On numerical solution of general fuzzy type-1 and type-2 arbitrary order dynamical systems

Abstract

This paper develops the mathematical framework and the solution of a system of type-1 and type-2 fuzzy fractional and arbitrary order dynamical systems. The theory of arbitrary order differential equation is developed with fuzzy initial values, fuzzy boundary values and fuzzy parameters. There are numerous natural phenomena which can be modelled as arbitrary order dynamical systems whose initial conditions and/or parameters may be imprecise in nature. The imprecision of initial values and/or parameters are generally modelled by fuzzy sets. Here, the concept arbitrary order dynamical systems is developed with the introduction of fuzzy fractional derivatives, fuzzy fractional partial derivatives, fuzzy stochastic process, fuzzy stochastic random variable and fuzzy Brownian motion. The generalized L^p-integrability, based on the extension of the class of differentiable and integrable fuzzy functions is applied and fuzzy arbitrary order differential equations are represented as fuzzy integral equations. A novel numerical scheme for simulations of numerical solution for fuzzy arbitrary order differential equations have also been developed. Some illustrative examples have been provided for different fuzzy type-1 and type-2 arbitrary order general dynamical systems models related to mathematical finance and problems in fluid flow.

Keywords

Fuzzy fractional calculus fuzzy arbitrary order ODE fuzzy arbitrary order PDE fuzzy arbitrary order SDE numerical solutions

1 Introduction

The non-local property of the fractional differential and integral operators made them very useful in mathematical model in recent times. However in general the analytical solution of fractional differential equations are quite difficult; hence most research works focused on obtaining the numerical solutions of fractional differential and integral operators [1 –4]. One of the most important usages of fractional differential equations is to analyse fractal sets, because fractals are irregular and nowhere differentiable functions. The properties local fractional integrals and derivatives are studied by [2] based on the local fractional calculus. The numerical methods for solving fractional order differential equations is developed by [5]. Smoothness of solution of caputo type fractional differential equation is studied in [6]; also Adams method to solve multi-term multi-order fractional order differential equations is developed by Diethelm in [7]. In [8] arbitrary real order differential equations are solved by matrix method which is based on discretization of the domain of given integro-differential operator as triangular strip matrix. In this approach the n-dimensional arbitrary integro-differential operator is discretised by Kronecker matrix product in an n-dimensional net of nodes [9]. Few research works are published in the recent time focusing numerical solution to fractional stochastic differential equations [10]. Runge kutta method for fractional stochastic differential equations was developed in [11].

In many real cases some parameters or functions in such modelling are found to be imprecise. The imprecision and uncertainty of knowledge are generally modelled by fuzzy functions. Zadeh initially introduced the idea of differentiability of fuzzy functions. It was further developed by using the extension principle and a few other concepts related to fuzzy derivative have been discussed by Puri [12]. The fuzzy initial value problem (FIVP) has been studied by Kaleva [13] and Seikkala [14]. A few alternative definitions of the differentiability of fuzzy functions are available amongst the Hukuhara differentiability [H-differentiability] and the strongly generalized differentiability [12, 15], became most acceptable and have attracted more attention than the other theories.

A more general concept to model imprecision and uncertainty is presented in [16], where type-2 fuzzy sets (T2FSs) is defined as a fuzzy set in which membership grades are Type-1 Fuzzy Sets (T1FSs). Moreover the parameters and state variables of different dynamic systems like environmental and biological systems are highly imprecise and could be modelled more appropriately by type 2 fuzzy variables and hence by type 2 fuzzy differential equations. The definition of derivative of type 2 fuzzy functions is first provided by Mazandarni with the notion of type-2 Hukuhara differentiability (H2-differentiability) [17] which is based on the distance measure between T2FSs known as Type-2 Hukuhara difference (H2-difference).

However, imprecision and chance are two fundamental manifestation of uncertainty in natural systems. The chance factor i.e. randomness of the uncertain system is often represented in terms of white noises or stochastic processes; and imprecision is modelled as fuzziness and both are found to appear simultaneously in real systems. In this context stochastic fuzzy differential equations, which incorporates randomness in terms of white noise and imprecision in terms of fuzzy variables is presented in [18]. Some extensive applications of fuzzy stochastic differential equations to model dynamics of population growth, stock price, short-term interest rate etc. are presented in papers [19, 20].

The paper [21] first used the local Lipschitz condition to obtain uniqueness and existence results for fuzzy stochastic dynamical system. Some theorem on local uniqueness are presented in [19, 22].

The idea of fuzzy Riemann-Liouville fractional differential equations were first presented in [23] base on the Hukuhara differentiability and facilitate solving uncertain fractional differential equations. The existence and uniqueness of solutions of Riemann-Liouville fuzzy fractional differential equations are studied in [24, 25]. In [26] Euler method was introduced to solve hybrid fuzzy differential equations. The explicit solutions of uncertain fractional differential equations under Riemann-Liouville H-differentiability is given in [27].

The organization of the paper is as follows. Section 2 contains the prerequisite theoretical backgrounds to understand the theory of general fuzzy arbitrary order differential equations. In Section 3 the main theoretical developments are described. Section 4 discusses the representation of fuzzy multi-term multi-order fractional order dynamics as fuzzy fractional order dynamical systems. In Section 5 the major contribution of the paper that is the development of a novel numerical schemes to solve type-1 and type-2 fuzzy arbitrary order differential equations, partial differential equations and stochastic differential equations are presented. In Section 6 different examples of type-1 and type-2 fuzzy arbitrary order dynamics are considered and its solution is obtained by the proposed method. The analysis of the numerical solution by the proposed method and major findings of the paper are presented in Section 7.

2 Mathematical prelims

In this section we introduce some basic concepts of type-1 and type-2 fuzzy sets, fuzzy calculus, elementary fractional calculus and stochastic calculus which are used in this paper to develop the theory of general fuzzy arbitrary order dynamics.

2.1 Type-1 fuzzy set (T1FS): Fuzzy sets and fuzzy calculus

Definition. A fuzzy set $\tilde{A}$ is a set such every element of A is characterized by its membership function μ_A (x), i.e., $\tilde{A} = {(x, μ_{A} (x)) : μ_{A} (x) \in [0, 1]}$ is a fuzzy set of elements x ∈ X. Definition of elementary operators in fuzzy sets are given in [16].

Throughout the paper let us denote the universe of type-1 and type-2 fuzzy set valued functions by E₁ and E₂ respectively.

Definition. If A ∈ Kⁿ then the ɛ-neighbourhood of A is defined by N (A, ɛ) = {x ∈ Rⁿ : d (x, A) < ɛ}, where $d (x, A) = inf_{a \in A} | | x - a | | and | | | |$ denotes the Euclidean distance in Rⁿ.

Definition. The Hausdorff separation ρ (A, B) of A, B ∈ Kⁿ is defined as ρ (A, B) = inf {ɛ > 0 : A ⊂ N (B, ɛ)}. The Hausdorff difference which is a metric on Kⁿ is defined as d_H (A, B) = max(ρ (A, B) , ρ (B, A)).

It is proved that (Kⁿ, h) is a complete and separable metric space [28].

Definition. The α-cut of a fuzzy set $\tilde{u}$ in Rⁿ is defined as u^α = {x ∈ Rⁿ : u (x) > α} and said to be α-level of $\tilde{u}$ , with α ∈ [0, 1]. Union of all α - cut of a fuzzy set u for α > 0 is defined as support of $\tilde{u}$ : u₀ = {x ∈ Rⁿ : u (x) >0} and it is denoted by $supp (\tilde{u})$ [16].

The definition of type-1 fuzzy number was provide in the papers [12 –14]. The T1FN $\tilde{u} \in E_{1}$ is represented in parametric form as the interval $[{\underline{u}}^{α}, {\bar{u}}^{α}], 0 \leq α \leq 1$ , where ${\underline{u}}^{α}$ and ${\bar{u}}^{α}$ are the lower bound and upper bound of $\tilde{u}$ .

Definition. Let $\tilde{u}, \tilde{v} \in E_{1}$ . If there exists a T1FN $\tilde{w} \in E_{1}$ such that, $\tilde{u} = \tilde{v} + \tilde{w}$ then $\tilde{w}$ is said to be Hukuhara difference (H - difference) of $\tilde{u}$ and $\tilde{v}$ ; and is denoted by $\tilde{u} -^{H} \tilde{v}$ .

Definition. $\tilde{f} : (a, b) \to E_{1} to \in T$ , be differentiable in the first form, if there exists an element ${\tilde{f}}^{'} (t_{0}) \in E_{1}$ such that for any sufficiently small h > 0, $lim_{h \to 0} \frac{\tilde{f} (t_{0} + h) -^{H} \tilde{f} (t_{0})}{h} = lim_{h \to 0} \frac{\tilde{f} (t_{0}) -^{H} \tilde{f} (t_{0} - h)}{h} = \tilde{f}' (t_{0})$ Or, $\tilde{f} (t)$ is differentiable at t₀, in the second form, if there exists an element ${\tilde{f}}^{'} (t_{0}) \in E_{1}$ , such that for any sufficiently small h > 0, [29] $lim_{h \to 0} \frac{\tilde{f} (t_{0}) -^{H} \tilde{f} (t_{0} + h)}{h} = lim_{h \to 0} \frac{\tilde{f} (t_{0} - h) -^{H} \tilde{f} (t_{0})}{h} = {\tilde{f}}^{'} (t_{0})$ .

The limits are realised in the metric space (E₁, d₁) where d₁ (u, v) = sup {d_H (u_α, v_α) , 0 ≤ α ≤ 1} and d_H is the Hausdorff distance.

If $\tilde{F} \in E_{1}$ is differentiable in T = [a, b] and if the derivative ${\tilde{F}}^{'}$ is integrable over T. Then for each s ∈ T we have the solution of fuzzy differential equation as $\tilde{F} (s) = \tilde{F} (a) + \int_{a}^{t} \tilde{F}' dt$ .

The fuzzy initial value problem and its α-cut representation is provided in [13 , 29].

2.2 Type-2 fuzzy sets (T2FSs)

A type 2 fuzzy set (T2FS) is a fuzzy set so that degree of membership of its elements are also fuzzy. Let E₂ denotes the set of all T2FS over R. The type-2 membership function of T2FS $\tilde{A_{\tilde{α}}} \in E_{2}$ is a mapping $μ_{\tilde{A_{\tilde{α}}}} : E_{2}^{\tilde{A_{\tilde{α}}}} \to [0, 1]$ ; where $E_{2}^{\tilde{A_{\tilde{a}}}}$ is the support set of $E_{2}^{\tilde{A_{\tilde{a}}}}$ . The 3D membership of $\tilde{A_{\tilde{α}}}$ is denoted by $μ_{\tilde{A_{\tilde{α}}}} (x, u), x \in E_{2}^{\tilde{A_{\tilde{α}}}}$ and u ∈ [0, 1] i.e. $\tilde{A_{\tilde{α}}} = {(x, u, μ_{\tilde{A_{\tilde{α}}}} (x, u)) : x \in E_{2}^{\tilde{A_{\tilde{α}}}}, u \in [0, 1]}$ , in which $0 \leq μ_{\tilde{A_{\tilde{α}}}} (x, u) \leq 1$ . $\tilde{A_{\tilde{α}}}$ can also be expressed as $\int_{x \in E_{2}} \tilde{A_{\tilde{α}}} \int_{u \in [0, 1]} \frac{μ_{\tilde{A_{\tilde{α}}}} (x, u)}{u (x)}$ where ∬denotes union over all admissible x and u.

