A modified Euler method for solving fuzzy differential equations under generalized differentiability

Abstract

In this paper, we intend to introduce a modified approach for solving fuzzy differential equations (FDEs) under generalized differentiability. Modified Euler method estimated FDEs by using a two-stage predictor–corrector algorithm with local truncation error of order two. The consistency, convergence, and stability of the proposed method are also investigated in detail. The acceptable accuracy of the Modified Euler method is illustrated by some examples.

1 Introduction

Fuzzy differential equations (FDEs) are applied for modeling problems in science and engineering under uncertainly and has been rapidly developing in recent years Allahviranloo (2020).

The concept of fuzzy derivatives is a very important notation in FDEs. Fuzzy derivatives were first initiated by Chang and Zadeh (1972), and followed up by Dubois and Prade (1982), Puri and Ralescu (1986) and Goetschel and Voxman (1987).

The concept of FDEs was introduced by Kaleva (1987), Seikkala (1987), and other researches. Many works have been done to study the numerical solution of FDEs under Hukuhara differentiability like Euler method, Taylor method, Runge–Kutta method, Predictor–Corrector and Improved Predictor–Corrector method (Ma et al. 1999; Abbasbandy and Allahviranloo 2002, 2004; Allahviranloo et al. 2007, 2009).

It is well-known that the solution of FDEs under Hukuhara derivatives has the property that the diameter is non-decreasing as t increase; to overcome this shortcoming, Bede and Gal introduced the strongly generalized differential of set value fuzzy function (Bede and Gal 2005) and studied more properties in Bede and Stefanini (2011), Bede and Gal (2006), Bede and Stefanini (2013).

Rapidly, the numerical methods for solving fuzzy differential equation and fuzzy integral equation under Hukuhara differentiability extended to fuzzy differential equation and fuzzy integral equation under strongly generalized differentiability (Rabiei et al. 2013; Hajighasemi et al. 2010; BaloochShahryari and Salashour 2012; Ahmadian et al. 2018, 2012).

Nieto et al. (2009) obtained solution of first-order fuzzy differential equations using generalized differentiability by interpreting the original fuzzy differential equations with two crisp ordinary differential equations.

Allahviranloo and Salahshour proposed a new approach for solving first order fuzzy differential equations under strongly generalized H-differentiability; the main part of the proposed technique was extending 1-cut solution of original FDEs by allocating some unknown spreads in Salahshour et al. (2018). Also Allahviranloo, Gouyandeh, and Armand by introducing the fuzzy Taylor expansion proposed the Euler method for solving FDEs under generalized differentiability; the local truncation error for the Euler method is o(h) (Allahviranloo et al. 2015). Tapaswini and Chakraverty proposed improved Euler methods for solving FDEs under Hukuhara differentiability in Tapaswini and Chakraverty (2012). In the present paper, we intend to improve the Euler method under generalized differentiability by two-stage predictor–corrector method, to get better accuracy without losing the simplicity of the Euler method. The convergence and stability of the proposed method are proved. By some examples, the capability of the proposed method is shown.

The paper is organized as follows: In Sect. 2, some basic definitions are brought. In the proposed method, moreover consistency, convergence, and stability of the Modified Euler method are introduced in Sect. 3. The numerical examples are presented in Sect. 4 and finally, conclusion is drawn.

2 Preliminaries

In this section the most basic notations and definitions used are introduced:

The set of fuzzy numbers, that is, normal, fuzzy convex, upper semi-continuous and compactly supported fuzzy sets which are defined over the real line and denoted by ${\mathbb {R}}_{{\mathcal {F}}}$. For $0 < \alpha \le 1$, set $[u]^{\alpha }= \Big \{ t \in {\mathbb {R}}\Big | u(t) \ge \alpha \Big \}$, and $[u]^{0} = \textit{cl}{\Big \{ t \in {\mathbb {R}}\Big | u(t) > 0 \Big \}}$. We represent $[u]^{\alpha } =[{u^{-}}{(\alpha )},~{u^{+}}{(\alpha )}]$, so if $u \in {\mathbb {R}}_{{\mathcal {F}}}$, the $\alpha$-level set $[u]^{\alpha }$ is a closed interval for all $\alpha \in [0,~1]$. For arbitrary $u,~v \in {\mathbb {R}}_{{\mathcal {F}}}$ and $k \in {\mathbb {R}}$, the addition and scalar multiplication are defined by $[u+v]^{\alpha } =[u]^{\alpha }+[v]^{\alpha }~,~[ku]^{\alpha }=k[u]^{\alpha },$ respectively.

