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Abstract

In this article, we propose a new approach to solve the hybrid fuzzy differential equations based on the feed-forward neural networks. We first replace it by a system of ordinary differential equations. A trial solution of this system involves two parts. The first part satisfies the initial condition and contains no adjustable parameters; however, the second part involves a feed-forward neural network containing adjustable parameters (the weights). This method shows that using neural networks provides solutions with good generalization and the high accuracy.

Keywords

Hybrid systems fuzzy differential equations neural network feed-forward artificial neural networks approximate solution

Introduction

Fuzzy differential equations (FDEs) are significant for studying and solving large proportions of problems in many topics of applied mathematics, particularly in relation to physics, geography, medicine, biology, control chaotic systems, bioinformatics and computational biology, synchronize hyperchaotic systems, economics and finance, and so on.^1–3 In lots of applications, some of the parameters are represented by fuzzy numbers rather than the crisp numbers, and hence, it is essential to develop mathematical models and numerical procedures which would have appropriately treated to general FDEs. The knowledge about differential equations is often incomplete or vague. The FDEs were formulated by Kaleva⁴ and Seikkala⁵ initially. Hybrid systems received too much attention in the recent literatures as well. Hybrid systems evolve in continuous time, like differential systems, but undergoing a fundamental change in their governing equations at a sequence of discrete times. When a continuous time dynamics of a hybrid system is given by a FDE, the system is called a hybrid fuzzy differential system. For analytical results on hybrid fuzzy differential equations (HFDEs), see Lakshmikantham and Liu⁶ and Lakshmikantham and Mohapatra.⁷ Hybrid systems are devoted for modeling, designing, and validating interactive systems of computer programs and continuous systems as well. These control systems which are capable of controlling complex systems have discrete event dynamics as well as continuous time dynamics that can be modeled by hybrid systems. The differential systems containing fuzzy-valued functions and interaction with a discrete time controller are named as hybrid fuzzy differential systems (HFDEs). Several numerical techniques have been applied for solving HFDEs. For instance, Pederson and Sambandham^8,9 investigated the numerical solution of these equations using the Euler and Runge–Kutta methods. Prakash and Kalaiselvi¹⁰ studied the predictor-corrector method for HFDEs. In addition, Fard and Bidgoli¹¹ solved HFDEs by the Nystrom method. Similarly, Kim and Sakthivel¹² studied the predictor–corrector method for HFDEs, and Allahviranloo and Salahshour¹³ investigated the numerical solution of HFDEs, using the Euler method under characterization theorem and Bede’s differentiability. Ahmadian et al.^14–16 applied Runge–Kutta method with lower function evaluations and the reduced the number of function evaluations for solving HFDEs. Paripour et al.¹⁷ applied Adomian decomposition method for solving HFDEs.

In this study, we develop numerical methods for HFDEs by an application of artificial neural network. Lagaris et al.¹⁸ designed artificial neural networks for solving ordinary differential equations (ODEs) and partial differential equations (PDEs). They used multilayer perceptron to estimate the solution of differential equations. In 2010, Effati and Pakdaman¹⁹ used artificial neural networks for solving FDEs. Similarly, Pakdaman et al.²⁰ solved differential equations of fractional order using an optimization technique based on training artificial neural network. Their neural network model was trained over an interval (over which the differential equation must be solved), so the inputs of the neural network model were the training points. Bede²¹ proved a characterization theorem which states that under certain conditions, an FDE under the Hukuhara differentiability is equivalent to a system of ODEs. Moreover, Bede also noticed that this characterization theorem can aid to solve FDEs numerically through converting the FDEs to a system of ODEs, which later could be solved by numerical methods. In this article, using the characterization theorem, we generalize a fourth-order Runge–Kutta method that originally presented to solve the HFDEs. That is, we substitute the original initial value problem with two parametric hybrid ordinary differential systems. Then, the extension of Bede’s characterization theorem for HFDEs, which was investigated by Pederson and Sambandham,²² is employed to generalize the derivatives. Finally, these results are applied to solve the HFDEs numerically by the fourth-order reduced Runge–Kutta (RRK) method.

Preliminaries

Definition 1

A fuzzy number is a function $\tilde{u} : R \to [0, 1]$ with the following properties:²³

$\tilde{u}$ is normal, that is, $\exists t_{0} \in R$ such that $\tilde{u} (t_{0}) = 1$ ;

$\tilde{u}$ is a fuzzy convex set, that is

\begin{array}{l} \tilde{u} (α t_{1} + (1 - α) t_{2}) \geq m i n {\tilde{u} (t_{1}), \tilde{u} (t_{2})} \\ \forall t_{1}, t_{2} \in ℝ, α \in [0, 1] \end{array}

$\tilde{u}$ is the upper semi-continuous on $R$ ;

$\bar{{t \in R : \tilde{u} (t) > 0}}$ is a compact set, where $\bar{S}$ denotes the closure of $S$ .

