Numerical Treatment of Hybrid Fuzzy Differential Equations Subject to Trapezoidal and Triangular Fuzzy Initial Conditions Using Picard’s and the General Linear Method

Abstract

We study hybrid fuzzy differential equations (HFDEs) under the Hukuhara derivative numerically using Picard’s and the general linear method (GLM). We use trapezoidal and triangular fuzzy numbers as the initial conditions. To demonstrate the efficiency of the proposed methods, the exact as well as the numerical solutions are presented numerically and graphically. In addition, a comparison is made between the results from applying the GLM and those obtained when applying the fifth order Runge–Kutta method as reported in the literature.

1. Introduction

Hybrid systems are often used in modeling systems that combine discrete event dynamics and continuous time dynamics. The term “hybrid fuzzy differential systems” refers to systems containing uncertainty—that is, fuzzy-valued functions interacting with a discrete time controller [1].

In most situations of real life, uncertainty can appear when modeling real-world problems. In such cases, fuzzy numbers have been used to obtain better results in problems where decision making and analysis are involved. A fuzzy number, which is an extension of a real number, has its own properties that can be related to number theory [2]. The concept of the fuzzy derivative was introduced by many researchers (see, for example [3]), while in [4], the authors proposed a new concept of derivatives based on the Hukuhara difference.

Differential equations also play a vital role in modeling real-world phenomena. In most cases, the initial conditions are assumed to be exact. In reality, there are certain errors in the initial conditions, which might be due to errors in observations, measurements or experiments. This may lead to fuzzy, incomplete or incorrect information. Fuzzy differential equations (FDEs) can be used to overrule such uncertainties or lack of precision; more on fuzzy differential equations and fuzzy Hukuhara derivatives and their generalizations can be found in [4,5,6].

Of particular interest is the use of hybrid fuzzy differential equations (HFDEs) as a natural way to model control systems with embedded uncertainty. In general, the initial value problems for hybrid differential equations have the form (HDE) [7,8,9],

$$\begin{array}{ccc}\hfill \frac{d}{dt}\left[y\left(t\right)-f\left(t,y\left(t\right)\right)\right]& =& g\left(t,x\left(t\right)\right),t\in \left[0,T\right]\hfill \\ \hfill y\left(t_0\right)& =& y_0\in R\hfill \end{array}$$

where $f,g:[0,T]\times \mathbb{R}\to \mathbb{R}$ are continuous functions.

The hybrid fuzzy differential systems under consideration have the form

$$\begin{array}{ccc}\hfill y^{{}^{\prime }}\left(t\right)& =& f\left(t,y,{\lambda }_k\left(y_k\right)\right),t\in \left[t_k,t_{k+1}\right],\hfill \\ \hfill y\left(t_k\right)& =& y_k\hfill \end{array}$$

where $0\le t_0<t_1<\cdots <t_k<\cdots ,$ and $t_k\to \infty$, $f\in C\left[{\mathbb{R}}^{+}\times E\times E,E\right],$${\lambda }_k\in C\left[E,E\right],$ and E is the set of all fuzzy numbers.

The solution of the HFDEs plays a major role in applied sciences and engineering. Of special importance is when modeling the interactive systems of computer of computer programs and continuous time dynamics. Due to this importance and since few of these differential equations can have a closed form solution, in real-world situations, numerical solutions are utilized to approximate the solutions of these problems.

Many authors have considered solving this system numerically; for example, in [10,11], the authors proposed Taylor’s method and the Runge–Kutta method for solving FDEs, and in [12,13] the authors used the Euler and Runge–Kutta methods for solving HFDEs, while [14,15,16] studied numerical methods for HFDEs by applying the Runge–Kutta method of order 5. In [17], the authors proposed an improved predictor–corrector (IPC) method to solve HFDEs, [1] used the Trapezoidal rule, and [18] used Picard’s method. The authors in [12,19] employed the Taylor series method, while [20] used differential transforms to solve the problem. Recently, reproducing the kernel Hilbert space technique was used by [21].

In this paper, we utilize Picard’s method and the GLM to solve the HFDE under generalized Hukuhara derivatives. For the initial conditions, we use triangular, triangular shaped and trapezoidal fuzzy numbers. To the best of our knowledge, this is the first study to use triangular and trapezoidal fuzzy numbers as initial conditions. We consider two examples and illustrate the results both numerically and graphically. To demonstrate the efficiency of the accuracy of the numerical results obtained, we compare them with the exact solution in addition to the results obtained using the fifth order Runge–Kutta method proposed by [14,15,16].

The outline of the paper is as follows: In Section 2, we define the fuzzy sets and fuzzy numbers and present some of their properties. In Section 3, we present the hybrid fuzzy differential system. The numerical methods are presented in Section 4 with our numerical experimentation and the comparison. Our conclusions are given in the last section.

2. Fuzzy Sets and Fuzzy Numbers

Before considering the hybrid fuzzy differential equations, we start with some basics of fuzzy set theory; refer to [6] for more.

Definition 1.

If X is a collection of objects denoted generically by x, then a fuzzy set A in X is a set of ordered pairs:

$$A=\left\{(x,{\mu }_A\left(x\right))\left|x\in X\right.\right\}$$

${\mu }_A\left(x\right)$ is called the membership function, where

$${\mu }_A\left(x\right)=\left\{\begin{array}{c}1,x\in A\\ 0,x\notin A.\end{array}\right.$$

Note that if A is a fuzzy subset of X, then the $\alpha$-cut set denoted by $A_{\alpha }$ is made up of members whose membership is not less than $\alpha$, where $A_{\alpha }=\{x\in X\left|{\mu }_A\left(x\right)\ge \alpha \right.\}$, and $\alpha$ can be chosen arbitrary between $0<\alpha \le 1$.

Definition 2.

A fuzzy number $\mathrm{A}$ is a fuzzy subset of the real numbers satisfying: $\left(i\right)\exists x:A\left(x\right)$$=1,$ and $\left(ii\right)$$A_{\alpha }$ is a closed and bounded interval for $0\le \alpha \le 1$.

The family of all fuzzy numbers is denoted by $R_F.$

Definition 3.

(Triangular fuzzy number) This is a fuzzy number represented with three points as follows: $A=\left(a_1,a_2,a_3\right)$

$${\mu }_{\left(A\right)}\left(x\right)=\left\{\begin{array}{c}0,x<a_1\hfill \\ \frac{x-a_1}{a_2-a_1},a_1\le x<a_2\hfill \\ \frac{a_3-x}{a_3-a_2},a_2\le x<a_3\hfill \\ 0,a_3\le x.\hfill \end{array}\right.$$

$A_{\alpha }=\left[\mathrm{a}\text{̠}\left(\alpha \right),\overline{a}\left(\alpha \right)\right]$, where

$$\mathrm{a}\text{̠}\left(\alpha \right)=a_1+\left(a_2-a_1\right)\alpha ,\overline{a}\left(\alpha \right)=a_3-\left(a_3-a_2\right)\alpha ,\forall \alpha \in \left[0,1\right].$$

Definition 4.