Definition 2.1

(see Kaleva 1987) The Hausdorff distance between fuzzy numbers is given by $D:{\mathbb {R}}_{{\mathcal {F}}} \times {\mathbb {R}}_{{\mathcal {F}}}\longrightarrow {{\mathbb {R}}}^{+} \cup \{0\}$ as

$$\begin{aligned} D(u,~v)= \sup _{\alpha \in [0,~1]} \max \Big \{ |u^{-} (\alpha ) -v^{-}(\alpha )| ,~ | u^{+}(\alpha ) - v^{+}(\alpha )|\Big \}. \end{aligned}$$

Consider $u,v,w,z \in {\mathbb {R}}_{{\mathcal {F}}}$ and $\lambda \in {\mathbb {R}}$; then the following properties are well-known for metric D:(see Negoita and Ralescu 1975):

• 1. $D(u\oplus w, ~v\oplus w)= D(u,~v);$

• 2. $D(\lambda u, ~\lambda v)=|\lambda |D(u,~v);$

• 3. $D(u\oplus v, ~w \oplus z)\le D(u,~w)+D(v,~z);$

• 4. $D(u\ominus v,~ w \ominus z)\le D(u,~w)+D(v,~z)$,  as long as $u\ominus v$ and $w\ominus z$ exist, where $u,~v,~w,~z \in {\mathbb {R}}_{{\mathcal {F}}}$,

where $\ominus$ is the Hukuhara difference(H-difference); it means that $w\ominus v =u$ if and only if $u\oplus v=w$.

Definition 2.2

(see Stefanini 2008) The generalized Hukuhara difference of two fuzzy numbers $u,~v \in {\mathbb {R}}_{{\mathcal {F}}}$ is defined as follows:

$$\begin{aligned} u \ominus v =w \Longleftrightarrow \left\{ \begin{array}{l} (a).~u=v+ w; \\ or~(b).~v=u+(-1)w. \end{array}\right. \end{aligned}$$

Proposition 2.1

(see Stefanini 2008) Let $A,B\in {K}_{n}^{C}$ be two compact convex set; then

• 1. if the gH-difference exists, it is unique and it is a generalization of the usual Hukuhara difference since $A \ominus B=A\sim _{h}B$, whenever $A\sim _{h}B$ exists,

• 2. $A \ominus A ={0}$,

• 3. if $A \ominus B$ exists in the sence (a), then $B \ominus A$ exists in sense (b) and vice versa,

• 4. $(A \oplus B)\ominus B= A,$

• 5. $\{0\}\ominus (A \ominus B)=(-B)\ominus (-A),$

• 6. we have $(A \ominus B)=(B \ominus A)=C$ if and only if $C=\{0\}$ and $A=B.$

Definition 2.3

(see Nieto et al. 2009) A fuzzy valued function $f:[a,b]\rightarrow {\mathbb {R}}_{{\mathcal {F}}}$ is said to be continuous at $t_0 \in [a,b]$ if for each $\epsilon > 0$ there is $\delta >0$ such that $D(f(t),f(t_0))<\epsilon$, whenever $t\in [a,b]$ and $|t-t_0|<\delta$. We say that f is fuzzy continuous on [a, b] if f is continuous at each $t_0 \in [a,~b]$.