The set of all fuzzy numbers is denoted by $E$ . Kaleva⁴ gave an alternative definition which yields the same $E$ .

Definition 2

An ordered pair of functions $(\underline{u_{1}} (r), \bar{u_{1}} (r)), 0 \leq r \leq 1$ presents an arbitrary fuzzy number, in parametric form, which satisfies the following requirements:²³

$\underline{u_{1}} (r)$ is a bounded left continuous non-decreasing on [0, 1];

$\bar{u_{1}} (r)$ is a bounded left continuous non-increasing function on [0, 1];

$\underline{u_{1}} (r) \leq \bar{u_{1}} (r), 0 \leq r \leq 1$ .

The following equations define the addition and the scalar multiplication of fuzzy numbers in $E$ :

${\tilde{u}}_{1} \oplus {\tilde{u}}_{2} = (\underline{u_{1}} (r) + \underline{u_{2}} (r), \bar{u_{1}} (r) + \bar{u_{2}} (r))$ ;

$(λ ⊙ {\tilde{u}}_{1}) = {\begin{matrix} (λ \underline{u_{1}} (r), λ \bar{u_{1}} (r)), λ \geq 0 \\ (λ \bar{u_{1}} (r), λ \underline{u_{1}} (r)), λ < 0 \end{matrix} .$

Definition 3

The Hausdorff distance between ${\tilde{u}}_{1}$ and ${\tilde{u}}_{2}$ for arbitrary fuzzy numbers, ${\tilde{u}}_{1} = (\underline{u_{1}} (r), \bar{u_{1}} (r))$ and ${\tilde{u}}_{2} = (\underline{u_{2}} (r), \bar{u_{2}} (r))$ , is²⁴

D ({\tilde{u}}_{1}, {\tilde{u}}_{2}) = sup_{r \in [0, 1]} m a x {| \underline{u_{1}} (r) - \underline{u_{2}} (r) |, | \bar{u_{1}} (r) - \bar{u_{2}} (r) |}

The following properties hold:

$(E, D)$ is a complete metric space;

$D ({\tilde{u}}_{1} \oplus {\tilde{u}}_{2}, {\tilde{u}}_{3} \oplus {\tilde{u}}_{2}) = D ({\tilde{u}}_{1}, {\tilde{u}}_{3}), \forall {\tilde{u}}_{1}, {\tilde{u}}_{2}, {\tilde{u}}_{3} \in E$ ;

$D (λ ⊙ {\tilde{u}}_{1}, λ ⊙ {\tilde{u}}_{2}) = | λ | D ({\tilde{u}}_{1}, {\tilde{u}}_{2}), \forall {\tilde{u}}_{1}, {\tilde{u}}_{2} \in E, \forall λ \in R$ ;

$D ({\tilde{u}}_{1} \oplus {\tilde{u}}_{2}, {\tilde{u}}_{3} \oplus {\tilde{u}}_{4}) \leq D ({\tilde{u}}_{1}, {\tilde{u}}_{3}) + D ({\tilde{u}}_{2}, {\tilde{u}}_{4}), \forall {\tilde{u}}_{1}, {\tilde{u}}_{2}, {\tilde{u}}_{3}, {\tilde{u}}_{4} \in E$ .

Definition 4

Let $f : R \to E$ is a fuzzy-valued function. If for arbitrary fixed $t_{0} \in R$ and $ε > 0, δ < 0$ such that²⁵

| t - t_{0} | < δ \Rightarrow D (f (t), f (t_{0})) < ε

$f$ is said to be continuous.

Definition 5

Suppose $x, y \in E$ . If there exists $z \in E$ such that $x = y \oplus z$ , then $z$ is called the Hukuhara difference (H-difference) of $x$ and $y$ , and it is denoted by $x ⊖ y$ .²⁶ In this article, the sign ⊖ stands for H-difference, and also note that $x ⊖ y \neq x + (- 1) y$ .

Definition 6

Let $f : T \to E$ is a fuzzy function and $T \subseteq R$ . We say $f$ is differentiable at $t_{0} \in T$ , if there exists an element $f' (t_{0}) \in E$ such that limits²⁶

\lim_{Δ t \to 0^{+}} \frac{f (t_{0} + Δ t) ⊖ f (t_{0})}{Δ t}

and

\lim_{Δ t \to 0^{+}} \frac{f (t_{0}) ⊖ f (t_{0} - Δ t)}{Δ t}

exist and are equal to $f' (t_{0})$ . Here, the limits are taken in the metric space $(E, D)$ .

The above definition is a generalization of the H-differentiability of a set-valued function. From Bhaskar et al.,²⁷ it follows that a H-differentiable function has increasing length of support, so this definition of a derivative is very restrictive. In this regard, Bede and Gal²⁸ introduced a more generalized definition of H-differentiability which is our interest in this article.