The α-cut representation of a T2FS $\tilde{A_{\tilde{α}}}$ at point x ∈ X is defined as $[\tilde{A}]_{\tilde{α}} = {u \in [0, 1] : μ_{\tilde{A_{\tilde{α}}}} (x, u) \geq \tilde{α}}$ [30, 31] and represented as the pair $[\underline{A_{\tilde{α}}}, \bar{A_{\tilde{α}}}]$ , where $\underline{A_{\tilde{α}}}$ and ${\bar{A}}_{\tilde{α}}$ are two T1FSs called Lower Membership Function (LMF), and Upper Membership Function (UMF) respectively.

The derivative of the type-2 fuzzy number-valued functions, is defined on the basis of corresponding metric space which is defined by Mazandarani [17]. The distance measure for T2FSs and type 2 metric space is defined as follows.

Definition. Let $\tilde{A_{\tilde{α}}}$ and $\tilde{B_{\tilde{α}}}$ be two T2FSs whose $\tilde{α}$ cuts of the secondary membership functions at x ∈ [a, b] are $μ_{\tilde{A_{\tilde{α}}}} (x, \tilde{α})$ and $μ_{\tilde{B_{\tilde{α}}}} (x, \tilde{α})$ respectively. Then the distance between $\tilde{A_{\tilde{α}}}$ and $\tilde{B_{\tilde{α}}}$ is defined as follows. $d_{2} (\tilde{A_{\tilde{α}}}, \tilde{B_{\tilde{α}}}) = \int_{a}^{b} H_{f} (μ_{\tilde{A_{\tilde{α}}}} (x, \tilde{α}), μ_{\tilde{B_{\tilde{α}}}} (x, \tilde{α})) dx,$ where $\begin{matrix} H_{f} (μ_{\tilde{A_{\tilde{α}}}} (x, \tilde{α}), μ_{\tilde{B_{\tilde{α}}}} (x, \tilde{α})) \\ = \frac{\int_{0}^{1} \tilde{α} d_{H} (μ_{\tilde{A_{\tilde{α}}}} (x, \tilde{α}), μ_{\tilde{B_{\tilde{α}}}} (x, \tilde{α})) d \tilde{α}}{\int_{0}^{1} \tilde{α} d \tilde{α}} \\ = 2 \int_{0}^{1} \tilde{α} d_{H} (μ_{\tilde{A_{\tilde{α}}}} (x, \tilde{α}), μ_{\tilde{B_{\tilde{α}}}} (x, \tilde{α})) d \tilde{α} . \end{matrix}$

Definition. Let $\tilde{x_{\tilde{α}}}, \tilde{y_{\tilde{α}}} \in E_{2}$ , if there exists $\tilde{z_{\tilde{α}}} \in E_{2}$ such that $\tilde{x_{\tilde{α}}} = \tilde{y_{\tilde{α}}} + \tilde{z_{\tilde{α}}}$ , then $\tilde{z_{\tilde{α}}}$ is called type-2 Hukuhara difference [H₂-difference] of $\tilde{x_{\tilde{α}}}$ and $\tilde{y_{\tilde{α}}}$ ; denoted by $\tilde{x_{\tilde{α}}} -^{H_{2}} \tilde{y_{\tilde{α}}}$ .

Definition. (Type 2 fuzzy derivatives) Let ${\tilde{f}}_{\tilde{α}} : T \to E_{2}$ and t₀ ∈ [a, b] ⊆ R. Then ${\tilde{f}}_{\tilde{α}}$ is differentiable at t₀ if there exists an element ${\tilde{f}}_{\tilde{α}} (t_{0}) \in E_{2}$ , such that for all h > 0, sufficiently small,

There are ${\tilde{f}}_{\tilde{α}} (t_{0} + h) -^{H_{2}} {\tilde{f}}_{\tilde{α}} (t_{0})$ , ${\tilde{f}}_{\tilde{α}} (t_{0}) -^{H_{2}} {\tilde{f}}_{\tilde{α}} (t_{0} - h)$ , and the limits converges: $\begin{matrix} lim_{h \to 0} \frac{{\tilde{f}}_{\tilde{α}} (t_{0} + h) -^{H_{2}} {\tilde{f}}_{\tilde{α}} (t_{0})}{h} \\ = lim_{h \to 0} \frac{{\tilde{f}}_{\tilde{α}} (t_{0} + h) -^{H_{2}} {\tilde{f}}_{\tilde{α}} (t_{0})}{h} \\ = {\tilde{f}}_{\tilde{α}} (t_{0}) Or \end{matrix}$

There are ${\tilde{f}}_{\tilde{α}} (t_{0}) -^{H_{2}} {\tilde{f}}_{\tilde{α}} (t_{0} + h), {\tilde{f}}_{\tilde{α}} (t_{0} - h) -^{H_{2}} {\tilde{f}}_{\tilde{α}} (t_{0})$ , and the limits converges: $\begin{matrix} lim_{h \to 0} \frac{{\tilde{f}}_{\tilde{α}} (t_{0}) -^{H_{2}} {\tilde{f}}_{\tilde{α}} (t_{0} + h)}{- h} \\ = lim_{h \to 0} \frac{{\tilde{f}}_{\tilde{α}} (t_{0} - h) -^{H_{2}} {\tilde{f}}_{\tilde{α}} (t_{0})}{- h} \\ = {\tilde{f}}_{\tilde{α}} (t_{0}) Or \end{matrix}$

There are ${\tilde{f}}_{\tilde{α}} (t_{0} + h) -^{H_{2}} {\tilde{f}}_{\tilde{α}} (t_{0})$ , ${\tilde{f}}_{\tilde{α}} (t_{0} - h) -^{H_{2}} {\tilde{f}}_{\tilde{α}} (t_{0})$ , and the limits converges: $\begin{matrix} lim_{h \to 0} \frac{{\tilde{f}}_{\tilde{α}} (t_{0} + h) -^{H_{2}} {\tilde{f}}_{\tilde{α}} (t_{0})}{h} \\ = lim_{h \to 0} \frac{{\tilde{f}}_{\tilde{α}} (t_{0} - h) -^{H_{2}} {\tilde{f}}_{\tilde{α}} (t_{0})}{- h} \\ = {\tilde{f}}_{\tilde{α}} (t_{0}) Or \end{matrix}$

There are ${\tilde{f}}_{\tilde{α}} (t_{0}) -^{H_{2}} {\tilde{f}}_{\tilde{α}} (t_{0} + h)$ , ${\tilde{f}}_{\tilde{α}} (t_{0}) -^{H_{2}} {\tilde{f}}_{\tilde{α}} (t_{0} - h)$ , and the limits converges: $\begin{matrix} lim_{h \to 0} \frac{{\tilde{f}}_{\tilde{α}} (t_{0}) -^{H_{2}} {\tilde{f}}_{\tilde{α}} (t_{0} + h)}{- h} \\ = lim_{h \to 0} \frac{{\tilde{f}}_{\tilde{α}} (t_{0}) -^{H_{2}} {\tilde{f}}_{\tilde{α}} (t_{0} - h)}{h} \\ = {\tilde{f}}_{\tilde{α}} (t_{0}) . [15, 17] . \end{matrix}$

2.3 Stochastic differential equation

Let I = [0, T] and a : I × R → R and b : I × R → R are two scalar functions. The standard form of a SDE could be written as follows: ${dX}_{t} = a (t, X_{t}) dt + b (t, X_{t}) {dW}_{t}$ (2.3.1)

Where W_t is a Wiener process, which is characterized as a Gaussian process with W₀ = 0 and W_t ∼ N (0, t) for each t > 0. A Wiener process is continuous throughout but not differentiable anywhere, therefore dW_t isn’t defined. So Equation 2.3.1 is the symbolic representation of the following integral equation $X_{t} = X_{t_{0}} + \int_{t_{0}}^{t} a (s, X_{s}) d s + \int_{t_{0}}^{t} b (s, X_{s}) d W_{s}$ (2.3.2)

It has been shown that a Wiener process satisfies the following properties: E (W_t) = 0, E ((W_t) ²) = t², and E ((W_{t
₄} - W_{t
₃}) (W_{t
₂} - W_{t
₁})) = 0 for all 0 ≤ t₁ ≤ t₂ ≤ t₃ ≤ t₄.

The second integral in Equation (2.3.2) containing the Wiener process) is not measurable in Riemann-Stieltjes sense, is called stochastic integral. For a class of suitable random functions h : [0, T] → R and partitions 0 = t₀ < t₁ < t₂ …… … < t_n = T with maximum spacing Δ; the Ito integral is defined as $\int_{0}^{T} h (t) d W_{t} = \lim_{Δ \to 0} \sum_{i = 1}^{n - 1} h (t) (W_{t_{i + 1}} - W_{t_{i}})$

Let Y_t = U (t, X_t) where U : [t₀, T] → R × R has continuous second order partial derivatives and X_t satisfies Ito stochastic Equation (2.3.1), then Y_t is a solution of ${dY}_{t} = L^{0} U (t, X_{t}) dt + L^{1} U (t, X_{t}) {dW}_{t}$ (2.3.3)

Where $L^{0} U : = \frac{\partial U}{\partial t} + a \frac{\partial U}{\partial x} + \frac{1}{2} b^{2} \frac{\partial^{2} U}{\partial x^{2}}$ and $L^{1} U : = b \frac{\partial U}{\partial x}$ . The Ito formula (2.3.3) can be interpreted as the following stochastic integral $Y_{t} = Y_{t_{0}} + \int_{t_{0}}^{t} L^{0} U (s, X_{s}) ds + \int_{t_{0}}^{t} L^{1} U (s, X_{s}) {dW}_{s}$ (2.3.4) for all t ∈ [t₀, T].