(Trapezoidal fuzzy number) A trapezoidal fuzzy number $A$ can be defined as $A=(a_1,a_2,a_3,a_4)$, where

$${\mu }_A\left(x\right)=\left\{\begin{array}{c}0,x<a_1\hfill \\ \frac{x-a_1}{a_2-a_1},a_1\le x<a_2\hfill \\ 1,a_2\le x<a_3\hfill \\ \frac{a_4-x}{a_4-a_3},a_3\le x<a_4\hfill \\ 0,a_4\le x.\hfill \end{array}\right.$$

The α- cut interval for this shape is

$$A_{\alpha }=\left[\left(a_2-a_1\right)\alpha +a_1,a_4-\left(a_4-a_3\right)\alpha \right],\forall \alpha \in \left[0,1\right].$$

Theorem 1.

If u and v are two fuzzy numbers and μ is real number, then, for each $\alpha \in [0,1]$, we have:

• ${\left[u+v\right]}_{\alpha }={\left[u\right]}_{\alpha }+{\left[v\right]}_{\alpha }=\left[u\text{̠}\left(\alpha \right)+v\text{̠}\left(\alpha \right),\overline{u}\left(\alpha \right)+\overline{v}\left(\alpha \right)\right]$
• ${\left[\mu u\right]}_{\alpha }=\mu {\left[u\right]}_{\alpha }=[min\{\mu u\text{̠}$$\left(\alpha \right),\mu \overline{u}\left(\alpha \right)\},max\{\mu u\text{̠}\left(\alpha \right),\mu \overline{u}\left(\alpha \right)\}]$.
• ${\left[uv\right]}_{\alpha }=[min\{u\text{̠}\left(\alpha \right)v\text{̠}$$\left(\alpha \right),u\text{̠}$$\left(\alpha \right)\overline{v}\left(\alpha \right),\overline{u}\left(\alpha \right)v\text{̠}$$\left(\alpha \right),$
  $\overline{u}\left(\alpha \right)\overline{v}\left(\alpha \right)\},max\{u\text{̠}\left(\alpha \right)v\text{̠}\left(\alpha \right),u\text{̠}\left(\alpha \right)\overline{v}\left(\alpha \right),\overline{u}\left(\alpha \right)v\text{̠}\left(\alpha \right),\overline{u}\left(\alpha \right)\overline{v}\left(\alpha \right)\}].$

Definition 5.

If $f:[a,b]\to R_F$ is a fuzzy function, then f is called Hukuhara-differentiable at $x_0$ (H-differentiable), and the derivative $f^{\prime }\left(x_0\right)$ is defined by

$$f^{\prime }\left(x_0\right)=\frac{f\left(x_0+h\right)⊝f\left(x_0\right)}{h},f^{\prime }\left(x_0\right)=\frac{f\left(x_0\right)⊝f(x_0-h)}{h}$$

If the limits exist and are equal, then the derivative $f^{\prime }\left(x_0\right)$ exists, where ⊝ is the Hukuhara difference.

Definition 6.

Let $f:[a,b]\to R_F$. Then, f is strongly generalized differentiable ($GH$-differentiable) at $x_0$ if the limits of some pair of the following exist and are equal:

$$1-\underset{h\to 0}{lim}\frac{f\left(x_0+h\right)⊝f\left(x_0\right)}{h}\mathit{and}\underset{h\to 0}{lim}\frac{f\left(x_0\right)⊝f\left(x_0-h\right)}{h}.$$

$$2-\underset{h\to 0}{lim}\frac{f\left(x_0\right)⊝f\left(x_0+h\right)}{-h}\mathit{and}\underset{h\to 0}{lim}\frac{f\left(x_0-h\right)⊝f\left(x_0\right)}{-h}.$$

$$3-\underset{h\to 0}{lim}\frac{f\left(x_0+h\right)⊝f\left(x_0\right)}{h}\mathit{and}\underset{h\to 0}{lim}\frac{f\left(x_0-h\right)⊝f\left(x_0\right)}{-h}.$$

$$4-\underset{h\to 0}{lim}\frac{f\left(x_0\right)⊝f\left(x_0+h\right)}{-h}\mathit{and}\underset{h\to 0}{lim}\frac{f\left(x_0\right)⊝f\left(x_0-h\right)}{-h}$$

Definition 7.

Let $f:[a,b]\to R_F$. Then, f is 1-differentiable on $[a,b]$ if f is differentiable in the sense (1) of Definition 7. Similarly, f is 2-differentiable on $[a,b]$ if f is differentiable in the sense (2) of Definition 7.

Theorem 2.

Let $f:[a,b]\to R_F$, where ${\left[f\right(x\left)\right]}_{\alpha }=[{\underline{f}}_{\alpha }\left(x\right),{\overline{f}}_{\alpha }\left(x\right)]$ for each $\alpha \in [0,1]$,

• If f is $1-$ differentiable, then $f{̠}_{\alpha }$ and ${\overline{f}}_{\alpha }$ are differentiable functions and ${\left[f^{\prime }\left(x\right)\right]}_{\alpha }$$=[f{̠}_{\alpha }^{\prime }\left(x\right),{{\overline{f}}^{\prime }}_{\alpha }\left(x\right)]$.
• If f is 2-differentiable, then $f{̠}_{\alpha }$ and ${\overline{f}}_{\alpha }$ are differentiable functions and ${\left[f^{\prime }\left(x\right)\right]}_{\alpha }$$=[{{\overline{f}}^{\prime }}_{\alpha }\left(x\right),{\underset{¯}{f}}_{\alpha }^{\prime }\left(x\right)]$.

3. Fuzzy Differential Equations

The first order differential equation will be explained for when using fuzzy numbers and fuzzy derivatives.

Let $y:I\to R_F$, where $I\subset \mathbb{R}$ is an interval. If $y\left(t,\alpha \right)=\left[\left(t,\alpha \right),\overline{y}\left(t,\alpha \right)\right]$ for all $t\in I$ and $\alpha \in [0,1],$ then $y^{{}^{\prime }}\left(t,\alpha \right)=\left[{\underset{¯}{y}}^{\prime }\left(t,\alpha \right),{\overline{y}}^{{}^{\prime }}\left(t,\alpha \right)\right]$ if $y^{{}^{\prime }}\in R_F$. Next, consider the initial value problem (IVP)

$$u\left(y\right)=\left\{\begin{array}{c}{y}^{{}^{\prime }}\left(t\right)=f\left(t,y\left(t\right)\right),\\ y\left(0\right)=y_0,\end{array}\right.$$

(1)

where $f:\left[0,\infty \right)\times \mathbb{R}\to \mathbb{R}$ is continuous. Let $y_0\left(\alpha \right)=\left[{\underset{¯}{y}}_0\left(\alpha \right),{\overline{y}}_0\left(\alpha \right)\right]$ and $y\left(t,\alpha \right)$$=\left[\underset{¯}{y}\left(t,\alpha \right),\overline{y}\left(t,\alpha \right)\right],$ and we obtain $f:[0,\infty )\times R_F\to R_F$, where

$${\left[f\left(t,y\right)\right]}^{\alpha }=[min\{f\left(t,u\right):u\in \left[{\underset{¯}{y}}^{\alpha }\left(t\right),{\overline{y}}^{\alpha }\left(t\right)\right]\},max\left\{f\left(t,u\right):u\in \left[{\underset{¯}{y}}^{\alpha }\left(t\right),{\overline{y}}^{\alpha }\left(t\right)\right]\right\}].$$

With $y_0\in R_F$, then $y:[0,\infty )\to R_F$ is a solution of (1) if

$$({{\underset{¯}{y}}^{\alpha })}^{{}^{\prime }}\left(t\right)=min\{f\left(t,u\right):u\in \left[{\underset{¯}{y}}^{\alpha }\left(t\right),{\overline{y}}^{\alpha }\left(t\right)\right]\},{\underset{¯}{y}}^{\alpha }\left(0\right)={\underset{¯}{y}}_0^{\alpha }$$

$${{(\overline{y}}^{\alpha })}^{{}^{\prime }}\left(t\right)=max\{f\left(t,u\right):u\in \left[{\underset{¯}{y}}^{\alpha }\left(t\right),{\overline{y}}^{\alpha }\left(t\right)\right]\},{\overline{y}}^{\alpha }\left(0\right)={\overline{y}}_0^{\alpha }$$

for all $t\in [0,\infty )$ and $\alpha \in$[0,1].