Definition 7

Let $f : (a, b) \to E$ and $x_{0} \in (a, b)$ . We say that $f$ is strongly generalized H-differentiable at $t_{0}$ if there exists an element $f' (t_{0}) \in E$ , such that²⁸

For all $Δ t > 0$ sufficiently small, $\exists f (t_{0} + Δ t) ⊖ f (t_{0}), \exists f (t_{0}) ⊖ f (t_{0} - Δ t)$ and limits (in the metric D)

\begin{array}{l} \lim_{Δ t \to 0} \frac{f (t_{0} + Δ t) ⊖ f (t_{0})}{Δ t} \\ = \lim_{Δ t \to 0} \frac{f (t_{0}) ⊖ f (t_{0} - Δ t)}{Δ t} = f^{'} (t_{0}) \end{array}

For all $Δ t > 0$ sufficiently small, $\exists f (t_{0}) ⊖ f (t_{0} + Δ t), \exists f (t_{0} - Δ t) ⊖ f (t_{0})$ and limits (in the metric D)

\begin{array}{l} \lim_{Δ t \to 0} \frac{f (t_{0}) ⊖ f (t_{0} + Δ t)}{- Δ t} \\ = \lim_{Δ t \to 0} \frac{f (t_{0} - Δ t) ⊖ f (t_{0})}{- Δ t} = f^{'} (t_{0}) \end{array}

For all $Δ t > 0$ sufficiently small, $\exists f (t_{0} + Δ t) ⊖ f (t_{0}), \exists f (t_{0} - Δ t) ⊖ f (t_{0})$ and limits (in the metric D)

\begin{array}{l} \lim_{Δ t \to 0} \frac{f (t_{0} + Δ t) ⊖ f (t_{0})}{Δ t} \\ = \lim_{Δ t \to 0} \frac{f (t_{0} - Δ t) ⊖ f (t_{0})}{- Δ t} = f^{'} (t_{0}) \end{array}

For all $Δ t > 0$ sufficiently small, $\exists f (t_{0}) ⊖ f (t_{0} + Δ t), \exists f (t_{0}) ⊖ f (t_{0} - Δ t)$ and limits (in the metric D)

\begin{array}{l} \lim_{Δ t \to 0} \frac{f (t_{0}) ⊖ f (t_{0} + Δ t)}{- Δ t} \\ = \lim_{Δ t \to 0} \frac{\exists f (t_{0}) ⊖ f (t_{0} - Δ t)}{Δ t} = f^{'} (t_{0}) \end{array}

Theorem 1 (Bede’s characterization theorem²¹)

Let us consider the fuzzy initial value problem (FIVP)

\begin{matrix} {\begin{matrix} y' (x) = f (x, y (x)) \\ y (x_{0}) = y_{0} \end{matrix} \end{matrix}

(1)

where $f : [x_{0}, x_{0} + a] \times E \to E$ is such that

$[f (x, y)]^{r} = [{\underline{f}}^{r} (x, \underline{y}, \bar{y}), {\bar{f}}^{r} (x, \underline{y}, \bar{y})]$ ;

${\underline{f}}^{r}$ and ${\bar{f}}^{r}$ are equicontinuous (i.e. for any $ε > 0$ there is a $δ > 0$ such that $| {\underline{f}}^{r} (x, y, z) - {\underline{f}}^{r} (x_{1}, y_{1}, z_{1}) | < ε$ and $| {\bar{f}}^{r} (x, y, z) - {\bar{f}}^{r} (x_{1}, y_{1}, z_{1}) | < ε$ for all $r \in [0, 1]$ , whenever $(x, y, z), (x_{1}, y_{1}, z_{1}) \in [x_{0}, x_{0} + a] \times R^{2}$ and $∥ (x, y, z) - (x_{1}, y_{1}, z_{1}) ∥ < δ$ ) and uniformly bounded on any bounded set;

There exists on an $L > 0$ such that

\begin{matrix} | {\underline{f}}^{r} (x, y, z) - {\underline{f}}^{r} (x, y_{1}, z_{1}) | \leq L m a x {| y_{1} - y |, \\ | z_{1} - z |} for all r \in [0, 1] \\ | {\bar{f}}^{r} (x, y, z) - {\bar{f}}^{r} (x, y_{1}, z_{1}) | \leq L m a x {| y_{1} - y |, \\ | z_{1} - z |} for all r \in [0, 1] \end{matrix}

Then, the FIVP (equation (1)) and system of ODEs

\begin{matrix} {\begin{matrix} (\underline{y^{r}} (x))' = {\underline{f}}^{r} (x, {\underline{y}}^{r}, {\bar{y}}^{r}) \\ (\bar{y^{r}} (x))' = {\bar{f}}^{r} (x, {\underline{y}}^{r}, {\bar{y}}^{r}) \\ {\underline{y}}^{r} (x_{0}) = {\underline{y}}_{0}^{r} \\ {\bar{y}}^{r} (x_{0}) = {\bar{y}}_{0}^{r} \end{matrix} \end{matrix}

(2)

are equivalent.