Definition 2.4

(see Bede and Gal 2005) Let $F:I\rightarrow {\mathbb {R}}$. Fix $t_{0} \in I$. We say F is strongly generalized differentiable at $t_{0}$, if there exists an element $F^{\prime }(t_{0})\in {\mathbb {R}}$ such that either

• 1. for all $h>0$ sufficiently closed to 0, the H-differences $F(t_{0}+h)\ominus F(t_{0})$, $F(t_{0})\ominus F(t_{0}-h)$ exist and limits (in the metric D)

  $$\begin{aligned} \lim _{h\rightarrow 0^{+}}\frac{F(t_{0}+h)\ominus F(t_{0})}{h}=\lim _{h\rightarrow 0^{+}}\frac{F(t_{0}) \ominus F(t_{0}-h)}{h}=F^{\prime }(t_{0}) \end{aligned}$$

  (2.1)

  or

• 2. for all $h>0$ sufficiently closed to 0, the H-differences $F(t_{0})\ominus F(t_{0}+h)$, $F(t_{0}-h)\ominus F(t_{0})$ exist and limits (in the metric D)

  $$\begin{aligned} \lim _{h\rightarrow 0^{+}}\frac{F(t_{0})\ominus F(t_{0}+h)}{-h}=\lim _{h\rightarrow 0^{+}}\frac{F(t_{0}-h) \ominus F(t_{0})}{-h}=F^{\prime }(t_{0}) \end{aligned}$$

  (2.2)

  or

• 3. for all $h>0$ sufficiently closed to 0, the H-differences $F(t_{0}+h)\ominus F(t_{0})$, $F(t_{0}-h)\ominus F(t_{0})$ exist and limits (in the metric D)

  $$\begin{aligned} \lim _{h\rightarrow 0^{+}}\frac{F(t_{0}+h)\ominus F(t_{0})}{h}=\lim _{h\rightarrow 0^{+}}\frac{F(t_{0}-h) \ominus F(t_{0})}{-h}=F^{\prime }(t_{0}) \end{aligned}$$

  (2.3)

  or

• 4. for all $h>0$ sufficiently closed to 0, the H-differences $F(t_{0})\ominus F(t_{0}+h)$, $F(t_{0})\ominus F(t_{0}-h)$ exist and limits (in the metric D)

  $$\begin{aligned} \lim _{h\rightarrow 0^{+}}\frac{F(t_{0})\ominus F(t_{0}+h)}{-h}=\lim _{h\rightarrow 0^{+}}\frac{F(t_{0}) \ominus F(t_{0}-h)}{h}=F^{\prime }(t_{0}) \end{aligned}$$

  (2.4)

Definition 2.5

Let $f:I\rightarrow {\mathbb {R}}_{F}$. We say f is $[(i)-gH]$-differentiable on I if f is differentiable in the sense (1) of Definition 2.4 and its derivative is denoted $f_{i.gH}$ and similarly for $[(ii)-gH]-$ differentiable we have $f_{ii.gH}$ if f is differentiable in the sense (2) of Definition 2.4.

Theorem 2.1

(see Chalco-Cano and Roman-Flores 2008) Let $f:I\rightarrow {\mathbb {R}}_{F}$ and put $[f(t)]^{\alpha }=[(f^{+})(t;\alpha )~,~(f^{-})(t;\alpha )]$ for each $\alpha \in [0,1].$

• 1. If f is $[(i)-gH]$ differentiable the $f^{+}$ and $f^{-}$ are differentiable functions and

  $$\begin{aligned} f_{i.gH}^{\prime }(t_0;\alpha ) =[(f^{-})^{\prime }(t;\alpha ),~(f^{+})^{\prime }(t;\alpha )], \end{aligned}$$

  (2.5)

• 2. If f is $[(ii)-gH]$ differentiable the $f^{+}$ and $f^{-}$ are differentiable functions and

  $$\begin{aligned} f_{ii.gH}^{\prime }(t_0;\alpha ) =[(f^{+})^{\prime }(t;\alpha ),~(f^{-})^{\prime }(t;\alpha )], \end{aligned}$$

  (2.6)

Definition 2.6

(see Chalco-Cano and Roman-Flores 2008) We say that a point $t_0 \in (a, b)$ is a switching point for the differentiability of f if in any neighborhood V of $t_0$ there exist points $t_1< t_0 < t_2$ such that

type (I):

at $t_1$ (2.5) holds while (2.6) does not hold and at $t_2$ (2.6) holds and (2.5) does not hold, or

type (II):

at $t_1$ (2.6) holds while (2.5) does not hold and at $t_2$ (2.5) holds and (2.6) does not hold.