Lastly, consider an $f:[0,\infty )\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$, which is continuous and satisfies the IVP:

$$\left\{\begin{array}{c}{y}^{{}^{\prime }}\left(t\right)=f\left(t,y\left(t\right),k\right),\\ y\left(0\right)=y_0.\end{array}\right.$$

With $x_0,k\in R_F$, we have $f:\left[0,\infty \right)\times R_F\times R_F\to R_F$, where

$${\left[f\left(t,y,k\right)\right]}^{\alpha }=[min\left\{f\left(t,u,u_k\right):u\in \left[{\underset{¯}{y}}^{\alpha }\left(t\right),{\overline{y}}^{\alpha }\left(t\right)\right],u_k\in \left[{\underset{¯}{k}}^{\alpha },{\overline{k}}^{\alpha }\right]\right\},$$

$$\left\{f\left(t,u,u_k\right):u\in \left[{\underset{¯}{y}}^{\alpha }\left(t\right),{\overline{y}}^{\alpha }\left(t\right)\right],u_k\in \left[{\underset{¯}{k}}^{\alpha },{\overline{k}}^{\alpha }\right]\right\}]$$

and $k^{\alpha }=[$${\underset{¯}{k}}^{\alpha },{\overline{k}}^{\alpha }]$. Then, $y:[0,\infty )\to R_F$ is a solution of (2), and $y_0,k\in R_F$ if

$${\left({\underset{¯}{y}}^{\alpha }\right)}^{{}^{\prime }}\left(t\right)=min\left\{\mathrm{f}\left(t,u,u_k\right):u\in \left[{\underset{¯}{y}}^{\alpha }\left(t\right),{\overline{y}}^{\alpha }\left(t\right)\right],u_k\in \left[{\underset{¯}{k}}^{\alpha },{\overline{k}}^{\alpha }\right]\right\},{\underset{¯}{y}}^{\alpha }\left(0\right)={\underset{¯}{y}}_0^{\alpha }$$

$${\left({\overline{y}}^{\alpha }\right)}^{\prime }\left(t\right)=max\left\{\mathrm{f}\left(t,u,u_k\right):u\in \left[{\underset{¯}{y}}^{\alpha }\left(t\right),{\overline{y}}^{\alpha }\left(t\right)\right],u_k\in \left[{\underset{¯}{k}}^{\alpha },{\overline{k}}^{\alpha }\right]\right\},{\overline{y}}^{\alpha }\left(0\right)={\overline{y}}_0^{\alpha }$$

for all $t\in [0,\infty )$ and $\alpha \in [0,1].$

4. Hybrid Fuzzy Differential Equations (HFDEs)

Here, the fuzzy differential equation is used when interacting with a discrete time controller. The HFDE has the form

$$\begin{array}{ccc}\hfill y^{{}^{\prime }}\left(t\right)& =& f\left(t,y\left(t\right),{\lambda }_k\left(y_k\right)\right),t\in \left[t_k,t_{k+1}\right],\hfill \\ \hfill y\left(t_k\right)& =& y_k\hfill \end{array}$$

(2)

where the prime “ ${}^{\prime }$ ” denotes fuzzy differentiation, $0\le t_0<t_1<\dots <t_k<\dots ,t_k\to \infty$, $f\in C\left[{\mathbb{R}}^{+}\times R_F\times R_F,R_F\right],$${\lambda }_k\in C\left[R_F,R_F\right].$ To be more specific, the system looks like

$$y^{{}^{\prime }}\left(t\right)=\left\{\begin{array}{c}{y}_0^{{}^{\prime }}\left(t\right)=f\left(t,y_0\left(t\right),{\lambda }_0\left(y_0\right)\right),y_0\left(t_0\right)=y_0,t_0\le t<t_1,\\ y_1^{{}^{\prime }}\left(t\right)=f\left(t,y_1\left(t\right),{\lambda }_1\left(y_1\right)\right),y_1\left(t_1\right)=y_1,t_1\le t<t_2,\\ \dots \\ y_k^{{}^{\prime }}\left(t\right)=f\left(t,y_k\left(t\right),{\lambda }_k\left(y_k\right)\right),y_k\left(t_k\right)=y_k,t_k\le t<t_{k+1}.\\ \dots \end{array}\right.$$

Assuming that the existence and uniqueness of solution of (2) hold for each $\left[t_k,t_{k+1}\right],$ by the solution of (2), we mean

$$y\left(t\right)=y\left(t,t_0,y_0\right)=\left\{\begin{array}{c}{y}_0\left(t\right),t_0\le t\le t_1\\ y_1\left(t\right),t_1\le t\le t_2\\ \dots \\ y_k\left(t\right),t_k\le t\le t_{k+1}\\ \dots \end{array}\right.$$

Note that the solutions are piece-wise differentiable on each sub-interval for $t\in \left[t_k,t_{k+1}\right]$ and a fixed $y_k\in R_F$ and $k=0,1,2,\dots$. Using the representation of fuzzy numbers, we may represent $y\in R_F$ by a pair of functions $\left(\underset{¯}{y}\left(\alpha \right),\overline{y}\left(\alpha \right)\right),0\le \alpha \le 1,$ such that

(i)

y$\left(\alpha \right)$ is bounded, left continuous and non-decreasing.

(ii)

$\overline{y}\left(\alpha \right)$ is bounded, left continuous and non-increasing.

(iii)

y$\left(\alpha \right)\le \overline{y}\left(\alpha \right),0\le \alpha \le 1.$

Therefore,

$$\left\{\begin{array}{c}{\underset{¯}{y}}^{\prime }\left(t\right)=\underset{¯}{f}\left(t,y,{\lambda }_k\left(y_k\right)\right)\equiv F_k\left(t,\underset{¯}{y},\overline{y}\right),\underset{¯}{y}\left(t_k\right)={\underset{¯}{y}}_k\\ {\overline{y}}^{{}^{\prime }}\left(t\right)=\overline{f}\left(t,y,{\lambda }_k\left(y_k\right)\right)\equiv G_k\left(t,\underset{¯}{y},\overline{y}\right),\overline{y}\left(t_k\right)={\overline{y}}_k\end{array}\right.$$

possesses a unique solution $(\underset{¯}{y},\overline{y})$, which is a fuzzy function—that is, for each t, the pair $\left[\underset{¯}{y}\left(t,\alpha \right),\overline{y}\left(t,\alpha \right)\right]$ is a fuzzy number, where $\underset{¯}{y}\left(t,\alpha \right)$ and $\overline{y}\left(t,\alpha \right)$ are, respectively, the solutions of the parametric form given by

$$\begin{array}{ccc}\hfill {\underset{¯}{y}}^{\prime }\left(t,\alpha \right)& =& F_k\left[t,\underset{¯}{y}\left(t,\alpha \right),\overline{y}\left(t,\alpha \right)\right],\underset{¯}{y}\left(t_k,\alpha \right)={\underset{¯}{y}}_k\left(\alpha \right)\hfill \\ \hfill {\overline{y}}^{{}^{\prime }}\left(t,\alpha \right)& =& G_k\left[t,\underset{¯}{y}\left(t,\alpha \right),\overline{y}\left(t,\alpha \right)\right],\overline{y}\left(t_k,\alpha \right)={\overline{y}}_k\left(\alpha \right)\hfill \end{array}$$

for $\alpha \in [0,1]$.