An alternative definition of stochastic integrals is defined by Stratonovich where the evolution of integration is carried out at the midpoint of each partitioned subinterval. The Stratonovich stochastic integral is defined as $\int_{0}^{T} h (t)^{\circ} {dW}_{t} = lim_{Δ \to 0} \sum_{i = 1}^{n - 1} (\frac{t_{i} + t_{i + 1}}{2}) (W_{t_{i + 1}} - W_{t_{i}})$

A Stratonovich SDE with scalar coefficients a and b is known as Stratonovich stochastic integral equation and symbolically written as ${dX}_{t} = a (t, X_{t}) dt + b (t, X_{t})^{\circ} {dW}_{t}$ (2.3.5) $X_{t} = X_{t_{0}} + \int_{t_{0}}^{t} a (s, X_{s}) ds + \int_{t_{0}}^{t} b (s, X_{s})^{\circ} {dW}_{s}$

Let U : [t₀, T] → R × R has continuous second order partial derivatives, then ${dY}_{t} = L^{0} U (t, X_{t}) dt + L^{1} U (t, X_{t})^{\circ} {dW}_{t}$ (2.3.6) where ${\underline{L}}^{0} U : \frac{\partial U}{\partial t} + a \frac{\partial U}{\partial x}$ and ${\underline{L}}^{1} U : = b \frac{\partial U}{\partial x}$ .

It could be shown that the Ito SDE

dX_t = a (t, X_t) dt + b (t, X_t) dW_t has the same solution with Stratonovich SDE

${dX}_{t} = \underline{a} (t, X_{t}) dt + b (t, X_{t})^{\circ} {dW}_{t}$ with modified drift coefficients $\underline{a} (t, X) = a (t, X) - \frac{1}{2} b (t, X) \frac{\partial b}{\partial x} (t, X) .$

2.4 Fractional derivatives

In order to define arbitrary order derivative the integral operator J which is defined as the fractional integral for a function f (x) is defined by Riemann and Liouville as follows.

$J_{a}^{α} f (x) = \frac{1}{Γ (α)} \int_{a}^{x} {(x - t)}^{α - 1} f (t) dt$ . Trivially

$J_{a} f (x) = \frac{1}{Γ (1)} \int_{a}^{x} f (t) dt$ or $\frac{d}{dx} J_{a} f (x) = f (x)$ Also if α is an integer and denoted by n then it could be written as $\begin{matrix} \frac{d^{n}}{d^{n} x} J^{n} f (x) \\ = \frac{d^{n}}{d^{n} x} \frac{1}{(n - 1)!} \int_{0}^{x} {(x - t)}^{n - 1} f (t) dt = f (x) \end{matrix}$

Let us assume that f is integrable and ∫f (t) dt = F (t) then it could be shown that $J_{a}^{n}$ is the inverse operator of $D^{n} f (x) = \frac{d^{n}}{d^{n} x} f (x) \forall n \in ℕ$ that is $J_{a}^{n} f (x) = D^{- n} f (x)$ or $D^{n} f (x) = J_{a}^{- n} J \frac{d}{dx} f (x)$ .

So $D^{n} f (x) = J_{a}^{- n} J_{a} \frac{d}{dx} f (x) = J^{1 - n} \frac{d}{dx} f (x)$ .

Therefore the n-th order derivative of a function could be represented as an integro-differential operator $J^{1 - n} \frac{d}{dx} f (x)$ . Hence the arbitrary order derivative of f (x) in Riemann and Liouville sense could be generalized as the an integro-differential operator and arbitrary order derivative of order α for f (x) is defined as $D_{a}^{α} f (x) = \frac{1}{Γ (1 - α)} \frac{d}{dx} \int_{a}^{x} \frac{f (t)}{{(x - t)}^{α}} dt or$ otherwise where n = [α] + 1 is the smallest integer greater than α it could be written as $^{RL} D_{a}^{α} f (x) = \frac{1}{Γ (n - α)} \frac{d^{n}}{{dx}^{n}} \int_{a}^{x} \frac{f (t)}{{(x - t)}^{α + 1 - n}} dt .$

Another version of definition of arbitrary order derivative was introduced by M. Caputo in his paper [32] as follows. $^{C} D_{a}^{α} f (x) = \frac{1}{Γ (n - α)} \int_{a}^{x} \frac{f^{n} (t)}{{(x - t)}^{α + 1 - n}} dt$ where n = [α] + 1 and fⁿ (t) is the nth integer order derivative of f (t).

2.5 Fractional partial derivatives

For more general case where f is a function of several variables the arbitrary order partial derivatives are similarly defined as the integrals over any particular variable. Let f : Rⁿ → Rⁿ then the partial derivative of f with respect to x_i in Riemann Liouville sense is defined as $\begin{matrix} ^{RL} D_{a}^{α} f (x_{1}, x_{2}, \dots \dots \dots, x_{i}, \dots \dots, x_{n}) \\ = \frac{1}{Γ (n - α)} \frac{\partial^{n}}{\partial x_{i}^{n}} \int_{a}^{x} \frac{f (x_{1}, x_{2}, \dots \dots, x_{i}, \dots \dots, x_{n})}{{(x_{i} - t)}^{α + 1 - n}} {dx}_{i} \end{matrix}$

Here the integral is taken over x_i for each fixed values of other x_rs, where r = 1, 2, …, n, i ≠ r.

Similarly the partial derivative of f with respect to x_i in Caputo sense is defined as $\begin{matrix} ^{C} {(D_{x_{i}})}_{a}^{α} f (x_{1}, x_{2}, \dots \dots \dots, x_{i}, \dots \dots, x_{n}) \\ = \frac{1}{Γ (n - α)} \int_{a}^{x} \frac{\frac{\partial^{n}}{\partial x_{i}^{n}} f (x_{1}, x_{2}, \dots \dots \dots, x_{i}, \dots \dots, x_{n})}{{(x_{i} - t)}^{α + 1 - n}} {dx}_{i} . \end{matrix}$

3 Fuzzy fractional derivatives and fuzzy fractional differential and integral equation

3.1 Fuzzy fractional derivatives

In order to define type-1 and type-2 fuzzy arbitrary order calculus and type-1 and type-2 fuzzy arbitrary order integro-differential operator first the type-1 and type-2 fuzzy arbitrary order derivatives should be defined.

Definition. Let $\tilde{f} : (a, b) \to E_{1}$ be a type-1 fuzzy valued function and $[\underline{f_{α}} (t), \bar{f_{α}} (t)], 0 \leq α \leq 1$ be the α-cut of $\tilde{f}$ ; then any arbitrary β^th order Riemann Liouville derivative of $\tilde{f}$ with respect to x is defined as: $^{RL} D_{a}^{β} \tilde{f} (x) = \frac{1}{Γ (n - β)} \frac{d^{n}}{{dx}^{n}} \int_{a}^{x} \frac{\tilde{f} (t)}{{(x - t)}^{β + 1 - n}} d_{1} t$ where n = [β] + 1. In general we denote arbitrary order β^th order Riemann Liouville derivative of type-1 fuzzy function f as $^{RL} D_{a}^{β} f (x) = \frac{1}{Γ (n - β)} \frac{d^{n}}{{dx}^{n}} \int_{a}^{x} \frac{f (t)}{{(x - t)}^{β + 1 - n}} dt$

Definition. Let $\tilde{f} : (a, b) \to E_{1}$ be a type-1 fuzzy valued function and $[\underline{f_{α}} (t), \bar{f_{α}} (t)], 0 \leq α \leq 1$ be the α-cut of $\tilde{f}$ ; then any arbitrary β^th order Caputo derivative of $\tilde{f}$ with respect to x is defined as: $^{C} D_{a}^{β} \tilde{f} (x) = \frac{1}{Γ (n - β)} \int_{a}^{x} \frac{{\tilde{f}}^{n} (t)}{{(x - t)}^{β + 1 - n}} d_{1} t$ where n = [β] + 1 and ${\tilde{f}}^{n} (t)$ is the nth integer order type-2 fuzzy derivative of $\tilde{f} (t)$ . In general we denote arbitrary order β^th order Caputo derivative of type-1 fuzzy function f as $^{C} D_{a}^{β} f (x) = \frac{1}{Γ (n - β)} \int_{a}^{x} \frac{f^{n} (t)}{{(x - t)}^{β + 1 - n}} dt .$

Definition. Let $\tilde{f_{\tilde{α}}} : (a, b) \to E_{2}$ be a type-2 fuzzy valued function and $[\underline{f_{\tilde{α}}} (t), \bar{f_{\tilde{α}}} (t)], 0 \leq α \leq 1$ be the α-cut of $\tilde{f_{\tilde{α}}}$ ; then any arbitrary β^th order Riemann Liouville derivative of $\tilde{f_{\tilde{α}}}$ is defined as follows. $^{RL} D_{a}^{β} \tilde{f_{\tilde{α}}} (x) = \frac{1}{Γ (n - β)} \frac{d^{n}}{{dx}^{n}} \int_{a}^{x} \frac{\tilde{f_{\tilde{α}}} (t)}{{(x - t)}^{β + 1 - n}} d_{2} t$ where n = [β] + 1. In general we denote arbitrary order β^th order Riemann Liouville derivative of type-2 fuzzy function f as $^{RL} D_{a}^{β} f (x) = \frac{1}{Γ (n - β)} \frac{d^{n}}{{dx}^{n}} \int_{a}^{x} \frac{\tilde{f_{\tilde{α}}} (t)}{{(x - t)}^{β + 1 - n}} dt .$

Definition. Let $\tilde{f_{\tilde{α}}} : (a, b) \to E_{2}$ be a type-2 fuzzy valued function and $[\underline{f_{\tilde{α}}} (t), \bar{f_{\tilde{α}}} (t)], 0 \leq α \leq 1$ be the α-cut of $\tilde{f_{\tilde{α}}}$ ; then any arbitrary β^th order Caputo derivative of $\tilde{f_{\tilde{α}}}$ is defined as follows. $^{C} D_{a}^{β} \tilde{f_{\tilde{α}}} (x) = \frac{1}{Γ (n - β)} \int_{a}^{x} \frac{{\tilde{f_{\tilde{α}}}}^{n} (t)}{{(x - t)}^{β + 1 - n}} d_{2} t$ where n = [β] + 1 and ${\tilde{f_{\tilde{α}}}}^{n} (t)$ is the nth integer order type-2 fuzzy derivative of $\tilde{f_{\tilde{α}}} (t)$ . In general we denote arbitrary order β^th order Caputo derivative of type-2 fuzzy function f as $^{C} D_{a}^{β} f (x) = \frac{1}{Γ (n - β)} \int_{a}^{x} \frac{{\tilde{f_{\tilde{α}}}}^{n} (t)}{{(x - t)}^{β + 1 - n}} d_{2} t .$

Similarly fuzzy fractional order partial derivatives could be defined as follows.

Definition. Let $\tilde{f} : R^{n} \to E_{1}^{n}$ be a type-1 fuzzy valued function and $[\underline{f_{\tilde{α}}} (t), \bar{f_{\tilde{α}}} (t)], 0 \leq α \leq 1$ be the α-cut of $\tilde{f}$ ; then any arbitrary β^th order Riemann Liouville partial derivative with respect to independent variable x_i of $\tilde{f}$ is defined as $\begin{matrix} ^{RL} {(D_{x_{i}})}_{a}^{β} \tilde{f} (x_{1}, x_{2}, \dots \dots \dots, x_{i}, \dots \dots, x_{n}) \\ = \frac{1}{Γ (n - β)} \frac{\partial^{n}}{\partial x_{i}^{n}} \int_{a}^{x} \frac{\tilde{f} (x_{1}, x_{2}, \dots \dots \dots, x_{i}, \dots \dots, x_{n})}{{(x - t)}^{β + 1 - n}} {dx}_{i} \end{matrix}$

where n = [β] + 1.