The hybrid fuzzy differential system

Consider the following hybrid fuzzy differential system

\begin{matrix} {\begin{matrix} y' (x) = f (x, y (x), λ_{k} (y_{k})), x \in [x_{k}, x_{k + 1}] \\ y (x_{k}) = y_{k} \end{matrix} \end{matrix}

(3)

where $0 \leq x_{0} < x_{1} < \dots < x_{k} < \dots, x_{k} \to \infty, f \in C [R^{+} \times E \times E, E], λ_{k} \in C [E, E]$ .

To be specific, the system would look like

\begin{matrix} y' (x) = {\begin{matrix} {y'}_{0} (x) = f (x, y_{0} (x), λ_{0} (y_{0})), & y_{0} (x_{0}) = y_{0}, x_{0} \leq x \leq x_{1} \\ {y'}_{1} (x) = f (x, y_{1} (x), λ_{1} (y_{1})), & y_{1} (x_{1}) = y_{1}, x_{1} \leq x \leq x_{2} \\ ⋮ & ⋮ \\ {y'}_{k} (x) = f (x, y_{k} (x), λ_{k} (y_{k})), & y_{k} (x_{k}) = y_{k}, x_{k} \leq x \leq x_{k + 1} \\ ⋮ & ⋮ \end{matrix} \end{matrix}

(4)

Assuming that the existence and uniqueness of solutions of equation (3) hold for each $[x_{k}, x_{k + 1}]$ , by the solution of equation (3), we obtain the following function

\begin{matrix} y (x) = y (x, x_{0}, y_{0}) = {\begin{matrix} y_{0} (x), & x_{0} \leq x \leq x_{1} \\ y_{1} (x), & x_{1} \leq x \leq x_{2} \\ ⋮ & ⋮ \\ y_{k} (x), & x_{k} \leq x \leq x_{k + 1} \\ ⋮ & ⋮ \end{matrix} \end{matrix}

(5)

We note that the solutions of equation (3) are piecewise differentiable in each interval for $x \in [x_{k}, x_{k + 1}]$ , for a fixed $y_{k} \in E$ and $k = 0, 1, 2, \dots$ . We replace equation (3) by the following equivalent system

\begin{matrix} {\begin{matrix} \underline{y}' (x) = \underline{f} (x, y (x), λ_{k} (y_{k})), \underline{y} (x_{k}) = \underline{y_{k}} \\ \bar{y}' (x) = \bar{f} (x, y (x), λ_{k} (y_{k})), \bar{y} (x_{k}) = \bar{y_{k}} \end{matrix} \end{matrix}

(6)

The parametric form of the above system is given by

\begin{matrix} {\begin{matrix} \underline{y}' (x, r) = F (x, \underline{y} (x, r), \bar{y} (x, r), \underline{λ_{k}} (y_{k}) (r), \bar{λ_{k}} (y_{k}) (r)), \underline{y} (x_{k}, r) = \underline{y_{k}} (r) \\ \bar{y}' (x, r) = G (x, \underline{y} (x, r), \bar{y} (x, r), \underline{λ_{k}} (y_{k}) (r), \bar{λ_{k}} (y_{k}) (r)), \bar{y} (x_{k}, r) = \bar{y_{k}} (r) \end{matrix} \end{matrix}

(7)

where $x \in [x_{k}, x_{k + 1}]$ and $r \in [0, 1]$ . Using Bede’s characterization theorem proposed by Bede,²¹ Pederson and Sambandham²² generalized the following characterization theorem for HFDEs.

Theorem 2

Consider the HFDE (3) expanded as equation (4) where for $k = 0, 1, 2, \dots$ , and each $f_{k} : [x_{k}, x_{k + 1}] \times E \to E$ , we have:²²

$[f_{k} (x, y)]^{r} = [{\underline{f}}_{k}^{r} (x, \underline{y}, \bar{y}), {\bar{f}}_{k}^{r} (x, \underline{y}, \bar{y})]$ ;

${\underline{f}}_{k}^{r}$ and ${\bar{f}}_{k}^{r}$ are equicontinuous and uniformly bounded on any bounded set;

There exists a $L_{k} > 0$ such that

\begin{matrix} | {\underline{f}}_{k}^{r} (x, y, z) - {\underline{f}}_{k}^{r} (x, y_{1}, z_{1}) | \leq L_{k} m a x {| y_{1} - y |, \\ | z_{1} - z |} for all r \in [0, 1] \\ | {\bar{f}}_{k}^{r} (x, y, z) - {\bar{f}}_{k}^{r} (x, y_{1}, z_{1}) | \leq L_{k} m a x {| y_{1} - y |, \\ | z_{1} - z |} for all r \in [0, 1] \end{matrix}