5. Numerical Details and Examples

5.1. Numerical Methods

To approximate the solution of the HFDE, we will employ two methods—namely, Picard’s method and the general linear method (GLM).

• Picard’s Method:
  This is an iterative method for solving initial value problems of the form

  $$y^{{}^{\prime }}=f\left(t,y\left(t\right)\right),t\ge t_0,y\left(t_0\right)=y_0=\alpha .$$

  (3)
  It uses successive approximations until convergence, thereby, resulting in better approximations with more iterations. To approximate the solution of the above initial value problem, the initial value problem (3) can be rewritten as an integral equation of the form

  $$y\left(t\right)=y_0+{\int }_{t_0}^{t}f\left(\tau ,y\left(\tau \right)\right)d\tau .$$
  From this equation a sequence of approximations can be obtained as

  $$y_{i+1}\left(t\right)=y_0+{\int }_{t_0}^{t}f\left(\tau ,y_i\left(\tau \right)\right)d\tau .$$

  (4)
  Given $y_i$, one can produce $y_{i+1}$. This process is continued to some tolerance. For our purpose, the 1-differentiable and 2-differentiable problems will have the forms:
  The 1-differentiable:

  $$\begin{array}{c}\hfill {\underline{y}}_{n+1}={\underline{y}}_0+{\int }_{t_0}^{t}f\left(\tau ,{\underline{y}}_n\left(\tau ,\alpha \right),{\lambda }_k\left(y_k\right)\right)d\tau \\ \hfill {\overline{y}}_{n+1}={\overline{y}}_0+{\int }_{t_0}^{t}f\left(\tau ,{\overline{y}}_n\left(\tau ,\alpha \right),{\lambda }_k\left(y_k\right)\right)d\tau .\end{array}$$

  (5)
  The 2-differentiable:

  $$\begin{array}{c}\hfill {\underline{y}}_{n+1}={\underline{y}}_0+{\int }_{t_0}^{t}f\left(\tau ,{\overline{y}}_n\left(\tau ,\alpha \right),{\lambda }_k\left(y_k\right)\right)d\tau \\ \hfill {\overline{y}}_{n+1}={\overline{y}}_0+{\int }_{t_0}^{t}f\left(\tau ,{\underline{y}}_n\left(\tau ,\alpha \right),{\lambda }_k\left(y_k\right)\right)d\tau ,\\ \hfill n=0,1,\cdots ,N\end{array}$$

  (6)
  Using results from theory of differential equations, it can be proven that the sequence of approximations converges to the exact solution of IVP.
• General Linear Method (GLM):
  The name “general linear method ” applies to a large group of numerical methods for ordinary differential equations; more on these methods can be found in [22,23,24]. We will consider r-value s-stage methods, where $r=1$ for the Runge–Kutta methods and $s=1$ for the linear multi-step methods. Each step of the computation takes, as input, a certain number $\left(r\right)$ of items of data and generates, for output, the same number of items. The output items correspond to the input items but are advanced through one time step $\left(h\right).$ Within a step, a certain number $\left(s\right)$ of stages of computations are performed.

We now present a GLM based on linear k-step Adams–Bashforth-type schemes for solving fuzzy initial value problems. Assume that, for equally spaced points $0=t_0<t_1<...<t_N=T$ at $t_n$, the exact solutions are indicated by $Y\left(t_n,\alpha \right)=\left[\underline{Y}\left(t_n,\alpha \right),\overline{Y}\left(t_n,\alpha \right)\right]$ and that $y\left(t_n,\alpha \right)=\left[\underline{y}\left(t_n,\alpha \right),\overline{y}\left(t_n,\alpha \right)\right]$ are the approximate solutions. The k-step Adams–Bashforth methods can be written as:

The 1-differentiable system:

$$\left\{\begin{array}{c}{\underline{y}}_{\alpha }\left(t_{n+k},\alpha \right)={\underline{y}}_{\alpha }\left(t_{n+k-1},\alpha \right)+h\sum _{j=0}^k{\beta }_{j}f\left(t_{n+j},{\underline{y}}_{\alpha }\left(t_{n+j},\alpha \right)\right)\\ {\overline{y}}_{\alpha }\left(t_{n+k},\alpha \right)={\overline{y}}_{\alpha }\left(t_{n+k-1},\alpha \right)+h\sum _{j=0}^k{\beta }_{j}f\left(t_{n+j},{\overline{y}}_{\alpha }\left(t_{n+j},\alpha \right)\right)\end{array}\right.$$

and the 2-differentiable system:

$$\left\{\begin{array}{c}{\underline{y}}_{\alpha }\left(t_{n+k},\alpha \right)={\underline{y}}_{\alpha }\left(t_{n+k-1},\alpha \right)+h\sum _{j=0}^k{\beta }_{j}f\left(t_{n+j},{\overline{y}}_{\alpha }\left(t_{n+j},\alpha \right)\right)\\ {\overline{y}}_{\alpha }\left(t_{n+k},\alpha \right)={\overline{y}}_{\alpha }\left(t_{n+k-1},\alpha \right)+h\sum _{j=0}^k{\beta }_{j}f\left(t_{n+j},{\underline{y}}_{\alpha }\left(t_{n+j},\alpha \right)\right)\end{array}\right.$$

Now, the input and output approximations for the general linear methods are, respectively,

$$y^{[n-1]}=\left[\begin{array}{c}{y}_{n+k-1}\\ {hf}_{n+k-1}\\ {hf}_{n+k-2}\\ ⋮\\ {hf}_{n+1}\\ {hf}_n\end{array}\right],y^{\left[n\right]}=\left[\begin{array}{c}{y}_{n+k}\\ {hf}_{n+k}\\ {hf}_{n+k-1}\\ ⋮\\ {hf}_{n+2}\\ {hf}_{n+1}\end{array}\right]$$