Definition. Let $\tilde{f} : (a, b) \to E_{1}^{n}$ be a type-1 fuzzy valued function and $[\underline{f_{\tilde{α}}} (t), \bar{f_{\tilde{α}}} (t)], 0 \leq α \leq 1$ be the α-cut of $\tilde{f}$ ; then any arbitrary β^th order Caputo partial derivative with respect to independent variable x_i of $\tilde{f}$ is defined as $\begin{matrix} ^{C} {(D_{x_{i}})}_{a}^{β} \tilde{f} (x_{1}, x_{2}, \dots \dots \dots, x_{i}, \dots \dots, x_{n}) \\ = \frac{1}{Γ (n - β)} \int_{a}^{x} \frac{\frac{\partial^{n}}{\partial x_{i}^{n}} \tilde{f} (x_{1}, x_{2}, \dots \dots, x_{i}, \dots \dots, x_{n})}{{(x - t)}^{β + 1 - n}} {dx}_{i} \end{matrix}$

where n = [β] + 1.

Definition. Let $\tilde{f_{\tilde{α}}} : R^{n} \to E_{2}^{n}$ be a type-2 fuzzy valued function and $[\underline{f_{\tilde{α}}} (t), \bar{f_{\tilde{α}}} (t)], 0 \leq α \leq 1$ be the α-cut of $\tilde{f_{\tilde{α}}}$ ; then any arbitrary β^th order Riemann Liouville partial derivative with respect to independent variable x_i of $\tilde{f_{\tilde{α}}}$ is defined as $^{RL} (D_{x_{i}})_{a}^{β} \tilde{f} (x_{1}, x_{2}, \dots \dots, x_{i}, \dots \dots, x_{n}) = \frac{1}{Γ (n - β)} \frac{\partial^{n}}{\partial x_{i}^{n}} \int_{a}^{x} \frac{\tilde{f} (x_{1}, x_{2}, \dots \dots, x_{i}, \dots \dots, x_{n})}{{(x - t)}^{β + 1 - n}} d x_{i} .$

Definition. Let $\tilde{f_{\tilde{α}}} : R^{n} \to E_{2}^{n}$ be a type-2 fuzzy valued function and $[\underline{f_{\tilde{α}}} (t), \bar{f_{\tilde{α}}} (t)], 0 \leq α \leq 1$ be the α-cut of $\tilde{f_{\tilde{α}}}$ ; then any arbitrary β^th order Caputo partial derivative with respect to independent variable x_i of $\tilde{f_{\tilde{α}}}$ is defined as $\begin{matrix} ^{C} D_{a}^{β} \tilde{f_{\tilde{α}}} (x_{1}, x_{2}, \dots \dots, x_{i}, \dots \dots, x_{n}) \\ = \frac{1}{Γ (n - β)} \int_{a}^{x} \frac{\frac{\partial^{n}}{\partial x_{i}^{n}} \tilde{f_{\tilde{α}}} (x_{1}, x_{2}, \dots \dots, x_{i}, \dots \dots, x_{n})}{{(x - t)}^{β + 1 - n}} {dx}_{i} . \end{matrix}$

In this paper, Caputo derivatives are used to express different arbitrary order dynamical systems as only integer order initial condition fⁿ (t₀) is necessary to solve these type of equation.

3.2 Type-1 fuzzy stochastic process and type-1 fuzzy stochastic integral

Definition. Let (Ω, A, P) denotes a complete probability space and M (Ω, A, Kⁿ) be the family of A - measurable set-valued (fuzzy) random variable with values in Kⁿ, i. e. the fuzzy mappings f : Ω → Kⁿ is such that {ω ∈ Ω : f (ω ∩ O ≠ φ) } ∈ A for every open set O ∈ Rⁿ.

It has been shown that f : Ω → Kⁿ is a set-valued random variable if, and only if, f is a A|B_{d
_H} measurable mapping, where B_{d
_H} denotes the Borel σ-algebra generated by the topology induced by the metric d_H in Kⁿ.

Definition. A fuzzy mapping $\tilde{x} : Ω \to E_{1}$ is called to be a fuzzy random variable, if every α- cut of $\tilde{x}$ is an A-measurable set-valued random variable.

Let us consider the following metric $d_{1} (u, v) = Infmax {\underset{t \in [0, 1]}{Sup} | ρ (t) - t |, \underset{t \in [0, 1]}{Sup} d_{H} (u^{1 - t}, v^{1 - ρ (t)})}$ u, v ∈ Kⁿ. Where ρ denotes the set of strictly increasing continuous functions ρ : [0, 1] → [0, 1] such that ρ (0) = 0, ρ (1) = 1, and B_{d
₁} is the sigma-algebra generated by the topology induced by the metric. It could be proved that the fuzzy mapping: $\tilde{x} : (Ω, A) \to (E_{1}, B_{d_{1}})$ is a fuzzy random variable if, and only if, $\tilde{x}$ is A|B_{d
₁} -measurable.

A fuzzy random variable $\tilde{f} \in M (Ω, A, K^{n})$ is called to be L^p integrably bounded (p ≥ 1), if there exists h ∈ (Ω, A, P, R⁺) such that $∥ \tilde{f} ∥ < h$ in P- almost everywhere i.e. P (d_H (X (ω) , { 0 }) ≤ h (ω)) = 1, where R⁺ =[0, ∞) , $∥ | A | ∥ = d_{H} (A, {0}) = \underset{a \in A}{Inf} ∥ a ∥$ for A ∈ Kⁿ and L^p (Ω, A, P, R⁺) is a space of equivalence classes with respect to equality in P almost everywhere of A-measurable random variables h : Ω → R⁺ such that $E (| h^{p} |) = \int_{Ω} | h^{p} | dP$ , in other words ${[\tilde{x}]}_{α} \in L^{p} (Ω, A, P, K^{n})$ for every α ∈ [0, 1].

Definition. Let I = [0, T] be an interval, and t ∈ [0, T]. Let {A_t } _t∈I be a filtration on the probability space (Ω, A, P); a fuzzy mapping $\tilde{x} : I \times Ω \to E_{1}$ is said to be a fuzzy stochastic process if for every t ∈ I the mapping $\tilde{x} (t, ω) : Ω \to E_{1}$ is a fuzzy random variable. A fuzzy stochastic process $\tilde{x}$ is d₁-continuous, if the mappings $\tilde{x} (t, ω) : I \times Ω \to E_{1}$ are d₁-continuous with respect to the probability measure P.

A fuzzy stochastic process $\tilde{x}$ is said to be measurable, if ${[\tilde{x} (t)]}_{α} : I \times Ω \to K^{n}$ is B (I) × A_t is measurable set valued random variable for all α ∈ [0, 1], where B (I) denotes the Borel σ-algebra of subsets of I. A fuzzy stochastic process x is called to be {A_t } _t∈I-adapted, if for every α ∈ [0, 1] the set valued random variable ${[\tilde{x} (t)]}_{α} : Ω \to K^{n}$ is {A_t } _t∈I-measurable for all t ∈ I. The fuzzy stochastic process $\tilde{x} : I \times Ω \to E_{1}$ which is {A_t } _t∈I-adapted and measurable is said to be nonanticipating [19, 20].

Let p ≥ 1, and N ={ A ∈ B [0, 1] × A^t : A^t ∈ A_t for every t ∈ [0, 1] } , A^t = { ω : (t, ω) ∈ A } for t ∈ [0, T] and h ∈ L^p (I × Ω, N, Rⁿ) : I × Ω → Rⁿ represents a nonanticipating stochastic processes such that $E (\int_{I} {∥ h (s) ∥}^{p} d_{1} s) < \infty$ . A fuzzy stochastic process $\tilde{x} : I \times Ω \to F (E_{1}^{n})$ , where is defined to be L^p-integrally bounded, if there exists a real-valued stochastic process h ∈ L^p (I × Ω, N, Rⁿ) such that $d_{1} (\tilde{x} (t, ω), 〈 0 〉) \leq h (t, ω)$ for almost all (t, ω) ∈ I × Ω. Then the fuzzy stochastic Lebesgue integral could be defined as $Ω ∋ ω = \int_{t_{1}}^{t_{2}} \tilde{x} (s, ω) d_{1} s \in E_{1}$ , which is again a fuzzy random variable. The properties of fuzzy stochastic Lebesgue integral are studied in papers [20 –22].

3.3 Type-2 fuzzy stochastic process and type-2 fuzzy stochastic integral

Definition. A type-2 fuzzy mapping $\tilde{x_{\tilde{α}}} : Ω \to E_{2}$ is a type-2 fuzzy random variable, if ${[\tilde{x}]}_{\tilde{α}}^{α} : Ω \to E_{2}^{n}$ is an $\tilde{A}$ -measurable fuzzy random variable for every α ∈ [0, 1]. If we consider the following metric: $\begin{matrix} d_{2} (\tilde{u}, \tilde{v}) \\ = Inf \max {\underset{t \in [0, 1]}{Sup} | ρ (t) - t |, \underset{t \in [0, 1]}{Sup} d_{1} ({\tilde{u}}^{1 - t}, {\tilde{v}}^{1 - ρ (t)})}, \\ \tilde{u}, \tilde{v} \in E_{1}^{n} . \end{matrix}$

Where ρ denotes the set of strictly increasing continuous functions ρ : [0, 1] → [0, 1] such that ρ (0) = 0, ρ (1) = 1, and B_{d
₂} is the sigma-algebra generated by the topology induced by the metric. It could be proved that the type-2 fuzzy mapping: $\tilde{x_{\tilde{α}}} : \tilde{A} \to (E_{2}, B_{d_{2}})$ is a fuzzy random variable if, and only if, $\tilde{x_{\tilde{α}}}$ is $\tilde{A} | B_{d_{2}}$ -measurable.