The following condition and notations are used in the reminder of paper:

We assume that in the whole paper, the generalized Hukuhara difference of two fuzzy numbers exists.

${\mathcal {C}}_{\mathcal {F}}([a,b], {\mathbb {R}}_{{\mathcal {F}}})$ is the set of fuzzy valued functions f which are defined on [a, b] and is fuzzy continuous from the interior points of [a, b] such that the continuity is one-sided at endpoints a, b.

${\mathcal {C}}^{k}_{gH} ([a,b], ~{\mathbb {R}}_{\mathcal {F}})$, is the space of functions f such that f and its first k, gH-derivtives are all in ${\mathcal {C}}_{\mathcal {F}}([a,b],{\mathbb {R}}_{{\mathcal {F}}})$.

Theorem 2.2

(see Allahviranloo et al. 2015) Consider $f : [a,b]\longrightarrow {\mathbb {R}}_{{\mathcal {F}}}$ is gH-differentiable such that type of differentiability f in [a, b] does not change. Then for $a\le s\le b$

(i):

if f(t) is $[(i)-gH]$-differentiable, then $f_{i.gH}^{\prime }(t)$ is (FR)-integrable over [a,  b] and

$$\begin{aligned} f(s)=f(a)\oplus \int _{a}^{s}f_{i.gH}^{\prime }(t){\text {d}}t. \end{aligned}$$

(2.7)

(ii):

If f(t) is $[(ii)-gH]$-differentiable, then $f_{ii.gH}^{\prime }(t)$ is (FR)-integrable over [a,  b] and

$$\begin{aligned} f(a)=f(s)\oplus (-1)\int _{a}^{s}f_{ii.gH}^{\prime }(t){\text {d}}t. \end{aligned}$$

(2.8)

Theorem 2.3

(see Allahviranloo et al. 2015) Let $f^{(i)}:[a,~b]\rightarrow {\mathbb {R}}_{{\mathcal {F}}}$ and $f \in {\mathcal {C}}^{4 }_{gH} ([a,b],~{\mathbb {R}}_{\mathcal {F}})$. For all $s \in [a,~b]$

(i):

Consider $f_{gH}^{(i)},~~i=1,\ldots ,n$ are $[(i)-gH]$-differentiable and type of differentiability does not change in interval [a,  b]; then

$$\begin{aligned} f_{i.gH}^{(i-1)}(s) = f_{i.gH}^{(i-1)}(a)\oplus \int _{a}^{s}f^{(i)}_{i.gH}(t){\text {d}}t. \end{aligned}$$

(2.9)

(ii):

If $f_{gH}^{(i)},~~i=1,\ldots ,n$ are $[(ii)-gH]$-differentiable and type of differentiability does not change in interval [a,  b], then

$$\begin{aligned} f_{ii.gH}^{(i-1)}(s)= f_{ii.gH}^{(i-1)}(a)\oplus \int _{a}^{s}f^{(i)}_{{\text {ii.gH}}}(t){\text {d}}t. \end{aligned}$$

(2.10)

(iii):

Suppose that $f^{(i)},~i=2k-1,~k \in {\mathbb {N}}$ are [(i)-gH]-differentiable and f(t), $f^{(i)}, ~i=2k,~k\in {\mathbb {N}}$ are [(ii)-gH]-differentiable, so

$$\begin{aligned} f_{i.gH}^{(i-1)}(s) =f_{{\text {i.gH}}}^{(i-1)}(a)\ominus (-1) \int _{a}^{s}f_{ii.gH}^{(i)}(t){\text {d}}t. \end{aligned}$$

(2.11)

(iv):