Under the 1- and 2-differentiability, the representations of the input vectors for the GLM are given by $y_{1\alpha }^{[n-1]}=\left[{\underline{y}}_{1\alpha }^{[n-1]},{\overline{y}}_{1\alpha }^{[n-1]}\right]$ and $y_{2\alpha }^{[n-1]}=\left[{\underline{y}}_{2\alpha }^{[n-1]},{\overline{y}}_{2\alpha }^{[n-1]}\right]$, while the corresponding output vectors are indicated by $y_{1\alpha }^{\left[n\right]}=\left[{\underline{y}}_{1\alpha }^{\left[n\right]},{\overline{y}}_{1\alpha }^{\left[n\right]}\right]$ and $y_{2\alpha }^{\left[n\right]}=\left[{\underline{y}}_{2\alpha }^{\left[n\right]},{\overline{y}}_{2\alpha }^{\left[n\right]}\right]$ under 1 and 2-differentiability, respectively. Now, we consider the input approximation of the general linear methods in terms of 1-differentiability as:

$${\underline{y}}_{1\alpha }^{[n-1]}=\left[\begin{array}{c}{{\underline{y}}_{n+k-1}}_{1\alpha }\\ {{h\underline{f}}_{n+k-1}}_{1\alpha }\\ {{h\underline{f}}_{n+k-2}}_{1\alpha }\\ ⋮\\ h{{\underline{f}}_{n+1}}_{1\alpha }\\ {{h\underline{f}}_n}_{1\alpha }\end{array}\right],{\overline{y}}_{1\alpha }^{[n-1]}=\left[\begin{array}{c}{{\overline{y}}_{n+k-1}}_{1\alpha }\\ {{h\overline{f}}_{n+k-1}}_{1\alpha }\\ {{h\overline{f}}_{n+k-2}}_{1\alpha }\\ ⋮\\ h{{\overline{f}}_{n+1}}_{1\alpha }\\ {{h\overline{f}}_n}_{1\alpha }\end{array}\right]$$

and under 2-differentiability, we obtain the following input vectors:

$${\underline{y}}_{2\alpha }^{[n-1]}=\left[\begin{array}{c}{{\underline{y}}_{n+k-1}}_{2\alpha }\\ {{h\underline{f}}_{n+k-1}}_{2\alpha }\\ {{h\underline{f}}_{n+k-2}}_{2\alpha }\\ ⋮\\ h{{\underline{f}}_{n+1}}_{2\alpha }\\ {{h\underline{f}}_n}_{2\alpha }\end{array}\right],{\overline{y}}_{2\alpha }^{[n-1]}=\left[\begin{array}{c}{{\overline{y}}_{n+k-1}}_{2\alpha }\\ {{h\overline{f}}_{n+k-1}}_{2\alpha }\\ {{h\overline{f}}_{n+k-2}}_{2\alpha }\\ ⋮\\ h{{\overline{f}}_{n+1}}_{2\alpha }\\ {{h\overline{f}}_n}_{2\alpha }\end{array}\right]$$

Using the above input vectors, the fuzzy general linear methods can be formulated under 1-differentiability as:

$$\left[\begin{array}{c}{Y}_{1\alpha }\\ y_{1\alpha }^{\left[n\right]}\end{array}\right]=\left[\begin{array}{c}\begin{array}{cc}A& U\end{array}\\ \begin{array}{cc}B& V\end{array}\end{array}\right]\left[\begin{array}{c}h{f}_{1\alpha }\left(Y_{1\alpha }\right)\\ y_{1\alpha }^{[n-1]}\end{array}\right]$$

and under 2-differentiability as:

$$\left[\begin{array}{c}{Y}_{2\alpha }\\ y_{2\alpha }^{\left[n\right]}\end{array}\right]=\left[\begin{array}{c}\begin{array}{cc}A& U\end{array}\\ \begin{array}{cc}B& V\end{array}\end{array}\right]\left[\begin{array}{c}h{f}_{2\alpha }\left(Y_{2\alpha }\right)\\ y_{2\alpha }^{[n-1]}\end{array}\right],$$

where $Y_{1\alpha }=\left[{\underline{Y}}_{1\alpha },{\overline{Y}}_{1\alpha }\right]$ and $Y_{2\alpha }=\left[{\underline{Y}}_{2\alpha },{\overline{Y}}_{2\alpha }\right]$ are internal stages under 1- and 2-differentiability, respectively. Additionally,

$$\left[\begin{array}{c}\begin{array}{cc}A& U\end{array}\\ \begin{array}{cc}B& V\end{array}\end{array}\right]=\left[\begin{array}{cccccc}0& 1& {\beta }_{k-1}& \cdots & {\beta }_1& {\beta }_0\\ 0& 1& {\beta }_{k-1}& \cdots & {\beta }_1& {\beta }_0\\ 1& 0& 0& \cdots & 0& 0\\ 0& 0& 1& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & 1& 0\end{array}\right]$$

We consider two examples of the Fuzzy GLMs form of k-step methods under strongly generalized differentiability for $k=4,5$. First, consider $k=4$. The input vectors for $k=4$ under 1- and 2-differentiability are, respectively,

$$\begin{gathered} {\underline{y}}_{\alpha 1}^{[n-1]}=\left[\begin{array}{c}{\underline{y}}_{\alpha 1}\left(t_{n+3}\right)\\ h{f}_{\alpha 1}\left(t_{n+3},{\underline{y}}_{\alpha 1}\left(t_{n+3}\right)\right)\\ h{f}_{\alpha 1}\left(t_{n+2},{\underline{y}}_{\alpha 1}\left(t_{n+2}\right)\right)\\ h{f}_{\alpha 1}\left(t_{n+1},{\underline{y}}_{\alpha 1}\left(t_{n+1}\right)\right)\\ h{f}_{\alpha 1}\left(t_n,{\underline{y}}_{\alpha 1}\left(t_n\right)\right)\end{array}\right],{\overline{y}}_{\alpha 1}^{[n-1]}=\left[\begin{array}{c}{\overline{y}}_{\alpha 1}\left(t_{n+3}\right)\\ h{f}_{\alpha 1}\left(t_{n+3},{\overline{y}}_{\alpha 1}\left(t_{n+3}\right)\right)\\ h{f}_{\alpha 1}\left(t_{n+2},{\overline{y}}_{\alpha 1}\left(t_{n+2}\right)\right)\\ h{f}_{\alpha 1}\left(t_{n+1},{\overline{y}}_{\alpha 1}\left(t_{n+1}\right)\right)\\ h{f}_{\alpha 1}\left(t_n,{\overline{y}}_{\alpha 1}\left(t_n\right)\right)\end{array}\right] \\ {\underline{y}}_{\alpha 2}^{[n-1]}=\left[\begin{array}{c}{\underline{y}}_{\alpha 2}\left(t_{n+3}\right)\\ h{f}_{\alpha 2}\left(t_{n+3},{\underline{y}}_{\alpha 2}\left(t_{n+3}\right)\right)\\ h{f}_{\alpha 2}\left(t_{n+2},{\underline{y}}_{\alpha 2}\left(t_{n+2}\right)\right)\\ h{f}_{\alpha 2}\left(t_{n+1},{\underline{y}}_{\alpha 2}\left(t_{n+1}\right)\right)\\ h{f}_{\alpha 2}\left(t_n,{\underline{y}}_{\alpha 2}\left(t_n\right)\right)\end{array}\right],{\overline{y}}_{\alpha 2}^{[n-1]}=\left[\begin{array}{c}{\overline{y}}_{\alpha 2}\left(t_{n+3}\right)\\ h{f}_{\alpha 2}\left(t_{n+3},{\overline{y}}_{\alpha 2}\left(t_{n+3}\right)\right)\\ h{f}_{\alpha 2}\left(t_{n+2},{\overline{y}}_{\alpha 2}\left(t_{n+2}\right)\right)\\ h{f}_{\alpha 2}\left(t_{n+1},{\overline{y}}_{\alpha 2}\left(t_{n+1}\right)\right)\\ h{f}_{\alpha 2}\left(t_n,{\overline{y}}_{\alpha 2}\left(t_n\right)\right)\end{array}\right] \end{gathered}$$

and

$$\left[\begin{array}{cccccc}0& 1& \frac{55}{24}& \frac{-59}{24}& \frac{37}{24}& \frac{-9}{24}\\ 0& 1& \frac{55}{24}& \frac{-59}{24}& \frac{37}{24}& \frac{-9}{24}\\ 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\end{array}\right]$$