A fuzzy random variable $\tilde{f_{\tilde{α}}} \in M (Ω, \tilde{A}, E_{2}^{n})$ is called to be L^p integrably bounded (p ≥ 1), if there exists $\tilde{h} \in (Ω, \tilde{A}, P, E_{2}^{+})$ such that $∥ \tilde{f_{\tilde{α}}} ∥$ $< \tilde{h}$ in P- almost everywhere i.e. $P (d_{2} (\tilde{X} (ω), {0}) \leq \tilde{h} (ω)) = 1$ , where $E_{2}^{+}$ is the type-2 fuzzy set over the support [0, ∞) , $∥ | \tilde{A} | ∥ = d_{2} (\tilde{A}, {0})$ for $\tilde{A} \in E_{2}^{n}$ and $L^{p} (Ω, \tilde{A}, P, E_{2}^{+})$ is a space of equivalence classes with respect to equality in P almost everywhere of $\tilde{A}$ -measurable random variables $\tilde{h} : Ω \to E_{2}^{+}$ such that $E (| {\tilde{h}}^{p} |) = \int_{Ω} | {\tilde{h}}^{p} | dP$ , in other words ${[\tilde{x}]}_{\tilde{α}}^{α} \in L^{p} (Ω, \tilde{A}, P, E_{2}^{n})$ for every α ∈ [0, 1].

Definition. Let I = [0, T] be an interval, and t ∈ [0, T]. Let ${\tilde{A_{t}}}_{t \in I}$ be a fuzzy filtration on the probability space $(Ω, \tilde{A}, P)$ ; a type-2 fuzzy mapping $\tilde{x_{\tilde{α}}} : I \times Ω \to E_{2}$ is said to be a fuzzy stochastic process if for every t ∈ I the mapping $\tilde{x_{\tilde{α}}} (t, ω) : Ω \to E_{2}$ is a fuzzy random variable. A fuzzy stochastic process $\tilde{x_{\tilde{α}}}$ is d₂-continuous, if the mappings $\tilde{x_{\tilde{α}}} (t, ω) : I \times Ω \to E_{2}$ are d₂-continuous with respect to the probability measure.

Definition. A type-2 fuzzy stochastic process $\tilde{x_{\tilde{α}}}$ is said to be measurable, if ${[\tilde{x} (t)]}_{\tilde{α}} : I \times Ω \to E_{2}^{n}$ is $B (I) \times \tilde{A_{t}}$ measurable set valued random variable for all $\tilde{α} \in [0, 1]$ , where B (I) denotes the Borel σ-algebra of subsets of I. A type-2 fuzzy stochastic process $\tilde{x_{\tilde{α}}}$ is called to be ${\tilde{A_{t}}}_{t \in I}$ -adapted, if for every $\tilde{α} \in [0, 1]$ the fuzzy random variable ${[\tilde{x} (t)]}_{\tilde{α}}^{α} : Ω \to E_{2}^{n}$ is ${\tilde{A_{t}}}_{t \in I}$ -measurable for all t ∈ I. The fuzzy stochastic process $\tilde{x_{\tilde{α}}} : I \times Ω \to E_{2}$ which is ${\tilde{A_{t}}}_{t \in I}$ -adapted and measurable is said to be nonanticipating.

Definition. Let p ≥ 1, and $N = {\tilde{A} \in B [0, 1] \times \tilde{A^{t}} : \tilde{A^{t}} \in \tilde{A_{t}} for every t \in [0, 1]}, \tilde{A^{t}} = {\tilde{ω} : (t, \tilde{ω}) \in \tilde{A}}$ for t ∈ [0, T] and $\tilde{h} \in L^{p} (I \times Ω, N, E_{2}^{n}) : I \times Ω \to E_{2}^{n}$ represents a nonanticipating fuzzy stochastic processes such that $E (\int_{I} {∥ \tilde{h} (s) ∥}^{p} d_{2} s) < \infty$ . A fuzzy stochastic process $\tilde{x_{\tilde{α}}} : I \times Ω \to E_{2}^{n}$ , is defined to be L^p-integrally bounded, if there exists a real-valued stochastic process $\tilde{h} \in L^{p} (I \times Ω, N, E_{2}^{n})$ such that $d_{2} (\tilde{x_{\tilde{α}}} (t, \tilde{ω}), 〈 0 〉) \leq \tilde{h} (t, \tilde{ω})$ for almost all $(t, \tilde{ω}) \in I \times Ω$ . Then the type-2 fuzzy stochastic Lebesgue integral could be defined as $Ω ∋ \tilde{ω} = \int_{t_{1}}^{t_{2}} \tilde{x_{\tilde{α}}} (s, \tilde{ω}) d_{2} s \in E_{2}$ .

3.4 Fuzzy stochastic differential equation

Thus the symbolic representation of fuzzy stochastic differential equation is as follows: $d \tilde{x} (t) = \tilde{f} (t, \tilde{x}) dt + \tilde{g} (t, \tilde{x}) dB (t)$ (3.4.1)

Where t ∈ I = [0, T] and $\tilde{f} : I \times Ω \times E_{1}^{n} \to E_{1}^{n}, \tilde{g} : I \times Ω \times E_{1}^{n} \to R^{n} \times R^{n}$ and ${\tilde{x}}_{0} : Ω \to E_{1}^{n}$ are fuzzy random functions and {B_t } _t∈I denotes a m-dimensional {A_t}-Brownian motion defined on (Ω, A, { A_t } _t∈I, P). The initial condition could be written as $P (\tilde{x} (0) = {\tilde{x}}_{0}) = 1$ .

B = (B¹, B², …… … , B^m) is defined as the m-dimensional Brownian motion where ${B_{t}^{1}}_{t \in I}, {B_{t}^{2}}_{t \in I}, \dots \dots \dots, {B_{t}^{m}}_{t \in I}$ are independent one dimensional {A_t } _t∈I Brownian motion. A fuzzy Brownian motion was defined by Li and Guan [33], however from their definition it has been shown that a fuzzy set–valued Brownian motion B_t, can be handle by an R^d – valued Wiener process b_t, in the sense that B_t = I_{b
_t}; i.e. it is actually the indicator function of a Wiener process [33]. In this context B_t is a fuzzy Brownian motion.

Also $\tilde{g} = ({\tilde{g}}^{1}, {\tilde{g}}^{2}, \dots, {\tilde{g}}^{m})$ where ${\tilde{g}}^{k} : I \times Ω \times E_{1}^{n} \to R^{n} \times R^{n}$ ; hence Equation (3.4.1) could be written as $d \tilde{x} (t) = \tilde{f} (t, \tilde{x}) dt + 〈 \sum_{k = 0}^{m} {\tilde{g}}^{k} (t, \tilde{x}) {dB}^{k} (t) 〉$ for t ∈ I and in P along with the initial condition $P (\tilde{x} (0) = {\tilde{x}}_{0}) = 1$ for t ∈ I.

Definition. A fuzzy stochastic process $\tilde{X} : I \times Ω \to E_{1}^{n}$ is said to be strong solution to (3.4.1) if $\tilde{X}$ belongs to $L^{2} (I \times Ω, N, E_{1}^{n})$ , is d₁-continuous in P almost everywhere satisfies

$\begin{matrix} \tilde{X} (t) & = & {\tilde{X}}_{0} + \int_{0}^{t} \tilde{f} (s, \tilde{X}) ds \\ + 〈 \sum_{k = 0}^{m} \int_{0}^{t} {\tilde{g}}^{k} (s, \tilde{X}) {dB}^{k} (s) 〉 \end{matrix}$ (3.4.2)

The integral involves fuzzy function is thus a fuzzy stochastic Lebesgue integral, while the integral involves fuzzy Brownian motion is the ensemble of the sum of Rⁿ-valued stochastic Ito integrals where the embedding $〈〉 : R^{n} \to E_{1}^{n}$ defined as the association with each a ∈ Rⁿ to its characteristic function [34].

Definition. If for any two strong solution $\tilde{x}, \tilde{y} : I \times Ω \to E_{1}^{n}$ of (3.4.1) and for every $t \in I, P (d_{1} (\tilde{x} (t), \tilde{y} (t)) = 0) = 1$ then the strong solution is said to be unique in P almost everywhere.

In the paper [22] the sufficient conditions for the existence and uniqueness of the solution to (3.4.1) are derived.

The sufficient condition for having strong solution to (3.4.1) has been studied in [22, 24].

3.5 Type-2 fuzzy stochastic differential equation

Thus the symbolic representation of type-2 fuzzy stochastic differential equation is as follows: $d \tilde{x_{\tilde{α}}} (t) = \tilde{f_{\tilde{α}}} (t, \tilde{x_{\tilde{α}}}) dt + 〈 \tilde{g_{\tilde{α}}} (t, \tilde{x_{\tilde{α}}}) dB (t) 〉$ (3.5.1)

Where t ∈ I = [0, T] and $\tilde{f_{\tilde{α}}} : I \times Ω \times E_{2}^{n} \to E_{2}^{n}, \tilde{g_{\tilde{α}}} : I \times Ω \times E_{2}^{n} \to R^{n} \times R^{n}$ and $\tilde{x_{0 \tilde{α}}} : Ω \to E_{1}^{n}$ are fuzzy random functions and {B_t } _t∈I denotes a m-dimensional {A_t}-Brownian motion defined on (Ω, A, { A_t } _t∈I, P). The initial condition could be written as $P (\tilde{x_{\tilde{α}}} (0) = \tilde{x_{0 \tilde{α}}}) = 1$ or in other word $\tilde{x_{\tilde{α}}} (0) = \tilde{x_{0 \tilde{α}}}$ in probability almost everywhere. B = (B¹, B², …… … , B^m) is defined as the m-dimensional Brownian motion where ${B_{t}^{1}}_{t \in I}, {B_{t}^{2}}_{t \in I}, \dots \dots \dots \dots, {B_{t}^{m}}_{t \in I}$ are independent one dimensional {A_t } _t∈I Brownian motion i.e. B_t is a fuzzy Brownian motion.

Also $\tilde{g_{\tilde{α}}} = ({\tilde{g_{\tilde{α}}}}^{1}, {\tilde{g_{\tilde{α}}}}^{2}, \dots \dots, {\tilde{g_{\tilde{α}}}}^{m})$ where ${\tilde{g_{\tilde{α}}}}^{k} : I \times Ω \times E_{2}^{n} \to R^{n} \times R^{n}$ ; hence Equation (3.5.1) could be written as $d \tilde{x_{\tilde{α}}} (t) = \tilde{f_{\tilde{α}}} (t, \tilde{x_{\tilde{α}}}) dt + 〈 \sum_{k = 0}^{m} {\tilde{g_{\tilde{α}}}}^{k} (t, \tilde{x_{\tilde{α}}}) {dB}^{k} (t) 〉$ for t ∈ I and in P along with the initial condition $P (\tilde{x_{\tilde{α}}} (0) = \tilde{x_{0 \tilde{α}}}) = 1$ for t ∈ I and

Definition. A type-2 fuzzy stochastic process $\tilde{X_{\tilde{α}}} : I \times Ω \to E_{2}^{n}$ is said to be strong solution to (3.5.1) if $\tilde{X}$ belongs to $L^{2} (I \times Ω, N, E_{2}^{n})$ , is d₂-continuous in P almost everywhere satisfies

$\begin{matrix} \tilde{X_{\tilde{α}}} (t) & = & \tilde{X_{0 \tilde{α}}} + \int_{0}^{t} \tilde{f_{\tilde{α}}} (s, \tilde{X_{\tilde{α}}}) ds \\ + 〈 \sum_{k = 0}^{m} \int_{0}^{t} {\tilde{g_{\tilde{α}}}}^{k} (s, \tilde{X_{\tilde{α}}}) {dB}^{k} (s) 〉 \end{matrix}$ (3.5.2)

This definition of type-2 fuzzy stochastic differential equations generalize the fuzzy stochastic differential equations and represented as type-2 fuzzy stochastic integral equations.