Consider for $i=2k-1, k\in {\mathbb {N}}$, $f^{(i)}$ are [(ii)-gH]-differentiable and f(t), $f^{(i)}$ are [(i)-gH]-differentiable for $i=2k,~k\in {\mathbb {N}}$, then

$$\begin{aligned} f_{{\text {ii.gH}}}^{(i-1)}(s) =f_{{\text {ii.gH}}}^{(i-1)}(a)\ominus (-1) \int _{a}^{s}f_{{\text {i.gH}}}^{(i)}(t){\text {d}}t. \end{aligned}$$

(2.12)

3 Proposed method

Now, let us consider the following fuzzy differential equation (FDE):

$$\begin{aligned} \left\{ \begin{array}{ll} y^{\prime }_{gH}(t)=f(t,y(t)), &{} t \in [0,T]; \\ y(0)={y_{0}}\in {\mathbb {R}}_{{\mathcal {F}}}; &{} \end{array}\right. \end{aligned}$$

(3.1)

here y(t) is an unknown fuzzy function of crisp variable t and $f:[0,~T]\times {\mathbb {R}}_{{\mathcal {F}}}\rightarrow {\mathbb {R}}_{{\mathcal {F}}}$ is continuous; also $y^{\prime }_{gH}(t)$ is the generalized Hukuhara derivative of y(t) such that the set of switching points is finite.

First, we are going to approximate $y^{\prime \prime }_{gH}(t)$ in four different cases by using concept of generalized differentiability.

Theorem 3.1

Let $f \in {\mathcal {C}}^{4}_{gH} ([a,b],~{\mathbb {R}}_{\mathcal {F}})$. Then

(i):

if $f_{gH}^{(i)},~~i=1,\ldots ,4$ are $[(i)-gH]$-differentiable and type of differentiability does not change in interval [a,  b], then

$$\begin{aligned} f_{i.gH}^{\prime \prime }(a) \approx \frac{f_{i.gH}^{'}(s) \ominus f_{i.gH}^{'}(a)}{s-a}, \end{aligned}$$

(3.2)

(ii):

let $f_{gH}^{(i)},~~i=1,\ldots ,4$ are $[(ii)-gH]$-differentiable and type of differentiability does not change in interval [a,  b], then

$$\begin{aligned} f_{ii.gH}^{\prime \prime }(a) \approx \frac{f_{ii.gH}^{'}(s) \ominus f_{ii.gH}^{'}(a)}{s-a}, \end{aligned}$$

(3.3)

(iii):

suppose that $f^{(i)},~i=1,3$ are $[(i)-gH]$-differentiable and f(t), $f^{(i)},~i=2,4$ are $[(ii)-gH]$-differentiable, so

$$\begin{aligned} (-1)f_{i.gH}^{\prime \prime }(a) \approx \frac{f_{ii.gH}^{'}(s)\ominus f_{ii.gH}^{'}(a)}{s-a}, \end{aligned}$$

(3.4)

(iv):

consider for $i=1,3$, $f^{(i)}$ are $[(ii)-gH]$-differentiable and f(t), $f^{(i)}$ are $[(i)-gH]$-differentiable for $i=2,4$, then

$$\begin{aligned} (-1)f_{ii.gH}^{\prime \prime }(a) \approx \frac{f_{i.gH}^{'}(s)\ominus f_{i.gH}^{'}(a)}{s-a}. \end{aligned}$$

(3.5)

Proof

Since $f \in {\mathcal {C}}^{4}_{gH} ([a,b],~{\mathbb {R}}_{\mathcal {F}})$, so $f^{(i)}_{gH}~,i=0,~1,~2,~3,~4,$ are (FR)-integrable on T.