Similarly, for $k=5$, we obtain

$${\underline{y}}_{1\alpha }^{[n-1]}=\left[\begin{array}{c}{\underline{y}}_{1\alpha }\left(t_{n+4}\right)\\ h{f}_{1\alpha }\left(t_{n+4},{\underline{y}}_{1\alpha }\left(t_{n+4}\right)\right)\\ h{f}_{1\alpha }\left(t_{n+3},{\underline{y}}_{1\alpha }\left(t_{n+3}\right)\right)\\ h{f}_{1\alpha }\left(t_{n+2},{\underline{y}}_{1\alpha }\left(t_{n+2}\right)\right)\\ h{f}_{1\alpha }\left(t_{n+1},{\underline{y}}_{1\alpha }\left(t_{n+1}\right)\right)\\ h{f}_{1\alpha }\left(t_n,{\underline{y}}_{1\alpha }\left(t_n\right)\right)\end{array}\right],{\overline{y}}_{1\alpha }^{[n-1]}=\left[\begin{array}{c}{\overline{y}}_{1\alpha }\left(t_{n+4}\right)\\ h{f}_{1\alpha }\left(t_{n+4},{\overline{y}}_{1\alpha }\left(t_{n+4}\right)\right)\\ h{f}_{1\alpha }\left(t_{n+3},{\overline{y}}_{1\alpha }\left(t_{n+3}\right)\right)\\ h{f}_{1\alpha }\left(t_{n+2},{\overline{y}}_{1\alpha }\left(t_{n+2}\right)\right)\\ h{f}_{1\alpha }\left(t_{n+1},{\overline{y}}_{1\alpha }\left(t_{n+1}\right)\right)\\ h{f}_{1\alpha }\left(t_n,{\overline{y}}_{1\alpha }\left(t_n\right)\right)\end{array}\right]$$

$${\underline{y}}_{2\alpha }^{[n-1]}=\left[\begin{array}{c}{\underline{y}}_{2\alpha }\left(t_{n+4}\right)\\ h{f}_{2\alpha }\left(t_{n+4},{\underline{y}}_{2\alpha }\left(t_{n+4}\right)\right)\\ h{f}_{2\alpha }\left(t_{n+3},{\underline{y}}_{2\alpha }\left(t_{n+3}\right)\right)\\ h{f}_{2\alpha }\left(t_{n+2},{\underline{y}}_{2\alpha }\left(t_{n+2}\right)\right)\\ h{f}_{2\alpha }\left(t_{n+1},{\underline{y}}_{2\alpha }\left(t_{n+1}\right)\right)\\ h{f}_{2\alpha }\left(t_n,{\underline{y}}_{2\alpha }\left(t_n\right)\right)\end{array}\right],{\overline{y}}_{2\alpha }^{[n-1]}=\left[\begin{array}{c}{\overline{y}}_{2\alpha }\left(t_{n+4}\right)\\ h{f}_{2\alpha }\left(t_{n+4},{\overline{y}}_{2\alpha }\left(t_{n+4}\right)\right)\\ h{f}_{2\alpha }\left(t_{n+3},{\overline{y}}_{2\alpha }\left(t_{n+3}\right)\right)\\ h{f}_{2\alpha }\left(t_{n+2},{\overline{y}}_{2\alpha }\left(t_{n+2}\right)\right)\\ h{f}_{2\alpha }\left(t_{n+1},{\overline{y}}_{2\alpha }\left(t_{n+1}\right)\right)\\ h{f}_{2\alpha }\left(t_n,{\overline{y}}_{2\alpha }\left(t_n\right)\right)\end{array}\right]$$

and

$$\left[\begin{array}{ccccccc}0& 1& \frac{1901}{720}& \frac{-2774}{720}& \frac{2616}{720}& \frac{-1274}{7200}& \frac{251}{720}\\ 0& 1& \frac{1901}{720}& \frac{-2774}{720}& \frac{2616}{720}& \frac{-1274}{720}& \frac{251}{720}\\ 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 1& 0\end{array}\right]$$

To use the FGLMs $(k=4,5)$ at step number n, denote the input items by $y_i^{[n-1]}$, $i=1,2,\dots ,r$ and denote the stages computed in the step and the stage derivatives by $Y_i$ and $F_i$, respectively,$i=1,2,\dots ,s.$ With this in mind, we present the methods in compact notation as

$$y^{\left[n-1\right]}=\left[\begin{array}{c}{y}_1^{\left[n-1\right]}\\ y_2^{\left[n-1\right]}\\ ⋮\\ y_r^{\left[n-1\right]}\end{array}\right],y^{\left[n\right]}=\left[\begin{array}{c}{y}_1^{\left[n\right]}\\ y_2^{\left[n\right]}\\ ⋮\\ y_r^{\left[n\right]}\end{array}\right],Y=\left[\begin{array}{c}{Y}_1\\ Y_2\\ ⋮\\ Y_s\end{array}\right],F=\left[\begin{array}{c}{F}_1\\ F_2\\ ⋮\\ F_s\end{array}\right].$$

The stages are computed by the formula

$$Y_i=\sum _{j=1}^s{a}_{\mathrm{ij}}h{F}_j+\sum _{j=1}^r{u}_{\mathrm{ij}}{y}_j^{[n-1]},i=1,2,\dots ,s$$

and the output approximations by the formula

$$y_i^{\left[n\right]}=\sum _{j=1}^s{b}_{\mathrm{ij}}h{F}_j+\sum _{j=1}^r{v}_{\mathrm{ij}}{y}_j^{[n-1]},i=1,2,\dots ,r$$

In each one of the above cases, the coefficients of the general linear formulation are presented in the $\left(s+r\right)\times \left(s+r\right)$ partitioned matrix

$$\left[\begin{array}{c}Y\\ y^{\left[n\right]}\end{array}\right]=\left[\begin{array}{cc}A& U\\ B& V\end{array}\right]\left[\begin{array}{c}hF\\ y^{\left[n-1\right]}\end{array}\right]$$

where $y^{[n-1]}$ and $y^{\left[n\right]}$ are the input and output approximations, respectively, and

$$A\in R^{s\times s},U\in R^{s\times r},B\in R^{r\times s},V\in R^{r\times r}.$$

For example, the matrices representing the Euler method and implicit Euler methods are, respectively,

$$\left[\begin{array}{cc}0& 1\\ 1& 1\end{array}\right]\mathrm{and}\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right].$$

5.2. Numerical Testing

For numerical testing, we consider HFDEs with selected types of fuzzy numbers as the initial conditions. Let us consider the following hybrid fuzzy IVP [14,17,18],

$$\begin{array}{ccc}\hfill y^{{}^{\prime }}\left(t\right)& =& y\left(t\right)+m\left(t\right){\lambda }_k\left(y\left(t_k\right)\right),t\in \left[t_k,t_{k+1}\right],\hfill \\ \hfill t_k& =& k,k=0,1,2,3,\dots )\hfill \end{array}$$