Then, the FIVP (equation (3)) and system of ODEs

\begin{matrix} {\begin{matrix} ({\underline{y}}_{k}^{r} (x))' =_{k}^{r} (x, {\underline{y}}_{k}^{r}, {\bar{y}}_{k}^{r}) \\ ({\bar{y}}_{k}^{r} (x))' = {\bar{f}}_{k}^{r} (x, {\underline{y}}_{k}^{r}, {\bar{y}}_{k}^{r}) \\ {\underline{y}}_{k}^{r} (x_{k}) = {\underline{y}}_{k - 1}^{r} (x_{k}), if k > 0, {\underline{y}}_{0}^{r} (x_{0}) = {\underline{y}}_{0}^{r} \\ {\bar{y}}_{k}^{r} (x_{k}) = {\bar{y}}_{k - 1}^{r} (x_{k}), if k > 0, {\bar{y}}_{0}^{r} (x_{0}) = {\bar{y}}_{0}^{r} \end{matrix} \end{matrix}

are equivalent.

Neural networks

Using neural networks provides solutions with very good generalizability (such as differentiability). However, an important feature of multilayer perceptrons is their utility to approximate functions, which leads to a wide applicability in most problems. In this article, the function approximation capability of feed-forward neural networks is used by expressing the trial solutions for system (7) as the sum of two terms (see equation (9)). The first term satisfies the initial conditions and does not contain adjustable parameters. The second term involves a feed-forward neural network to be trained, so satisfies the differential equations. Since it is known as a multilayer perceptron with one hidden layer which can approximate any function to arbitrary accuracy, the multilayer perceptron is used as the type of the network architecture.

If ${\underline{y}}_{T} (x, r, \underline{p})$ is a trial solution for the first equation in system (3) and ${\bar{y}}_{T} (x, r, \bar{p})$ is a trial solution for the second equation in system (3) where $\underline{p}$ and $\bar{p}$ are adjustable parameters (indeed ${\underline{y}}_{T} (x, r, \underline{p})$ and ${\bar{y}}_{T} (x, r, \bar{p})$ are approximations of ${\underline{y}}_{T} (x, r)$ and ${\bar{y}}_{T} (x, r)$ , respectively), then a discretized issue of system (3) might be converted to the optimization problem

\min_{\vec{v}} \sum_{i = 1}^{m} {\begin{matrix} {({\underline{y'}}_{T} (x_{i}, r, \underline{v}) - F [x_{i}, {\underline{y}}_{T} (x_{i}, r, \underline{v}), {\bar{y}}_{T} (x_{i}, r, \bar{v}), \underline{λ_{k}} (y_{k}) (r), \bar{λ_{k}} (y_{k}) (r)])}^{2} \\ + {({\bar{y'}}_{T} (x_{i}, r, \bar{v}) - G [x_{i}, {\underline{y}}_{T} (x_{i}, r, \underline{v}), {\bar{y}}_{T} (x_{i}, r, \bar{v}), \underline{λ_{k}} (y_{k}) (r), \bar{λ_{k}} (y_{k}) (r)])}^{2} \end{matrix}}

(8)

where $\vec{v} = (\underline{v}, \bar{v})$ includes all adjustable parameters with the initial conditions

{\underline{y}}_{T} (x_{0}, r, \underline{v}) = {\underline{y}}_{0} (r), {\bar{y}}_{T} (x_{0}, r, \bar{v}) = {\bar{y}}_{0} (r)

Each trial solution ${\underline{y}}_{T}$ and ${\bar{y}}_{T}$ employs one feed-forward neural network for which the corresponding networks are denoted by $\underline{N}$ and $\bar{N}$ , with adjustable parameters $\underline{v}$ and $\bar{v}$ , respectively. Thus, ${\underline{y}}_{T}$ and ${\bar{y}}_{T}$ can be selected as follows

\begin{matrix} {\begin{matrix} \underline{y}'_{T} (x, r, \underline{v}) = \underline{y} (x_{0}, r) + (x - x_{0}) \underline{N} (x, r, \underline{v}) \\ {\bar{y}}_{T} (x, r, \bar{v}) = \bar{y} (x_{0}, r) + (x - x_{0}) \bar{N} (x, r, \bar{v}) \end{matrix} \end{matrix}

(9)

where $\underline{N}$ and $\bar{N}$ are single-output feed-forward neural networks with adjustable parameters $\underline{v}$ and $\bar{v}$ , respectively. Here, $x$ and $r$ are the network inputs. It is easy to see that in equation (9), ${\underline{y}}_{T}$ and ${\bar{y}}_{T}$ satisfy the initial conditions. From equation (9), it is easy to show that

\begin{matrix} {\begin{matrix} \underline{y}'_{T} (x, r, \underline{v}) = \underline{N} (x, r, \underline{v}) + (x - x_{0}) \frac{\partial \underline{N}}{\partial x} \\ \bar{y}'_{T} (x, r, \bar{v}) = \bar{N} (x, r, \bar{v}) + (x - x_{0}) \frac{\partial \bar{N}}{\partial x} \end{matrix} \end{matrix}

(10)