Definition. If for any two strong solution $\tilde{x_{\tilde{α}}}, \tilde{y_{\tilde{α}}} : I \times Ω \to E_{2}^{n}$ of (3.5.1) and for every $t \in I, P (d_{2} (\tilde{x_{\tilde{α}}} (t), \tilde{y_{\tilde{α}}} (t)) = 0) = 1$ then the strong solution is said to be unique in P almost everywhere.

4 Fuzzy arbitrary order dynamical systems

4.1 Type-1 fuzzy arbitrary order initial value problem

The fuzzy initial value problem with arbitrary order β could be written as $^{C} D_{a}^{β} \tilde{x} (t) = \tilde{F} (t, x (t)), \tilde{x} (0) = {\tilde{x}}_{0}, {\tilde{x}}^{1} (0) = {\tilde{x}}_{0}^{1}, \dots \dots \dots, {\tilde{x}}^{n} (0) = {\tilde{x}}_{0}^{n}$ where ${\tilde{x}}^{r} (0) = \frac{d^{r}}{{dt}^{r}} \tilde{x} (t)$ where $\tilde{F} : [0, T] \times E_{1} \to E_{1}$ is a continuous fuzzy mapping, $\tilde{x}$ is a fuzzy vector and ${\tilde{x}}_{0}, {\tilde{x}}_{0}^{1}, \dots, {\tilde{x}}_{0}^{n}$ are fuzzy number valued vector initial condition. In terms of α-cut this could be written as ${[x (t)]}_{α} = [\underline{x_{α}} (t), \bar{x_{α}} (t)] [x_{0}]_{α} = [\underline{x_{0 α}}, \bar{x_{0 α}}]$ and ${[F (t, x (t))]}_{α} = [\underline{F_{α}} (t, \underline{x_{α}} (t), \bar{x_{α}} (t)), \bar{F_{α}} (t, \underline{x_{α}} (t), \bar{x_{α}} (t))]$ .

Case I. If we consider $^{C} D_{a}^{β} x (t)$ by using the caputo derivative in the first form, then the arbitrary order initial value problem is written as follows. $\begin{matrix} (i) & ^{C} D_{a}^{β} \underline{x_{α}} (t) = \underline{F_{α}} (t, \underline{x_{α}} (t), \bar{x_{α}} (t)), \underline{x_{α}} (0) \\ = \underline{x_{0 α}}, \underline{x_{α}} (0) = \underline{x_{0 α}}, \underline{x_{α}^{1}} (0) \\ = \underline{x_{0 α}^{1}}, \dots \dots, \underline{x_{α}^{n}} (0) = \underline{x_{0 α}^{n}} \\ ^{C} D_{a}^{β} \bar{x_{α}} (t) = \bar{F_{α}} (t, \underline{x_{α}} (t), \bar{x_{α}} (t)), \bar{x_{α}} (0) \\ = \bar{x_{0 α}}, \bar{x_{α}^{1}} (0) = \bar{x_{0 α}^{1}}, \dots, \bar{x_{α}^{n}} (0) = \bar{x_{0 α}^{n}} \end{matrix}$

Case II. If we consider $^{C} D_{a}^{β} x (t)$ by using the derivative in the second form, then the arbitrary order initial value problem is written as follows. $\begin{matrix} (ii) & ^{C} D_{a}^{β} \underline{x_{α}} (t) = \underline{F_{α}} (t, \underline{x_{α}} (t), \bar{x_{α}} (t)), \underline{x_{α}} (0) \\ = \bar{x_{0 α}}, \underline{x_{α}^{1}} (0) = \bar{x_{0 α}^{1}}, \dots \dots, \underline{x_{α}^{n}} (0) = \bar{x_{0 α}^{n}} \\ ^{C} D_{a}^{β} \bar{x_{α}} (t) = \bar{F_{α}} (t, \underline{x_{α}} (t), \bar{x_{α}} (t)), \bar{x_{α}} (0) \\ = \underline{x_{0 α}}, \bar{x_{α}^{1}} (0) = \underline{x_{0 α}^{1}}, \dots, \bar{x_{α}^{n}} (0) = \underline{x_{0 α}^{n}} \end{matrix}$

Hence the solution could be written as $\tilde{x} (t) =^{C} D_{a}^{- β} \tilde{F} (t, x (t))$ with $\tilde{x} (0) = {\tilde{x}}_{0}, {\tilde{x}}^{1} (0) = {\tilde{x}}_{0}^{1}, \dots \dots \dots, {\tilde{x}}^{n} (0) = {\tilde{x}}_{0}^{n}$ .

4.2 Type-2 fuzzy arbitrary order initial value problem

The fuzzy initial value problem with arbitrary order β could be written as $^{C} D_{a}^{β} \tilde{x_{\tilde{α}}} (t) = \tilde{F_{\tilde{α}}} (t, x (t)), \tilde{x_{\tilde{α}}} (0) = \tilde{x_{\tilde{α} 0}}, {\tilde{x_{\tilde{α}}}}^{1} (0) = {\tilde{x}}_{\tilde{α} 0}^{1}, \dots \dots \dots, {\tilde{x_{\tilde{α}}}}^{n} (0) = {\tilde{x}}_{\tilde{α} 0}^{n}$ where $x^{r} (0) = \frac{d^{r}}{{dt}^{r}} x (t)$ where $\tilde{F_{\tilde{α}}} : [0, T] \times E_{2} \to E_{2}$ is a continuous fuzzy mapping, $\tilde{x_{\tilde{α}}}$ is a fuzzy vector and $\tilde{x_{\tilde{α} 0}}, {\tilde{x}}_{\tilde{α} 0}^{1}, \dots \dots, {\tilde{x}}_{\tilde{α} 0}^{n}$ are fuzzy number valued vector. In terms of α-cut this could be written as ${[x (t)]}_{\tilde{α}} = [\underline{x_{\tilde{α}}} (t), \bar{x_{\tilde{α}}} (t)]$ $[x_{0}]_{\tilde{α}} = [\underline{x_{0 \tilde{α}}}, \bar{x_{0 \tilde{α}}}]$ and ${[F (t, x (t))]}_{\tilde{α}} = [\underline{F_{\tilde{α}}} (t, \underline{x_{\tilde{α}}} (t), \bar{x_{\tilde{α}}} (t)), \bar{F_{\tilde{α}}} (t, \underline{x_{\tilde{α}}} (t), \bar{x_{\tilde{α}}} (t))]$ .

Case I. If we consider $^{C} D_{a}^{β} x (t)$ by using the caputo derivative in the first form, then the arbitrary order initial value problem is written as follows. $\begin{matrix} (i) & ^{C} D_{a}^{β} \underline{x_{\tilde{α}}} (t) = \underline{F_{\tilde{α}}} (t, \underline{x_{\tilde{α}}} (t), \bar{x_{\tilde{α}}} (t)), \underline{x_{\tilde{α}}} (0) \\ = \underline{x_{\tilde{α} 0}}, \underline{x_{\tilde{α}}^{1}} (0) = \underline{x_{\tilde{α} 0}^{1}}, \dots, \underline{x_{\tilde{α}}^{n}} (0) = \underline{x_{\tilde{α} 0}^{n}}; \\ ^{C} D_{a}^{β} \bar{x_{\tilde{α}}} (t) = \bar{F_{\tilde{α}}} (t, \underline{x_{\tilde{α}}} (t), \bar{x_{\tilde{α}}} (t)), \bar{x_{\tilde{α}}} (0) \\ = \bar{x_{\tilde{α} 0}}, \bar{x_{\tilde{α}}^{1}} (0) = \bar{x_{\tilde{α} 0}^{1}}, \dots \dots \dots, {\bar{x_{\tilde{α}}}}^{n} (0) \\ = \bar{x_{\tilde{α} 0}^{n}} \end{matrix}$

Case II. If we consider $^{C} D_{a}^{β} x (t)$ by using the derivative in the second form, then the arbitrary order initial value problem is written as follows. $\begin{matrix} (ii) & ^{C} D_{a}^{β} \underline{x_{\tilde{α}}} (t) = \underline{F_{\tilde{α}}} (t, \underline{x_{\tilde{α}}} (t), \bar{x_{\tilde{α}}} (t)), \underline{x_{\tilde{α}}} (0) \\ = \bar{x_{\tilde{α} 0}}, \underline{x_{\tilde{α}}^{1}} (0) = \bar{x_{\tilde{α} 0}^{1}}, \dots, {\underline{x_{\tilde{α}}}}^{n} (0) = \bar{x_{\tilde{α} 0}^{n}}; \\ ^{C} D_{a}^{β} \bar{x_{\tilde{α}}} (t) = \bar{F_{\tilde{α}}} (t, \underline{x_{\tilde{α}}} (t), \bar{x_{\tilde{α}}} (t)), \bar{x_{\tilde{α}}} (0) \\ = \underline{x_{\tilde{α} 0}}, \bar{x_{\tilde{α}}^{1}} (0) = \underline{x_{\tilde{α} 0}^{1}}, \dots \dots \dots, {\bar{x_{\tilde{α}}}}^{n} (0) \\ = \underline{x_{\tilde{α} 0}^{n}} \end{matrix}$

Hence the solution could be written as $\tilde{x_{\tilde{α}}} (t) =^{C} D_{a}^{- β} \tilde{F_{\tilde{α}}} (t, x (t))$ with $\tilde{x_{\tilde{α}}} (0) = \tilde{x_{\tilde{α} 0}}, {\tilde{x_{\tilde{α}}}}^{1} (0) = {\tilde{x}}_{\tilde{α} 0}^{1}, \dots \dots \dots, {\tilde{x_{\tilde{α}}}}^{n} (0) = {\tilde{x}}_{\tilde{α} 0}^{n}$ .