(7)

where

$$\begin{array}{ccc}\hfill m\left(t\right)& =& \left\{\begin{array}{c}2\left(t\left(mod1\right)\right)\mathrm{if}t\left(mod1\right)\le 0.5;\\ 2\left(1-t\left(mod1\right)\right)\mathrm{if}t\left(mod1\right)>0.5;\end{array}\right.\hfill \\ \hfill {\lambda }_k\left(\mu \right)& =& \left\{\begin{array}{c}0ifk=0\\ \mu ifk\in \left\{1,2,\dots \right\}\end{array}\right.\hfill \end{array}$$

We will impose initial fuzzy conditions on (7), which are given as:

• Triangular Fuzzy Number: Let $y\left(0\right)=(0.75,1,1.125)$ and

  $$y\left(0,\alpha \right)=\left[\left(0.75+0.25\alpha \right),\left(1.125-0.125\alpha \right)\right],0\le \alpha \le 1$$
  .
• Trapezoidal Fuzzy Number: Let $y\left(0\right)=(0.5,0.75,1,1.125)$ and

  $$y\left(0,\alpha \right)=\left[0.5+0.25\alpha ,1.125-0.125\alpha \right],0\le \alpha \le 1.$$
• Triangular Shaped Fuzzy Number: Let

  $$y\left(0,\alpha \right)=\left[0.75+0.25{\alpha }^2,1.125-0.125{\alpha }^2\right],0\le \alpha \le 1.$$

5.2.1. Triangular Fuzzy Numbers

We solve the hybrid fuzzy differential equations (7) using Picard’s method and the general linear method under 1- and 2-differentiable Hukuhara differentiation.

Now, consider the following examples:

Example 1.

Let $y\left(0\right)=(0.75,1,1.125)$ and

$$y\left(0,\alpha \right)=\left[\left(0.75+0.25\alpha \right),\left(1.125-0.125\alpha \right)\right],0\le \alpha \le 1$$

Picard’s Method

Case 1: Under 1-differentiability:

We apply the Picard Method for hybrid fuzzy differential Equation (5) with $N=50$ when $\alpha =0$ and $t\in \left[0,1\right]$, and then $f\left(\tau ,y_n\left(\tau ,\alpha \right)\right)=y_n\left(\tau ,\alpha \right)$ and $t_0=0.$ We obtain ${\underline{y}}_{50}$ and ${\overline{y}}_{50}$ when $t=1,t=1.5$ and $t=2.$ The exact and approximate solutions for this example under $1-$ differentiability are given in

Table 1. Exact and numerical values (Picard’s method) for the 1-solution at $t=1$.

$\mathit{\alpha }$	$\underset{¯}{\mathit{y}}$	$\overline{\mathit{y}}$	$\underset{¯}{\mathit{Y}}$	$\overline{\mathit{Y}}$
0	2.038711371	3.058067057	2.038711371	3.058067057
0.2	2.174625462	2.990110011	2.174625462	2.990110011
0.4	2.310539554	2.922152965	2.310539554	2.922152965
0.6	2.446453645	2.854195919	2.446453645	2.854195919
0.8	2.582367737	2.786238874	2.582367737	2.786238874
1	2.718281828	2.718281828	2.718281828	2.718281828

,

Table 2. Exact and numerical values (Picard’s method) for the 1-solution at $t=1.5$.

$\mathit{\alpha }$	$\underset{¯}{\mathit{y}}$	$\overline{\mathit{y}}$	$\underset{¯}{\mathit{Y}}$	$\overline{\mathit{Y}}$
0	3.967666294	5.951499441	3.967666294	5.951499441
0.2	4.232177380	5.819243898	4.232177380	5.819243898
0.4	4.496688466	5.686988355	4.496688466	5.686988355
0.6	4.761199553	5.554732811	4.761199553	5.554732811
0.8	5.025710639	5.422477268	5.025710639	5.422477268
1	5.290221725	5.290221725	5.290221725	5.290221725

and

Table 3. Exact and numerical values (Picard’s method) for the 1-solution at $t=2$.

$\mathit{\alpha }$	$\underset{¯}{\mathit{y}}$	$\overline{\mathit{y}}$	$\underset{¯}{\mathit{Y}}$	$\overline{\mathit{Y}}$
0	7.257731754	10.88659763	7.257731754	10.88659763
0.2	7.741580537	10.64467323	7.741580537	10.64467323
0.4	8.225429321	10.40274884	8.225429321	10.40274884
0.6	8.709278105	10.16082445	8.709278105	10.16082445
0.8	9.193126888	9.918900064	9.193126888	9.918900064
1	9.676975672	9.676975672	9.676975672	9.676975672

and in Figure 1. Note that the exact and approximate solutions agree up to ${10}^{-16}$.

Case 2: Under 2-differentiability:

We again apply the Picard method for the hybrid fuzzy differential Equation (6) with $N=50$ when $\alpha =0$ and $t\in \left[0,1\right]$, and then $f\left(\tau ,y_n\left(\tau ,\alpha \right)\right)=y_n\left(\tau ,\alpha \right)$ and $t_0=0$. The results obtained are given in

Table 4. Exact and numerical values (Picard’s method) for the 2-solution at $t=1$.

$\mathit{\alpha }$	$\underset{¯}{\mathit{y}}$	$\overline{\mathit{y}}$	$\underset{¯}{\mathit{Y}}$	$\overline{\mathit{Y}}$
0	2.479411818	2.617366609	2.479411818	2.617366609
0.1	2.503298819	2.627458131	2.503298819	2.627458131
0.3	2.551072821	2.647641175	2.551072821	2.647641175
0.5	2.598846823	2.667824218	2.598846823	2.667824218
0.8	2.670507826	2.698098784	2.670507826	2.698098784
1.0	2.718281828	2.718281828	2.718281828	2.718281828

,

Table 5. Exact and numerical values (Picard’s method) for the 2-solution at $t=1.5$.

$\mathit{\alpha }$	Picard		Exact
	$\underset{¯}{\mathit{y}}$	$\overline{\mathit{y}}$	$\underset{¯}{\mathit{Y}}$	$\overline{\mathit{Y}}$
0	4.932442377	4.986723357	4.932442377	4.986723357
0.1	4.968220312	5.017073194	4.968220312	5.017073194
0.3	5.039776182	5.077772868	5.039776182	5.077772868
0.5	5.111332051	5.138472541	5.111332051	5.138472541
0.8	5.218665856	5.229522052	5.218665856	5.229522052
1	5.290221725	5.290221725	5.290221725	5.290221725

and

Table 6. Exact and numerical values (Picard’s method) for the 2-solution at $t=2$.

$\mathit{\alpha }$	Picard		Exact
	$\underset{¯}{\mathit{y}}$	$\overline{\mathit{y}}$	$\underset{¯}{\mathit{Y}}$	$\overline{\mathit{Y}}$
0	9.068147228	9.076182156	9.0681472287	9.076182156
0.1	9.129030073	9.136261508	9.129030073	9.136261508
0.3	9.250795761	9.256420211	9.250795761	9.256420211
0.5	9.372561450	9.376578914	9.372561450	9.376578914
0.8	9.555209983	9.556816969	9.555209983	9.556816969
1	9.676975672	9.676975672	9.676975672	9.676975672

and in Figure 2.

Example 2.

Let $y\left(0\right)=(0.25,0.75,4)$

$$y\left(0,\alpha \right)=\left[(0.25+0.5\alpha ),(4-3.25\alpha )\right],0\le \alpha \le 1$$

Case 1: Under 1-differentiability:

The results obtained are presented in

Table 7. Exact and numerical values (Picard’s method) for the 1-solution at $t=1$.

