to the negative growth rate. Conversely, if $x<\mathcal {K}$ then the population growth should be positive. Using the above model of the population growth, we consider the following differential equation that is known as the Verhulst equations or logistic equations

$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{ll} \dot{\Phi}_{gr}(t)&=r {\Phi}(t)\cdot\left( 1.5\ominus^{gr}{\Phi}(t)\right)\\ {\Phi}(0)&=\mathcal{A}, \end{array}\right.\qquad t\in[0,7], \end{array} $$

where r = 0.8 and $\mathcal {A}=(0.1,0.3,0.5;0.1,0.2,0.3;0,0.1,0.2)$. Here, due to the uncertainty of available information about the initial population of the species when modeling this real-world problems, neutrosophic value presentation has been considered as a better description in the formulation of the mathematical model.

In addtion, based on the approach mentioned in previous section, we have

$$ \frac{\partial {\Phi}^{gr}(t,\alpha,\beta,\gamma,\mu)}{\partial t}= 0.8{\Phi}^{gr}(t,\alpha,\beta,\gamma,\mu)\left( 1.5-{\Phi}^{gr}(t,\alpha,\beta,\gamma,\mu)\right) $$

(7)

subject to the initial condition

$$\begin{array}{@{}rcl@{}} {\Phi}^{gr}(0,\alpha,\beta,\gamma,\mu)=\left( 0.3 + 0.2\alpha+ 0.4(1-\alpha)\mu_{1};0.2-0.1\beta+ 0.2\beta\mu_{2};0.1-0.1\gamma+ 0.2\gamma\mu_{3}\right), \end{array} $$

where α, β, γ ∈ [0, 1] and μ_i ∈ [0, 1] (i = 1,2,3).

The solution of the granular differential (1) is

$$\begin{array}{@{}rcl@{}} {\Phi}^{gr}(t,\alpha,\beta,\gamma,\mu)=\frac{1.5 {\Phi}^{gr}(0,\alpha,\beta,\gamma,\mu)}{{\Phi}^{gr}(0,\alpha,\beta,\gamma,\mu)+\left[1.5-{\Phi}^{gr}(0,\alpha,\beta,\gamma,\mu)\right]e^{-1.2t}}, \end{array} $$

whose (α, β, γ) − cuts is given as follows

$$\begin{array}{@{}rcl@{}} [{\Phi}(t)]_{(\alpha,\beta,\gamma)}&=&\left[\frac{0.15 + 0.3\alpha}{0.1 + 0.2\alpha+\left( 1.4-0.2\alpha\right)e^{-1.2t}},\frac{0.75-0.3\alpha}{0.5-0.2\alpha+\left( 1 + 0.2\alpha\right)e^{-1.2t}}\right.\\ &&\quad\left.\frac{0.3-0.15\beta}{0.2-0.1\beta+\left( 1.3 + 0.1\beta\right)e^{-1.2t}},\frac{0.3 + 0.15\beta}{0.2 + 0.1\beta+\left( 1.3-0.1\beta\right)e^{-1.2t}}\right.\\ &&\quad\left.\frac{0.15-0.15\gamma}{0.1-0.1\gamma+\left( 1.4 + 0.1\gamma\right)e^{-1.2t}},\frac{0.15 + 0.15\gamma}{0.1 + 0.1\gamma+\left( 1.4-0.1\gamma\right)e^{-1.2t}}\right]. \end{array} $$

The (α, β, γ) − level sets of the solution of the logistic (3.3) with respect to the initial condition Φ(0) = (0.1,0.3,0.5;0.1,0.2,0.3;0,0.1,0.2) is presented in Fig. 8

Fig. 8

The (α, β, γ) − cuts of the solution Φ(t) of the logistic (3.3).

Remark 4.2

From the above figure, we see that if at the initial time, the population of species is in the carrying capacity of the environment then the population will approach to the carrying capacity value as time increases.

Example 4.3

The inverted pendulum system is a popular demonstration of using feedback control to stabilize an open-loop unstable system. In this example, we consider the following mechanical system which model an inverted pendulum on the cart.

By applying Newton’s second law to mechanical system including two masses m₁, m₂, we have following nonlinear model

$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{ll} (m_{1}+m_{2})\ddot{y}+m_{2}\ell\ddot{\theta}\cos\theta-m_{2}\ell\dot{\theta}^{2}\sin\theta+\mu \dot{y}&=u\\ \ell\ddot{\theta}-g\sin\theta+\ddot{y}\cos \theta&= 0. \end{array}\right. \end{array} $$

Let x₁ = y, x₂ = 𝜃, $x_{3}=\dot {y}$, $x_{4}=\dot {\theta }$. Then, we obtain the state equation corresponding to the mechanical system in Fig. 9

$$\begin{array}{@{}rcl@{}} \left[\begin{array}{ll} \dot{x}_{1}\\\dot{x}_{2}\\\dot{x}_{3}\\\dot{x}_{4} \end{array}\right]=\left[\begin{array}{cc} x_{3}\\x_{4}\\\left[\begin{array}{cc} m_{1}+m_{2}&m_{2}\ell\cos x_{2}\\ \cos x_{2}&\ell \end{array}\right]^{-1}\left( \left[\begin{array}{cc} m_{2}\ell {x_{4}^{2}}\sin x_{2}-\mu x_{3}\\ g\sin x_{2} \end{array}\right]+\left[\begin{array}{cc} 1\\0 \end{array}\right]u\right) \end{array}\right]. \end{array} $$