4.3 Type-1 fuzzy arbitrary order boundary value problem

Let $\tilde{f} : R^{n} \to E_{1}^{n}$ be a type-1 fuzzy valued function and $[\underline{f}, \bar{f}], 0 \leq α \leq 1$ be the α-cut of $\tilde{f}$ . Let F (D) be any arbitrary integro-differential operator. Then the type-2 fuzzy boundary value problem could be written as $F (D) \tilde{U} = \tilde{f} (x_{1}, x_{2}, \dots \dots, x_{i}, \dots \dots, x_{n})$ with initial conditions $\tilde{I}$ and boundary conditions $\tilde{B^{+}}$ and $\tilde{B^{-}}$ as follows

i.e. $F (D) \tilde{U} = \tilde{f} (x_{1}, x_{2}, \dots \dots, x_{i}, \dots \dots, x_{n});$ $\begin{matrix} \tilde{U} (x_{1}, x_{2}, \dots, x_{i}^{0}, \dots, x_{n}) \\ = \tilde{I} (x_{1}, x_{2}, \dots, x_{i}^{0}, \dots, x_{n}) \\ \tilde{U} (x_{1}, x_{2}, \dots, x_{r}^{+}, \dots, x_{n}) \\ = \tilde{B^{+}} (x_{1}, x_{2}, \dots, x_{r}^{+}, \dots, x_{n}) \\ \tilde{U} (x_{1}, x_{2}, \dots, x_{r}^{-}, \dots, x_{n}) \\ = \tilde{B^{-}} (x_{1}, x_{2}, \dots, x_{r}^{-}, \dots, x_{n}) \end{matrix}$

4.4 Type-2 fuzzy arbitrary order boundary value problem

Let $\tilde{f_{\tilde{α}}} : R^{n} \to E_{2}^{n}$ be a type-2 fuzzy valued function and $[\underline{f_{\tilde{α}}}, \bar{f_{\tilde{α}}}], 0 \leq α \leq 1$ be the α-cut of $\tilde{f_{\tilde{α}}}$ . Let F (D) be any arbitrary integro-differential operator. Then the type-2 fuzzy boundary value problem could be written as $F (D) \tilde{U_{\tilde{α}}} = \tilde{f_{\tilde{α}}} (x_{1}, x_{2}, \dots \dots \dots, x_{i}, \dots \dots, x_{n})$ with initial conditions ${\tilde{I}}_{\tilde{α}}$ and boundary conditions $\tilde{B_{\tilde{α}}^{+}}$ and $\tilde{B_{\tilde{α}}^{-}}$ as $F (D) \tilde{U_{\tilde{α}}} = \tilde{f_{\tilde{α}}} (x_{1}, x_{2}, \dots, x_{i}, \dots, x_{n});$ $\begin{matrix} {\tilde{U}}_{\tilde{α}} (x_{1}, x_{2}, \dots, x_{i}^{0}, \dots, x_{n}) \\ = {\tilde{I}}_{\tilde{α}} (x_{1}, x_{2}, \dots, x_{i}^{0}, \dots, x_{n}) \\ \tilde{U_{\tilde{α}}} (x_{1}, \dots, x_{r}^{+}, \dots, x_{n}) \\ = \tilde{B_{\tilde{α}}^{+}} (x_{1}, \dots, x_{r}^{+}, \dots, x_{n}) \\ \tilde{U_{\tilde{α}}} (x_{1}, \dots, x_{r}^{-}, \dots, x_{n}) \\ = \tilde{B_{\tilde{α}}^{-}} (x_{1}, \dots, x_{r}^{-}, \dots, x_{n}) . \end{matrix}$

In general we denote a fuzzy number valued function (both type-1 and type-2 together) by $\bar{z}$ . Then the general fuzzy arbitrary order dynamical system could be written as $\begin{matrix} F (D) \bar{z} (t) & = & \bar{f} (t, x (t)), \bar{z} (0) = \overset{\leftrightarrow}{z_{0}}, {\bar{z}}^{1} (0) \\ = & \overset{\leftrightarrow}{z_{0}^{1}}, \dots \dots \dots, \overset{\leftrightarrow}{z^{n}} (0) = \overset{\leftrightarrow}{z_{0}^{n}} . \end{matrix}$

The integral may be the type-1 or type-2 fuzzy integral according to the nature of the variable $\overset{\leftrightarrow}{z}$ .

5 Numerical solution to fuzzy fractional differential and integral equation

Consider a system of fuzzy arbitrary order dynamical systems in the vector form where state variables and parameters are fuzzy numbers. Let n and k be the dimension of the system and the number of fuzzy parameters. If X = (x₁, x₂, …, x_i, …, x_n) and $\overset{\leftrightarrow}{w}$ represent any type-1 or type-2 fuzzy variable together; then the fuzzy arbitrary order dynamical systems could be represented in terms of integral-differential operator L, fuzzy functions $\overset{\leftrightarrow}{f} (X)$ , initial condition $\overset{\leftrightarrow}{I}$ and boundary conditions $\overset{\leftrightarrow}{B_{r}}$ as follows: ${[L (D) \overset{\leftrightarrow}{U}]}_{α} = {[\overset{\leftrightarrow}{f} (X)]}_{α}$

with $\begin{matrix} {[\bar{U} (x_{1}, \dots, x_{i}^{0}, \dots, x_{n})]}_{α} \\ = {[\bar{I} (x_{1}, \dots, x_{i}^{0}, \dots, x_{n})]}_{α} \end{matrix}$ and $\begin{matrix} {[\overset{\leftrightarrow}{U} (x_{1}, \dots, x_{r}^{+}, \dots, x_{n})]}_{α} \\ = {[\overset{\leftrightarrow}{B_{r}^{+}} (x_{1}, \dots, x_{r}^{+}, \dots, x_{n})]}_{α} \\ {[\overset{\leftrightarrow}{U} (x_{1}, \dots, x_{r}^{-}, \dots, x_{n})]}_{α} \\ = {[\overset{\leftrightarrow}{B_{r}^{-}} (x_{1}, \dots, x_{r}^{-}, \dots, x_{n})]}_{α}, \\ r = 1, \dots, i - 1, i + 1, \dots, n . \end{matrix}$

Where $\overset{\leftrightarrow}{I}$ is the vector of fuzzy initial conditions, $\overset{\leftrightarrow}{B_{r}^{-}}$ and $\overset{\leftrightarrow}{B_{r}^{+}}$ the fuzzy boundary condition in general. ${[\overset{\leftrightarrow}{P}]}_{α} = {[\overset{\leftrightarrow}{P_{1}}, \overset{\leftrightarrow}{P_{2}}, \dots, \overset{\leftrightarrow}{P_{k}}]}_{α}$ be the parameter vector where the parameters are generally fuzzy numbers. Then similarly N = n + k interval valued entities would generate the region of uncertainty and the foot print of uncertainty of the solution of fuzzy arbitrary order dynamical systems thus is the union of the solution of the corresponding ordinary arbitrary order dynamical systems with any initial condition within the region of uncertainty. That is any solution of arbitrary order fuzzy boundary problem must be a solution of the corresponding ordinary arbitrary order boundary value problem by the fuzzy differential inclusion.

Now for each $\tilde{α}$ -cut the region of uncertainty is constituted by ${[\tilde{u} (X)]}_{\tilde{α}}$ and ${[\tilde{P_{1}}, \tilde{P_{2}}, \dots \dots, \tilde{P_{n}}]}_{\tilde{α}}$ i.e. $[\underline{u_{\tilde{α}} (X)}, \bar{u_{\tilde{α}} (X)}]$ and $[{\underline{P}}_{\tilde{α} i}, \bar{P_{\tilde{α} i}}], i = 1, 2, \dots, k$ .

The evolution of surface of uncertainty is carried out by taking the values of the functions and parameters as any arbitrary footprint of uncertainty of the fuzzy variables i.e. $I \in [\underline{I_{l α}}, \bar{I_{u α}}]$ or $I \in [\underline{I_{l α}}, \underline{I_{u α}}, \bar{I_{l α}}, \bar{I_{u α}}], B_{r}^{-} \in [\underline{B_{l α}^{- r}}, \bar{B_{u α}^{- r}}]$ or $B_{r}^{-} \in [\underline{B_{l α}^{- r}}, \underline{B_{u α}^{- r}}, \bar{B_{l α}^{- r}}, \bar{B_{u α}^{- r}}]$ , $B_{r}^{+} \in [\underline{B_{l α}^{+ r}}, \bar{B_{u α}^{+ r}}]$ or $B_{r}^{+} \in [\underline{B_{l α}^{+ r}}, \underline{B_{u α}^{+ r}}, \bar{B_{l α}^{+ r}}, \bar{B_{u α}^{+ r}}], r = 1, 2, \dots, n$ and $P_{i} \in [\underline{P_{l α}^{i}}, \bar{P_{u α}^{i}}]$ or $P_{i} \in [\underline{P_{l α}^{i}}, \underline{P_{u α}^{i}}, \bar{P_{l α}^{i}}, \bar{P_{u α}^{i}}]$ . Then any solution of the embedded system $\begin{matrix} F (D) u (X) & = & f (X), u (x^{1}, \dots, x_{0}^{i} \dots, x^{n}) \\ = & I (x^{1}, \dots, x^{i - 1}, x^{i + 1} \dots, x^{n}) \end{matrix}$ and $\begin{matrix} u (x^{1}, \dots, x_{r}^{-} \dots, x^{n}) \\ = B_{r}^{-} (x^{1}, \dots, x^{r - 1}, x^{r + 1} \dots, x^{n}), \\ u (x^{1}, \dots, x_{r}^{+} \dots, x^{n}) \\ = B_{r}^{+} (x^{1}, \dots, x^{r - 1}, x^{r + 1} \dots, x^{n}), \\ r = 1, \dots \dots, i - 1, i + 1, \dots, n . \end{matrix}$

Would be a solution of the above fuzzy arbitrary order dynamical systems with $f (X), I, B_{r}^{+}, B_{r}^{-}$ and P satisfies the inclusion. Consequently the solution of the required fuzzy arbitrary order dynamics $\overset{\leftrightarrow}{u} (X)$ is the union of all solution u (X).

6 Numerical examples

In this section we have considered few real world examples of ordinary dynamical systems (6.1), partial dynamical systems (6.2) and stochastic dynamical systems (6.3). The corresponding dynamics are extended into type-1 and type-2 fuzzy dynamics by introducing imprecision in different functions and parameters and analysed in this section.

6.1 Fuzzy Fractional ordinary differential equation: Fuzzy fractional brusselator system

The reaction brusselator system studies the mechanism of an autocatalytic, oscillating chemical reaction. In an autocatalytic reaction some derivative species is created in the reaction which acts to increase the rate of the reaction. The fuzzy reaction brusselator dynamics with arbitrary order β could be written as

$\begin{matrix} D_{t}^{β} \overset{\leftrightarrow}{x_{1}} & = & A - (μ + 1) \overset{\leftrightarrow}{x_{1}} + \overset{\leftrightarrow}{x_{1}}^{2} \overset{\leftrightarrow}{x_{2}} \\ D_{t}^{β} \overset{\leftrightarrow}{x_{2}} & = & μ \overset{\leftrightarrow}{x_{1}} - \overset{\leftrightarrow}{x_{1}}^{2} \overset{\leftrightarrow}{x_{2}} \end{matrix}$ (6.1)

Where $\overset{\leftrightarrow}{x_{1}}$ and $\overset{\leftrightarrow}{x_{2}}$ represents the fuzzy concentration of two reactants. Here we considered A = 1 and μ = 4 and we represent the solution of type-1 and type-2 fuzzy systems for β = 0.8 in Figs. (1 and 2) respectively. Here the corresponding ordinary fractional differential equations is solved by fractional Adams method [7].