$\mathit{\alpha }$	Picard		Exact
	$\underset{¯}{\mathit{y}}$	$\overline{\mathit{y}}$	$\underset{¯}{\mathit{Y}}$	$\overline{\mathit{Y}}$
0	0.6795704573	10.87312731	0.6795704571	10.87312731
0.1	0.8154845485	9.989685719	0.8154845485	9.989685719
0.3	1.087312731	8.222802531	1.087312731	8.222802531
0.5	1.359140914	6.455919342	1.359140914	6.455919342
0.8	1.766883188	3.805594559	1.766883188	3.805594559
1	2.038711371	2.038711371	2.038711371	2.038711371

,

Table 8. Exact and numerical values (Picard’s method) for the 1-solution at $t=1.5$.

$\mathit{\alpha }$	Picard		Exact
	$\underset{¯}{\mathit{y}}$	$\overline{\mathit{y}}$	$\underset{¯}{\mathit{Y}}$	$\overline{\mathit{Y}}$
0	1.322555431	21.16088690	1.322555431	21.16088690
0.1	1.587066517	19.44156484	1.587066517	19.44156484
0.3	2.116088690	16.00292072	2.116088690	16.00292072
0.5	2.645110862	12.56427659	2.645110862	12.56427659
0.8	3.438644121	7.406310415	3.438644121	7.406310415
1	3.967666294	3.967666294	3.967666294	3.967666294

and

Table 9. Exact and numerical values (Picard’s method) for the 1-solution at $t=2$.

$\mathit{\alpha }$	Picard		Exact
	$\underset{¯}{\mathit{y}}$	$\overline{\mathit{y}}$	$\underset{¯}{\mathit{Y}}$	$\overline{\mathit{Y}}$
0	2.419243918	38.70790268	2.419243918	38.70790268
0.1	2.903092701	35.56288559	2.903092701	35.56288559
0.3	3.870790268	29.27285140	3.870790268	29.27285140
0.5	4.838487836	22.98281722	4.838487836	22.98281722
0.8	6.290034187	13.54776594	6.290034187	13.54776594
1	7.257731754	7.257731754	7.257731754	7.257731754

and in Figure 3.

Thus, the Picard method with triangular fuzzy number as the initial conditions gave highly accurate results when used with high numbers of iterations.

General Linear Method (GLM):

We solved the same Examples 1 and 2 as in the previous subsection using the general linear method for hybrid fuzzy differential equations with $N=100$ and $K=5.$ Similar results and similar accuracy are observed, and the results obtained for Example 1 for 1-differentiability and 2-differentiability are given in Figure 4 and Figure 5, respectively, and those for Example 2 for 1-differentiability are given in Figure 6.

We also performed a comparison of the proposed GLM for Example 1 with the Runge–Kutta of order 5 method proposed by [14,15,16]. The results for both models are presented in

Table 10. Comparison between the GLM and the Runge–Kutta fifth order for the 1-solution at $t=1$.

$\mathit{\alpha }$	GLM		Runge–Kutta Fifth Order
	$\underset{¯}{\mathit{y}}$	$\overline{\mathit{y}}$	$\underset{¯}{\mathit{Y}}$	$\overline{\mathit{Y}}$
0	2.038711371	3.058067057	2.038711370	3.058067056
0.2	2.174625462	2.990110011	2.174625462	2.990110010
0.4	2.310539554	2.922152965	2.310539553	2.922152964
0.6	2.446453645	2.854195919	2.446453645	2.854195919
0.8	2.582367737	2.786238874	2.582367736	2.786238873
1	2.718281828	2.718281828	2.718281827	2.718281827

and

Table 11. Comparison between the GLM and the Runge–Kutta fifth order for the 1-solution at $t=2$.

$\mathit{\alpha }$	GLM		Runge–Kutta Fifth Order
0	7.257731754	10.88659763	7.257731754	10.88659763
0.2	7.741580537	10.64467323	7.741580537	10.64467323
0.4	8.225429321	10.40274884	8.225429321	10.40274884
0.6	8.709278105	10.16082445	8.709278105	10.16082445
0.8	9.193126888	9.918900064	9.193126888	9.918900064
1	9.676975672	9.676975672	9.676975672	9.676975672

for the 1-solution and in

Table 12. Comparison between the GLM and the Runge–Kutta fifth order for the 2-solution at $t=1$.

$\mathit{\alpha }$	GLM		Runge–Kutta Fifth Order
0	2.479411818	2.617366609	2.479411818	2.617366608
0.2	2.527185820	2.637549653	2.527185820	2.637549652
0.4	2.574959822	2.657732697	2.574959822	2.657732696
0.6	2.622733824	2.677915740	2.622733824	2.677915740
0.8	2.670507826	2.698098784	2.670507825	2.698098784
1	2.718281828	2.718281828	2.718281827	2.718281827

and

Table 13. Comparison between the GLM and Runge–Kutta fifth order for the 2-solution at $t=2$.

$\mathit{\alpha }$	GLM		Runge–Kutta Fifth Order
0	9.068147228	9.076182156	9.068147223	9.076182152
0.2	9.189912917	9.196340859	9.189912912	9.196340855
0.4	9.311678606	9.316499563	9.311678601	9.316499558
0.6	9.433444294	9.436658266	9.433444289	9.436658261
0.8	9.555209983	9.556816969	9.555209978	9.556816964
1	9.676975672	9.676975672	9.676975667	9.676975667

for the 2-solution.

The results are almost identical except for slight differences in the 10th digits. This means that the GLM competes well with other methods in terms of accuracy, such as the Runge–Kutta, while in terms of the amount of work, it requires less computations. This is due to the fact that the GLM passes more than one piece of information between steps. Each step of the computation takes a certain number of data items as input and generates the same numbers of data items as output [23,24].

5.2.2. Triangular Shaped and Trapezoidal Fuzzy Numbers

We solve the hybrid fuzzy differential equation subject to triangular and trapezoidal fuzzy numbers.

1. Trapezoidal Fuzzy Numbers:

Let $y\left(0\right)$ = (0.5, 0.75, 1, 1.125) and

$$y\left(0,\alpha \right)=\left[\left(0.75+0.25\alpha \right),\left(1.125-0.125\alpha \right)\right],0\le \alpha \le 1$$

The solutions obtained using Picard’s method and the general linear method for t = 1.0, 1.5 and 2.0 are given in Figure 7 and Figure 8, respectively.

When using the triangular fuzzy numbers of the form

$$y\left(0,\alpha \right)=\left[\left(0.75+0.25{\alpha }^2\right),\left(1.125-0.125{\alpha }^2\right)\right],0\le \alpha \le 1$$

The solutions obtained for Picard’s method and the general linear method are given in Figure 9 and Figure 10, respectively.

Again, accurate results are reported when using Picard’s method or the GLM with trapezoidal fuzzy numbers and triangular fuzzy numbers as the initial conditions. The general linear method gave accurate results with a lower number of steps compared with the Runge–Kutta method as reported by other authors.

6. Conclusions

We applied Picard’s method and the general linear method successfully to solve hybrid fuzzy differential equations subject to trapezoidal and triangular fuzzy numbers. We determined that both Picard’s method and the GLM gave results with high accuracy for each of the initial conditions. All numerical computations were performed using MATLAB. The results obtained are accurate and compare well with the exact solution and the fifth order Runge–Kutta method and found the GLM to have greater advantages over the other methods reported in the literature.