(i):

Let $f^{(i)}$ be $[(i)-gH]$-differentiable; by Theorem 2.3, we can write

$$\begin{aligned} f_{i.gH}^{\prime }(s)=f_{i.gH}^{\prime }(a)\oplus \int _{a}^{s}f_{i.gH}^{\prime \prime }(s_{1}){\text {d}}s_{1}, \end{aligned}$$

where

$$\begin{aligned} f_{i.gH}^{\prime \prime }(s_1)=f_{i.gH}^{\prime \prime }(a) \oplus \int _{a}^{s_1}f_{i.gH}^{\prime \prime \prime }(s_2){\text {d}}s_2. \end{aligned}$$

(3.6)

by integration of Eq. (3.6), we get

$$\begin{aligned} \int _{a}^{s}f_{i.gH}^{\prime \prime }(s_1){\text {d}}s_1&\quad =\int _{a}^{s}f_{i.gH}^{\prime \prime }(a){\text {d}}s_1 \oplus \int _{a}^{s}\bigg (\int _{a}^{s_1}f_{i.gH}^{\prime \prime \prime } (s_2){\text {d}}s_2\bigg ) {\text {d}}s_1, \\&\quad =f_{i.gH}^{\prime \prime }(a)\odot (s-a)\oplus \int _{a}^{s} \bigg (\int _{a}^{s_1}f_{i.gH}^{\prime \prime \prime }(s_2){\text {d}}s_2\bigg ){\text {d}}s_1, \end{aligned}$$

where the last double (FR)-integral belongs to ${\mathbb {R}}_{{\mathcal {F}}}$. So

$$\begin{aligned} f_{i.gH}^{\prime }(s)= & {} f_{i.gH}^{\prime }(a) \oplus f_{i.gH}^{\prime \prime }(a)\odot (s-a)\\&\oplus \int _{a}^{s}\bigg (\int _{a}^{s_1}f_{i.gH}^{\prime \prime \prime } (s_2){\text {d}}s_2 \bigg ){\text {d}}s_1. \end{aligned}$$

Consequently,

$$\begin{aligned} f_{i.gH}^{\prime \prime \prime }(s_2)=f_{i.gH}^{\prime \prime \prime }(a) \oplus \int _{a}^{s_2}f_{i.gH}^{(4)}(s_3){\text {d}}s_3; \end{aligned}$$

applying the (FR)-integral operator to $f_{i.gH}^{\prime \prime \prime }(s_2)$, gives

$$\begin{aligned}&\int _{a}^{s_1}f_{i.gH}^{\prime \prime \prime }(s_2){\text {d}}s_2\\&\quad =f_{i.gH}^{\prime \prime \prime }(a) \odot (s_1-a) \oplus \int _{a}^{s_1}\bigg (\int _{a}^{s_2}f_{i.gH}^{(4)}(s_3){\text {d}}s_3\bigg ){\text {d}}s_2; \end{aligned}$$

furthermore,

$$\begin{aligned} \int _{a}^{s}\bigg (\int _{a}^{s_1}f_{i.gH}^{\prime \prime \prime } (s_2){\text {d}}s_2\bigg ){\text {d}}s_1= & {} f_{i.gH}^{\prime \prime \prime }(a) \odot \int _{a}^{s} (s_1-a){\text {d}}s_1\\&\oplus \int _{a}^{s}\bigg (\int _{a}^{s_1}\bigg (\int _{a}^{s_2} f_{i.gH}^{(4)}(s_3){\text {d}}s_3\bigg ){\text {d}}s_2\bigg ){\text {d}}s_1, \end{aligned}$$

where the last triple integral belongs to ${\mathbb {R}}_{{\mathcal {F}}}$. Hence

$$\begin{aligned} f_{i.gH}^{\prime }(s)= & {} f_{i.gH}^{\prime }(a)\oplus f_{i.gH}^{\prime \prime } (a)\odot (s-a)\oplus f_{i.gH}^{\prime \prime \prime }(a)\\&\odot \frac{(s-a)^2}{2!} \oplus \int _{a}^{s} \bigg (\int _{a}^{s_1}\bigg (\int _{a}^{s_2}f_{i.gH}^{(4)}(s_3){\text {d}}s_3\bigg ){\text {d}}s_2 \bigg ){\text {d}}s_1. \end{aligned}$$