Now suppose a multilayer perceptron has a hidden layer with $H$ sigmoid units and a linear output unit (Figure 1). Therefore, we have

\begin{matrix} {\begin{matrix} \underline{N} = \sum_{i = 1}^{m} {\underline{w}}_{i} σ ({\underline{t}}_{i}), {\underline{t}}_{i} = {\underline{a}}_{i 1} x + {\underline{a}}_{i 2} r + {\underline{b}}_{i} \\ \bar{N} = \sum_{i = 1}^{m} {\bar{w}}_{i} σ ({\bar{t}}_{i}), {\bar{t}}_{i} = {\bar{a}}_{i 1} x + {\bar{a}}_{i 2} r + {\bar{b}}_{i} \end{matrix} \end{matrix}

(11)

where $σ (t)$ is the sigmoid transfer function, $\underline{a}$ and $\bar{a}$ ( $m \times 2$ matrices) are the weights of input layers, and $\underline{b}$ and $\bar{b}$ ( $m \times 1$ matrices) are the bias vectors of input units. $\underline{w}$ and $\bar{w}$ ( $m \times 1$ matrices) are the weight vectors of output units, and $σ (t) = 1 / (1 + e^{- t})$ is the sigmoid transfer function. The following is obtained

\begin{matrix} {\begin{matrix} \frac{\partial \underline{N}}{\partial x} = \sum_{i = 1}^{m} {\underline{w}}_{i} {\underline{a}}_{i 1} σ' ({\underline{t}}_{i}) \\ \frac{\partial \bar{N}}{\partial x} = \sum_{i = 1}^{m} {\bar{w}}_{i} {\bar{a}}_{i 1} σ' ({\bar{t}}_{i}) \end{matrix} \end{matrix}

(12)

where $σ' ({\bar{t}}_{i})$ is the first derivative of the sigmoid function. Now, if we substitute equation (10) in (8), the constrained optimization problem (8) might be changed with the unconstrained optimization problem as follow

\min_{\vec{v}} \sum_{i = 1}^{n} {\begin{matrix} (\underline{N} (x_{i}, r, \underline{v}) + (x_{i} - x_{0}) \frac{\partial \underline{N}}{\partial x} - F [x_{i}, {\underline{y}}_{T} (x_{i}, r, \underline{v}), {\bar{y}}_{T} (x_{i}, r, \bar{v}), \underline{λ_{k}} (y_{k}) (r), \bar{λ_{k}} (y_{k}) (r) {])}^{2} \\ + ({\bar{N}}_{T} (x_{i}, r, \bar{v}) + (x_{i} - x_{0}) \frac{\partial \bar{N}}{\partial x} - G [x_{i}, {\underline{y}}_{T} (x_{i}, r, \underline{v}), {\bar{y}}_{T} (x_{i}, r, \bar{v}), \underline{λ_{k}} (y_{k}) (r), \bar{λ_{k}} (y_{k}) (r) {])}^{2} \end{matrix}}

(13)

Figure 1.

Architecture of the perceptron.

Numerical example

We use the Broyden–Fletcher–Goldfarb–Shanno (BFGS) quasi-Newton method to minimize the objective function in the MATLAB optimization toolbox. We take $\underline{e} (x, r) = {\underline{y}}_{T} (x, r) - {\underline{y}}_{a} (x, r)$ and $\bar{e} (x, r) = {\bar{y}}_{T} (x, r) - {\bar{y}}_{a} (x, r)$ , where $y_{T} = ({\underline{y}}_{T}, {\bar{y}}_{T})$ is the approximated solution and $y_{a} = ({\underline{y}}_{a}, {\bar{y}}_{a})$ is the known exact solution. We use multilayer perceptron consisting of 1 hidden layer with 10 hidden units and 1 linear output unit.

Example

Assume the hybrid FIVP

\begin{matrix} {\begin{matrix} y' (x) = y (x) + m (x) λ_{k} (y_{x_{k}}), x \in [x_{k}, x_{k + 1}], x_{k} = k, k = 0, 1, \dots \\ y (0) = [r - 0.25, 2 - 1.25 r] \end{matrix} \end{matrix}

(14)

where

\begin{matrix} m (x) = {\begin{matrix} 2 (t (m o d 1)), & x (m o d 1) \leq 0.5 \\ 2 (1 - t (m o d 1)), & x (m o d 1) > 0.5 \end{matrix} \end{matrix}

(15)

and

\begin{matrix} λ_{k} (μ) = {\begin{matrix} \hat{0}, & k = 0 \\ μ, & k \in {1, 2, \dots} \end{matrix} \end{matrix}

(16)

for which $\hat{0} \in E^{n}$ is defined as $\hat{0} (y) = 1$ if $y = 0$ and $\hat{0} (y) = 0$ if $y \neq 0$ .