Next, by using linearization method, we obtain the linearized system of inverted pendulum model as follows

$$ \left[\begin{array}{ccccc} \dot{x}_{1}\\\dot{x}_{2}\\\dot{x}_{3}\\\dot{x}_{4} \end{array}\right]=\left[\begin{array}{ccccc} 0&0&1&0\\ 0&0&0&1\\ 0&-\frac{m_{2}}{m_{1}}g&-\frac{\mu}{m_{1}}&0\\ 0&\frac{m_{1}+m_{2}}{m_{1}\ell}g&\frac{\mu}{m_{1}\ell}&0 \end{array}\right]\left[\begin{array}{cc} x_{1}\\x_{2}\\x_{3}\\x_{4} \end{array}\right]+\left[\begin{array}{cc} 0\\0\\\frac{1}{m_{1}}\\-\frac{1}{m_{1}\ell} \end{array}\right]u, $$

(8)

in which the parameters and their values are given in Table 4.

Fig. 9

Inverted pendulum on cart

Table 4 Parameter’s value

Here, we consider the acceleration of gravity g = (9.6,9.8,10;0.5,1.5,2.5;1.5,2,2.5) $\in \mathcal {T}$ is an uncertain quantity due to errors in measurement and influence of environmental factors such temperature, humidity, meteorology, etc. From the uncertainty of g, it follows that the matrix’s coefficients of (8) is also uncertain, that is equivalent to the uncertainty in the form of solution.

In this example, our main aim is to show that the open-loop system is unstable, i.e., the system (8) is considered under assumption that external force u ≡ 0. Indeed, the system (8) then becomes

$$ \left[\begin{array}{cc} \dot{x}_{1}\\\dot{x}_{2}\\\dot{x}_{3}\\\dot{x}_{4} \end{array}\right]=\left[\begin{array}{cccccc} 0&0&1&0\\ 0&0&0&1\\ 0&-\frac{g}{3}&-\frac{166}{117}&0\\ 0&\frac{2g}{3}&\frac{83}{117}&0 \end{array}\right]\left[\begin{array}{cc} x_{1}\\x_{2}\\x_{3}\\x_{4} \end{array}\right]. $$

(9)

Based on the concepts of gr-differentiability and horizontal membership function approach, the differential system (9) can be transformed into following form

$$ \left[\begin{array}{cc} \frac{\partial x_{1}^{gr}(t,\alpha,\beta,\gamma,\mu_{x_{1}})}{\partial t}\\\frac{\partial x_{2}^{gr}(t,\alpha,\beta,\gamma,\mu_{x_{2}})}{\partial t}\\\frac{\partial x_{3}^{gr}(t,\alpha,\beta,\gamma,\mu_{x_{3}})}{\partial t}\\\frac{\partial x_{4}^{gr}(t,\alpha,\beta,\gamma,\mu_{x_{4}})}{\partial t} \end{array}\right]=\left[\begin{array}{cccc} 0&0&1&0\\ 0&0&0&1\\ 0&-\frac{g^{gr}(\alpha,\beta,\gamma,\mu_{g})}{3}&-\frac{166}{117}&0\\ 0&\frac{2g^{gr}(\alpha,\beta,\gamma,\mu_{g})}{3}&\frac{83}{117}&0 \end{array}\right]\left[\begin{array}{cc} x_{1}^{gr}(t,\alpha,\beta,\gamma,\mu_{x_{1}})\\x_{2}^{gr}(t,\alpha,\beta,\gamma,\mu_{x_{2}})\\x_{3}^{gr}(t,\alpha,\beta,\gamma,\mu_{x_{3}})\\x_{4}^{gr}(t,\alpha,\beta,\gamma,\mu_{x_{4}}) \end{array}\right], $$

(10)

Let us choose $\mu _{x_{1}}=\mu _{x_{2}}=\mu _{x_{3}}=\mu _{x_{4}}=\mu _{g}=\mu$ and denote

$$\begin{array}{@{}rcl@{}} A=\left[\begin{array}{cccc} 0&0&1&0\\ 0&0&0&1\\ 0&-\frac{g^{gr}(\alpha,\beta,\gamma,\mu)}{3}&-\frac{166}{117}&0\\ 0&\frac{2g^{gr}(\alpha,\beta,\gamma,\mu)}{3}&\frac{83}{117}&0 \end{array}\right]. \end{array} $$

Then, since the stability of system (9) is equivalent to the stability of corresponding granular linear differential system, the rest of proof is to show that the linear system (10) is unstable for each (α, β, γ) − cuts and μ ∈ [0, 1]. By using MATLAB’s tool, we obtain following tables about the truth, indeterminacy and falsity membership function of Reλ(A) (Tables 5, 6 and 7).