If s is very closed to a and $(s-a)\rightarrow 0$,

$$\begin{aligned} D(f_{i.gH}^{\prime }(s),f_{i.gH}^{\prime }(a)\oplus f_{i.gH}^{\prime \prime }(a)\odot (s-a))\rightarrow 0, \end{aligned}$$

and

$$\begin{aligned} D(f_{i.gH}^{\prime }(s)\ominus f_{i.gH}^{\prime }(a),f_{i.gH}^{\prime \prime }(a) \odot (s-a))\rightarrow 0, \end{aligned}$$

this means where $(s-a)\rightarrow 0$, we obtain:

$$\begin{aligned} f_{i.gH}^{\prime \prime }(a)\approx & {} \frac{f_{i.gH}^{'}(s) \ominus f_{i.gH}^{'}(a)}{s-a}. \end{aligned}$$

(ii):

Suppose that $f^{(i)}$ is $[(ii)-gH]$-differentiable for $i=1,2,3,4$; by Theorem 2.3, we have

$$\begin{aligned} f_{ii.gH}^{\prime }(s)=f_{ii.gH}^{\prime }(a)\oplus \int _{a}^{s}f_{ii.gH}^{\prime \prime }(s_{1}){\text {d}}s_{1}, \end{aligned}$$

where $f_{ii.gH}^{\prime \prime }$ is gained by

$$\begin{aligned} f_{ii.gH}^{\prime \prime }(s_1)=f_{ii.gH}^{\prime \prime }(a) \oplus \int _{a}^{s_1}f_{ii.gH}^{\prime \prime \prime }(s_2){\text {d}}s_2. \end{aligned}$$

It is easy to see that

$$\begin{aligned} \int _{a}^{s}f_{ii.gH}^{\prime \prime }(s_1){\text {d}}s_1&\quad =f_{ii.gH}^{\prime \prime }(a)\odot (s-a) \oplus \int _{a}^{s}\bigg (\int _{a}^{s_1} f_{ii.gH}^{\prime \prime \prime }(s_2){\text {d}}s_2 \bigg ){\text {d}}s_1, \end{aligned}$$

where the last double (FR)-integral belongs to ${\mathbb {R}}_{{\mathcal {F}}}$. Thus we conclude

$$\begin{aligned} f_{ii.gH}^{\prime }(s)= & {} f_{ii.gH}^{\prime }(a) \oplus f_{ii.gH}^{\prime \prime }(a) \odot (s-a)\\&\oplus \int _{a}^{s}\bigg (\int _{a}^{s_1}f_{ii.gH}^{\prime \prime \prime } (s_2){\text {d}}s_2 \bigg ){\text {d}}s_1. \end{aligned}$$

Consequently $f_{ii.gH}^{\prime \prime \prime }$ can be written as

$$\begin{aligned} f_{ii.gH}^{\prime \prime \prime }(s_2)=f_{ii.gH}^{\prime \prime \prime }(a) \oplus \int _{a}^{s_2}f_{ii.gH}^{(4)}(s_3){\text {d}}s_3, \end{aligned}$$

applying the (FR)-integral operator to $f_{ii.gH}^{\prime \prime \prime }(s_2)$, gives

$$\begin{aligned} \int _{a}^{s_1}f_{ii.gH}^{\prime \prime \prime }(s_2){\text {d}}s_2 =f_{ii.gH}^{\prime \prime \prime }(a) \odot (s_1-a) \oplus \int _{a}^{s_1}\bigg (\int _{a}^{s_2}f_{ii.gH}^{(4)}(s_3){\text {d}}s_3\bigg ){\text {d}}s_2; \end{aligned}$$

furthermore,

$$\begin{aligned} \int _{a}^{s}\bigg (\int _{a}^{s_1}f_{ii.gH}^{\prime \prime \prime } (s_2){\text {d}}s_2\bigg ){\text {d}}s_1= & {} f_{ii.gH}^{\prime \prime \prime }(a) \odot \int _{a}^{s} (s_1-a){\text {d}}s_1\\&\oplus \int _{a}^{s}\bigg (\int _{a}^{s_1}\bigg (\int _{a}^{s_2} f_{ii.gH}^{(4)}(s_3){\text {d}}s_3\bigg ){\text {d}}s_2\bigg ){\text {d}}s_1, \end{aligned}$$