The hybrid fuzzy initial problem (equation (14)) is equivalent to the following system of FIVPs

\begin{matrix} {\begin{matrix} {y'}_{0} (x) = y_{0} (x), x \in [0, 1] \\ y (0) = [r - 0.25, - 1.125 r + 1.875] \\ y_{i}' (x) = x_{i} (t) + m (x) y_{i} (x_{i}), x \in [x_{i}, x_{i + 1}] \end{matrix} \end{matrix}

In equation (14), $y (x) + m (x) λ_{k} (y_{x_{k}})$ is a continuous function of $x$ , $y$ , and $λ_{k} (y_{x_{k}})$ . Therefore, referring example by Kaleva,⁴ for each $k = 0, 1, 2, \dots$ the FIVP

\begin{matrix} {\begin{matrix} y' (x) = f (x, y (x), λ_{k} (y_{k})), x \in [x_{k}, x_{k + 1}] \\ y (x_{k}) = y_{k} \end{matrix} \end{matrix}

has a unique solution on $[x_{k}, x_{k + 1}]$ .

Exact solution for $x = 1$ is

y (1, r) = [(r - 0.25) e, (- 1.125 r + 1.875) e], r \in [0, 1]

The trial solutions in the neural form are as follows for $x \in [0, 1]$

\begin{matrix} {\begin{matrix} {\underline{y}}_{T} (x) = (r - 0.25) + x \underline{N} (x, r, \underline{v}) \\ {\bar{y}}_{T} (x) = (- 1.125 r + 1.875) + x \bar{N} (x, r, \bar{v}) \end{matrix} \end{matrix}

The exact solution for $x = 1.5$ is

\begin{array}{l} y (1.5, r) = [(5.291 r - 1.3227), \\ (- 5.9523 r + 9.9202)], r \in [0, 1] \end{array}

The trial solutions for $x \in [1, 1.5]$ are

\begin{matrix} {\begin{matrix} {\underline{y}}_{T} (x) = (5.291 r - 1.3227) + (x - 1) \underline{N} (x, r, \underline{v}) \\ {\bar{y}}_{T} (x) = (- 5.9523 r + 9.9202) + (x - 1) \bar{N} (x, r, \bar{v}) \end{matrix} \end{matrix}

The exact solution for $x = 2$ is

y (2, r) = [(9.5992 r - 2.3415), (- 10.4312 r + 17.6889)], r \in [1.5, 2]

The trial solutions for $x \in [1.5, 2]$ are

\begin{matrix} {\begin{matrix} {\underline{y}}_{T} (x) = (9.5992 r - 2.3415) + (x - 1.5) \underline{N} (x, r, \underline{v}) \\ {\bar{y}}_{T} (x) = (- 10.4312 r + 17.6889) + (x - 1.5) \bar{N} (x, r, \bar{v}) \end{matrix} \end{matrix}

The numerical results of example are given in Tables 1 –3. The exact and approximate solutions by artificial neural network are compared and plotted at $x = 1$ , $x = 1.5$ , and $x = 2$ in Figures 2 –4, respectively. It is deduced that the results are very close to the exact solutions which confirm the validity and feasibility of this method.

Table 1.

Comparison of the approximated solutions and the exact solutions on [0, 1].

$r$	Exact solution $({\underline{y}}_{a} (1, r), {\bar{y}}_{a} (1, r))$	Approximated solution $({\underline{y}}_{T} (1, r), {\bar{y}}_{T} (1, r))$	Absolute error
0.0	(–0.67957045, 5.0967784)	(–0.67957036, 5.0967784)	(6.215545e−07, 1.875812e−06)
0.1	(–0.40774227, 4.7909717)	(–0.40774282, 4.7909717)	(2.279674e−06, 8.964055e−07)
0.2	(–0.13591409, 4.4851650)	(–0.13591370, 4.4851650)	(1.967348e−07, 2.704311e−07)
0.3	(0.13591409, 4.1793583)	(0.13591389, 4.1793583)	(1.503723e−06, 2.986754e−07)
0.4	(0.40774227, 3.8735516)	(0.40774239, 3.8735516)	(1.733932e−06, 1.385048e−06)
0.5	(0.67957045, 3.5677448)	(0.67957043, 3.5677448)	(7.522059e−07, 1.104163e−06)
0.6	(0.95139863, 3.2619381)	(0.95139853, 3.2619382)	(2.363650e−07, 5.354381e−07)
0.7	(1.22322682, 2.9561314)	(1.12232269, 2.9561314)	(1.381800e−07, 5.435571e−07)
0.8	(1.49505500, 2.6503247)	(1.4950549, 2.6503247)	(2.570600e−07, 2.209716e−06)
0.9	(1.76688318, 2.3445180)	(1.7668831, 2.3445180)	(6.189354e−07, 7.521389e−08)
1.0	(2.03871137, 2.03871137)	(2.0387113, 2.0387113)	(4.528218e−06, 4.528218e−06)

Table 2.

Comparison of the approximated solutions and the exact solutions on [1, 1.5].