Fig.1

Type-1 Brusselator system.

Fig.2

Type-2 Brusselator system.

6.2 Fuzzy fractional advection diffusion equation

The advection–diffusion equation represents the combination of the diffusion and advection process which describes different physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to diffusion and convection. Where u is the state variable; d is called the diffusivity or diffusion coefficient, $\vec{v}$ represents the average velocity of the moving substance. Whereas R stands for sources or sinks of the quantity u. The fractional order general fuzzy advection diffusion equation with three fractional order derivatives β, γ and δ respectively could be written as $\frac{\partial^{β} \overset{\leftrightarrow}{u}}{\partial t} = d \frac{\partial^{γ} \overset{\leftrightarrow}{u}}{\partial x^{2}} - v \frac{\partial^{δ} \overset{\leftrightarrow}{u}}{\partial x} - λ \overset{\leftrightarrow}{u} - R$ (6.2)

The type-1 and type-2 fuzzy arbitrary order advection diffusion equation is solved by the proposed method for β = 0.8, γ = 0.75, δ = 0.75 and the solution is represented in Figs. (3 –5).

Fig.3

Surface plot for fuzzy fractional ADE at α= 1.

Fig.4

Type-1 fuzzy fractional ADE upper and lower bounds for concentration at different α-cuts.

Fig.5

Type-2 fuzzy fractional ADE upper and lower bound for concentration at different α-cuts.

6.3 Fuzzy Fractional stochastic differential equation

6.3.1 Fuzzy Fractional asset price model

The fractional asset price model with arbitrary order h could be written as $d^{h} {\tilde{X}}_{t} = \tilde{λ} {\tilde{X}}_{t} dt + σ \sqrt{{\tilde{X}}_{t}} d^{h} {\tilde{W}}_{t}$ (6.3)

Here $\tilde{λ}$ represents the average drift in the price of a commodity whereas σ represents the volatility of the market which have been consider imprecise in this example. For the simulation we consider X (0) = X₀ = 1 ; σ = 0.8 and ${\tilde{W}}_{t}$ is a fuzzy Brownian motion. To analyse the impact of imprecise parameters we consider the parameter $\tilde{λ}$ to be a fuzzy parameter. For Euler model $\tilde{λ}$ represents the expected return and σ represents the standard deviation. We have considered $\tilde{λ}$ to be $\tilde{λ} = [0.03 0.05 0.07]$ .

Also as a separate case we extend the fuzzy stochastic differential equation into type-2 fuzzy stochastic equation by extending the parameter $\tilde{λ}$ into type-2 fuzzy parameter $\tilde{\tilde{λ}}$ taken to be $\tilde{\tilde{λ}} = [0.03 0.04 0.05 0.07]$ and fuzzy Brownian motion ${\tilde{W}}_{t}$ . The arbitrary h^th order type-2 fuzzy Euler model could be written as $d^{h} \tilde{\tilde{X_{t}}} = \tilde{\tilde{λ}} \tilde{\tilde{X_{t}}} dt + σ \sqrt{\tilde{\tilde{X_{t}}}} d {\tilde{W}}_{t}$ (6.4)

The type-1 and type-2 fuzzy Euler model is solved by the proposed method for h = 0.6 and the solution is represented in Figs. (6 and 7).

Fig.6

Type-1 fuzzy fractional stock price model.

Fig.7

Type-2 fuzzy fractional stock price model.

6.3.2 Fuzzy fractional European option pricing model

The arbitrary h^th order European option model could be written as

$\begin{matrix} d^{h} X_{1}^{t} & = & \tilde{λ} X_{1}^{t} dt + X_{2}^{t} \sqrt{X_{t}} d^{h} {\tilde{W}}_{t}^{1} \\ d^{h} X_{2}^{t} & = & (σ_{0} - X_{2}^{t}) dt + \sqrt{X_{2}^{t}} d^{h} {\tilde{W}}_{t}^{2} \end{matrix}$ (6.5)

Here the first equation is same as the asset price model and $X_{1}^{t}$ represents the value of the option. The second variable $X_{2}^{t}$ represents the volatility of the option price. $\tilde{λ}$ represents the average drift in the price of a option. For the simulation we consider σ₀ = 0.75 ; $X_{1}^{0} = 1, X_{2}^{0} = σ_{0}$ and ${\tilde{W}}_{t}^{1}$ and ${\tilde{W}}_{t}^{2}$ are two independent fuzzy Brownian motions. To analyse the impact of imprecise parameters we consider the parameter $\tilde{λ}$ to be a fuzzy parameter. For European option model $\tilde{λ}$ represents the expected return and σ₀ represents the standard deviation. We have considered $\tilde{λ}$ to be $\tilde{λ} = [0.02 0.03 0.04]$ . Also we extend the fuzzy stochastic differential equation into type-2 fuzzy stochastic equation by extending the parameter $\tilde{λ}$ into type-2 fuzzy parameter $\tilde{\tilde{λ}}$ taken to be $\tilde{\tilde{λ}} = [0.02 0.025 0.035 0.04]$ and type-2 fuzzy Brownian motions ${\tilde{W}}_{t}^{1}$ and ${\tilde{W}}_{t}^{2}$ . The arbitrary h^th order type-2 fuzzy European option model could be written as

$\begin{matrix} d^{h} X_{1}^{t} & = & \tilde{\tilde{λ}} X_{1}^{t} dt + X_{2}^{t} \sqrt{X_{t}} d {\tilde{W}}_{t}^{1} \\ d^{h} X_{2}^{t} & = & (0 - X_{2}^{t}) dt + \sqrt{X_{2}^{t}} d {\tilde{W}}_{t}^{2} \end{matrix}$ (6.6)

The type-1 and type-2 fuzzy European option price model is solved by the proposed method for h = 0.6 and the solution is represented in Figs. (8 and 9).

Fig.8

Type-1 fuzzy fractional European option model.

Fig.9

Type-2 fuzzy fractional European option model.

7 Results and discussion

The numerical simulations of different class of fuzzy arbitrary order dynamical systems are carried out to obtain the solution of corresponding fuzzy integro-differential equations. The solution of fuzzy type-1 and type-2 ordinary dynamical systems, partial dynamical systems and stochastic dynamical systems are represented graphically in the above sections. The solution of fuzzy type-1 and type-2 fractional reaction brusselator system is shown in Figs. (1 and 2) respectively. The solution of fuzzy type-1 and type-2 fractional advection diffusion equation is shown in Figs. (3 –5) respectively. In Figs. (6 and 7) solution of fuzzy type-1 and type-2 fractional asset price model is represented whereas the solution of fuzzy type-1 and type-2 fractional European option model is represented in Figs. (8 and 9). Here for the simulation purpose and for the sake of simplicity we consider the initial conditions and parameter values as type-1 and type-2 triangular fuzzy numbers. In the above figures $\tilde{y} = {[yl, yu]}_{α}$ and $\tilde{z} = {[zll, zlu, zul, zuu]}_{α}$ are plotted with respect independent variables for α = 0, 0.2, 0.4, 0.6, 0.8, 1. Thus for six nested α-cuts nested solution intervals are represented in different shades shown in legends. In Section 6.1 we extend the fractional reaction brusselator system into FIVP by introducing fuzzy initial condition and the temporal distribution of two reactant is shown in Figs. (1 and 2) for type-1 and type-2 fuzzy initial condition respectively. Whereas Section 6.2 extends fractional advection diffusion equation into FBVP by introducing fuzzy initial condition. Here in Fig. (3) the spatio-temporal distribution of concentration of the fuzzy fractional advection diffusion equation for α = 1 i.e. the solution of the ordinary fractional advection diffusion equation. The spatio-temporal distribution of concentration is shown in Figs. (5 and 6) for type-1 and type-2 fuzzy initial condition respectively. In Section 6.3 illustrative examples of general fuzzy arbitrary order stochastic dynamical systems are considered, namely fuzzy fractional asset price model and fuzzy fractional European option model. In these case the initial value of a stock or option is generally well known in a particular time; but the parameters of the system which are some statistical ensembles in general may not be precisely known. Therefore in this section we considered type-1 and type-2 fuzzy fractional SDE with fuzzy parameters and precise initial conditions.

It is noted that at α= 1 the solution for all preceding fuzzy type-2 arbitrary order dynamical systems converges with their respective type-1 fuzzy solution, whereas fuzzy type-1 arbitrary order dynamical systems converges to the solution of the corresponding crisp arbitrary order dynamical systems models.

The proposed method illustrate the impact of the imprecision and uncertainty on the solution for uncertain initial condition and/or boundary conditions and/or parameter values. Moreover, at the absolute precision level or grade of membership is at α= 1 the solution coincide with the ordinary dynamical system. In this sense the solution obtained by solving fuzzy type-2 arbitrary order dynamical systems is the generalization of corresponding system of fuzzy type-1 arbitrary order dynamical systems. The solution of a fuzzy type-1 and fuzzy type-2 arbitrary order dynamical systems are obtained by solving the corresponding embedded system i.e. to solve crisp system with all possible values of the parameters and functions within the uncertainty region. It is already noted that the direct use of interval algebra and interval calculus to solve fuzzy dynamics generally produces unacceptable and incorrect results. Whereas the method proposed in this paper, which is based on fuzzy differential inclusion provides an acceptable solution of general fuzzy arbitrary order dynamical systems.

8 Conclusions

Fuzziness is one of the best alternative to model imprecision and uncertainty in the dynamics. Moreover, if the membership of some uncertain function or parameter are also imprecise, T2FSs are one of the best alternative to model the dynamics. The main objective of this paper is to define the type-1 and type-2 fuzzy arbitrary order dynamical systems and develop a methodology to obtain numerical solution of type-1 as well as type-2 fuzzy arbitrary order dynamical systems. The representation of general fuzzy arbitrary order dynamical systems are based on integrability of the general fuzzy number-valued functions and fuzzy random variables.

General fuzzy arbitrary order dynamical systems are the generalization of ordinary arbitrary order dynamical systems. However for general fuzzy arbitrary order SDE the stochastic processes is extended to E₁ and E₂ the universe of fuzzy set valued functions. Consequently, type-2 fuzzy arbitrary order dynamical systems reduces to system of type-1 fuzzy arbitrary order dynamical systems, which again reduces to system of ordinary arbitrary order embedded systems.

In this paper, we have presented the definition of fuzzy type-1 and type-2 fuzzy arbitrary order dynamical systems and proposed a methodology to solve system any fuzzy type-1 and type-2 fuzzy arbitrary order dynamical systems. A number of numerical examples is solved and the result is shown and analysed. In future studies, we may investigate the effect of fuzzy order of an arbitrary order dynamical systems.

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