Table 5 Truth membership function of Reλ(A)

Table 6 Indeterminacy membership function of Reλ(A)

Table 7 Falsity membership function of Reλ(A)

As a result, we can see that Reλ(A) is always positive for each α, β, γ ∈ [0, 1] and μ ∈ [0, 1], that means the granular linear differential system (10) is unstable. Therefore, it implies that the open-loop system of inverted pendulum model is an unstable system.

Example 4.4 (Mass - Spring - Damper)

Consider a mechanical system containing two masses that are hung from the ceiling by two strings. Here, each string can be modeled as a combination of a spring and a dashpot for friction (see Fig. 10). If we act to the system an external force u then by Hook’s law, we can deduce that the forces are linearly proportional to the corresponding displacements, while the forces due to the frictions depend on both displacements and velocities. By applying Newton’s second law to two masses m₁ and m₂, we obtain that

$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{ll} M_{1}\ddot{y}_{1}=u-K_{1}(y_{1}-y_{2})-D_{1}(\dot{y}_{1}-\dot{y}_{2})\\ M_{2}\ddot{y}_{2}=K_{1}(y_{1}-y_{2})+D_{1}(\dot{y}_{1}-\dot{y}_{2})-K_{2}y_{2}-D_{2}\dot{y}_{2}, \end{array}\right. \end{array} $$

where $\dot {y}_{i}$, $\ddot {y}_{i}$ represent for gr-derivative and second gr-derivative of y_i, respectively.

Fig. 10

Mass - Spring - Damper model

To obtain the state equations, let us denote x₁ = y₁, $x_{2}=\dot {y}_{1}$, x₃ = y₂, $x_{4}=\dot {y}_{2}$. Then, the state equations of the system can be represented by following matrix form

$$ \left[\begin{array}{cccc} \dot{x}_{1}\\\dot{x}_{2}\\\dot{x}_{3}\\\dot{x}_{4} \end{array}\right]=\left[\begin{array}{cccccc} 0&1&0&0\\ -\frac{K_{1}}{M_{1}}&-\frac{D_{1}}{M_{1}}&\frac{K_{1}}{M_{1}}&\frac{D_{1}}{M_{1}}\\ 0&0&1&0\\ \frac{K_{1}}{M_{2}}&\frac{D_{1}}{M_{2}}&-\frac{K_{1}+K_{2}}{M_{2}}&-\frac{D_{1}+D_{2}}{M_{2}} \end{array}\right]\left[\begin{array}{cc} x_{1}\\x_{2}\\x_{3}\\x_{4} \end{array}\right]+\left[\begin{array}{cc} 0\\\frac{1}{M_{1}}\\0\\0 \end{array}\right]u, $$

(11)

where u is external force and the coefficients D_i, K_i, m_i are determined in Table 8:

Table 8 Parameter values

In addition, the initial state of this mechanic system is given as

$$ \left\{\begin{array}{ll} x_{1}(0)=\left( 0.05,0.1,0.15;0,0.1,0.2;0.05,0.15,0.25\right),\\ x_{2}(0)=(-0.8,-0.6,-0.4;-0.1,0,0.1;-0.2,-0.1,0),\\ x_{3}(0)=\left( -0.1,0,0.1;0,0.1,0.2;0,0.05,0.1\right),\\ x_{4}(0)=(0.6,0.8,1;0.5,0.8,1.1;0.3,0.4,0.5). \end{array}\right. $$

(12)

Since the initial states and the external force acting to the mechanical system cannot be certain values due to the lack of specialized measure equipment and the errors in experiment and computation, it follows that the mechanical system becomes a complex system containing uncertainties in both coefficients and conditions and hence, it is necessary to introduce uncertainty in the solution.

For the initial problem to the system (11) subject to the conditions (12), by using MATLAB’s program for Runge Kutta numerical method, we obtain that Figs. 11, 12 and 13 show the graphical representation of solution of mechanical system (11) with initial state (12) with respect to some different values of (α, β, γ) − cuts.

Fig. 11

The (α, β, γ) −cuts of solution of the problem (11)-(12) with α = β = γ = 0 and μ = 1

Fig. 12

The (α, β, γ) −cuts of solution of the problem (11)-(12) with α = β = γ = 0.5 and μ = 1

Fig. 13

The (α, β, γ) −cuts of solution of the problem (11)-(12) with α = β = γ = 1 and μ = 1

5 Conclusions

In this work, by using horizontal membership functions approach, a new representation of triangular neutrosophic number is introduced. Additionally, the metric on space of single valued triangular neutrosophic numbers and the continuity of neutrosophic valued functions are also presented. Especially, the concept of derivative of neutrosophic valued function, namely granular derivative, is firstly defined based on granular difference beside the foundation of the concept granular integral. Under these concepts, the neutrosophic differential equations have been investigated. To solve this kind of equations, the horizontal membership function approach is used. The next step of our future research, we will study the controllability and stabilizability for some classes of linear time-invariant neutrosophic systems, neutrosophic dynamic system of fractional order with applications to signal processing.