$r$	Exact solution $({\underline{y}}_{a} (1.5, r), {\bar{y}}_{a} (1.5, r))$	Approximated solution $({\underline{y}}_{T} (1.5, r), {\bar{y}}_{T} (1.5, r))$	Absolute error
0.0	(–1.3227000, 9.9202000)	(–1.3226999, 9.9202001)	(5.689375e−07, 1.803075e−07)
0.1	(–0.79360000, 9.93249699)	(–0.79359992, 9.3249704)	(5.323851e−07, 1.929814e−07)
0.2	(–0.26450000, 8.7297400	(–0.26450009, 8.7297400)	(8.652878e−08, 3.078635e−07)
0.3	(0.26460000, 8.1345099)	(0.26459997, 8.1345101)	(1.503723e−06, 2.907790e−07)
0.4	(0.79370000, 7.5392800)	(0.79369997, 7.5392796)	(3.160333e−08, 4.194941e−07)
0.5	(1.32280000, 6.9440499)	(1.3227999, 6.9440496)	(1.548584e−07, 1.045301e−06)
0.6	(1.85190000, 6.3488200)	(1.8519000, 6.3488203)	(1.394677e−07, 7.646418e−07)
0.7	(2.38100000, 5.7535900)	(2.3809999, 5.7535900)	(4.624456e−08, 3.001244e−07)
0.8	(2.91010000, 5.1583599)	(2.9101000, 5.1583601)	(7.806130e−08, 4.992620e−07)
0.9	(3.43920000, 4.5631299)	(3.4392000, 4.5631302)	(2.682584e−07, 1.344480e−07)
1.0	(3.96830000, 3.9678999)	(3.9682999, 3.9678999)	(2.320238e−07, 0.256288e−08)

Table 3.

Comparison of the approximated solutions and the exact solutions on [1.5, 2].

$r$	Exact solution $({\underline{y}}_{a} (2, r), {\bar{y}}_{a} (2, r))$	Approximated solution $({\underline{y}}_{T} (2, r), {\bar{y}}_{T} (2, r))$	Absolute error
0.0	(–2.341500, 17.688900)	(–2.341499, 17.688902)	(6.215545e−07, 1.875812e−06)
0.1	(–1.381580, 16.645780)	(–1.381582, 16.645789)	(2.279674e−06, 8.584055e−07)
0.2	(–0.421660, 15.602660	(–0.421661, 15.602662)	(1.967348e−07, 2.350311e−07)
0.3	(0.538260, 14.559540)	(0.538262, 14.559541)	(1.503723e−06, 0.211054e−07)
0.4	(1.498180, 13.516420)	(1.498185, 13.516421)	(1.733932e−06, 1.315011e−06)
0.5	(2.458100, 12.473300)	(2.458099, 12.473301)	(1.531159e−07, 1.104163e−06)
0.6	(3.418019, 11.430180)	(3.418017, 11.430185)	(2.301750e−07, 5.354311e−07)
0.7	(4.377940, 10.387060)	(4.377947, 10.387059)	(7.801521e−07, 8.412501e−07)
0.8	(5.337860, 9.343940)	(5.337862, 9.343942)	(1.301100e−07, 2.201216e−06)
0.9	(6.297780, 8.300820)	(6.297786, 8.300821)	(6.189354e−07, 1.521389e−08)
1.0	(7.257731, 7.257731)	(7.257732, 7.257732)	(1.719323e−07, 1.719218e−07)

Figure 2.

Comparison exact solution and approximated solution ∘ in $x = 1$ .

Figure 3.

Comparison exact solution and approximated solution ∘ in $x = 1.5$ .

Figure 4.

Comparison exact solution and approximated solution ∘ in $x = 2$ .

Conclusion

In summary, we defined a new method for solving HFDEs. We demonstrated the ability of neural networks for approximating the solutions of FDEs. By comparing our achievements with the results obtained using numerical methods, it is clear that our proposed method gives more accurate approximations. Also better results (specially in nonlinear cases) might be possible if we use more neurons or training points. In addition, after solving a FDE, we obtained the solution at any arbitrary point in the training interval (even between training points). Applicability in function approximations of neutral networks is the main reason for using neural networks. More research is in progress for applying and extending this new approach for solving nth-order FDEs as well as a system of FDEs. The numerical results showed that the method has good accuracy and it is efficient.

Footnotes

Academic Editor: Dumitru Baleanu

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The second author, Massimiliano Ferrara, received a financial support from the iritMED, Istituto di Ricerca per l Innovazione e la Tecnologia nel Mediterraneo, for publication of this article.

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Approximate solutions by artificial neural network of hybrid fuzzy differential equations

Abstract

Keywords

Introduction

Preliminaries

Definition 1

Definition 2

Definition 3

Definition 4

Definition 5

Definition 6

Definition 7

Theorem 1 (Bede’s characterization theorem 21 )

The hybrid fuzzy differential system

Theorem 2

Neural networks

Numerical example

Example

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References

Theorem 1 (Bede’s characterization theorem²¹)