where the last triple integral belongs to ${\mathbb {R}}_{{\mathcal {F}}}$. Hence

$$\begin{aligned} f_{ii.gH}^{\prime }(s)= & {} f_{ii.gH}^{\prime }(a)\oplus f_{ii.gH}^{\prime \prime }(a) \odot (s-a)\oplus f_{ii.gH}^{\prime \prime \prime }(a)\\&\odot \frac{(s-a)^2}{2!} \oplus \int _{a}^{s}\bigg (\int _{a}^{s_1} \bigg (\int _{a}^{s_2}f_{ii.gH}^{(4)}(s_3){\text {d}}s_3 \bigg ){\text {d}}s_2 \bigg ){\text {d}}s_1. \end{aligned}$$

If s is very closed to a and $(s-a)\rightarrow 0$,

$$\begin{aligned} D(f_{ii.gH}^{\prime }(s)\ominus f_{ii.gH}^{\prime }(a), f_{i.gH}^{\prime \prime }(a)\odot (s-a))\rightarrow 0, \end{aligned}$$

this means where $(s-a)\rightarrow 0$ we have

$$\begin{aligned} f_{ii.gH}^{\prime \prime }(a)\approx & {} \frac{f_{ii.gH}^{'}(s) \ominus f_{ii.gH}^{'}(a)}{s-a}. \end{aligned}$$

(iii):

Suppose that $f^{(i)},~i=1,3$, are $[(i)-gH]$-differentiable and $f^{(i)},~i=2,4$, are $[(ii)-gH]$-differentiable, so

$$\begin{aligned} f_{ii.gH}^{\prime }(s)=f_{ii.gH}^{\prime }(a)\ominus (-1) \odot \int _{a}^{s} f_{i.gH}^{\prime \prime }(s_1){\text {d}}s_{1}, \end{aligned}$$

As it is known, $f_{i.gH}^{\prime \prime }$ is obtained from

$$\begin{aligned} f_{i.gH}^{\prime \prime }(s_1)=f_{i.gH}^{\prime \prime }(a) \ominus (-1)\odot \int _{a}^{s_{1}}f_{ii.gH}^{\prime \prime \prime }(s_{2}){\text {d}}s_{2}; \end{aligned}$$

by integration we obtain

$$\begin{aligned} \int _{a}^{s}f_{i.gH}^{\prime \prime }(s_1){\text {d}}s_{1}=f_{i.gH}^{\prime \prime }(a) \odot (s-a)\ominus (-1)\odot \int _{a}^{s}\bigg (\int _{a}^{s_1} f_{ii.gH}^{\prime \prime \prime }(s_2){\text {d}}s_2\bigg ) d s_1; \end{aligned}$$

therefore,

$$\begin{aligned} f_{ii.gH}^{\prime }(s)=f_{ii.gH}^{\prime }(a) \oplus (-1)f_{i.gH}^{\prime \prime }(a)\odot (s-a)\ominus \int _{a}^{s}\bigg (\int _{a}^{s_1}f_{ii.gH}^{\prime \prime \prime }(s_2){\text {d}}s_2\bigg ); \end{aligned}$$

similarly, in order to find $f_{ii.gH}^{\prime \prime \prime }$ by Theorem 2.3, we have

$$\begin{aligned} f_{ii.gH}^{\prime \prime \prime }(s_2)=f_{ii.gH}^{\prime \prime \prime }(a) \ominus (-1)\int _{a}^{s_2}f_{i.gH}^{(4)}(s_3){\text {d}}s_3, \end{aligned}$$

and

$$\begin{aligned} \int _{a}^{s}\bigg (\int _{a}^{s_1}f_{ii.gH}^{\prime \prime \prime }(s_2){\text {d}}s_2\bigg ){\text {d}}s_1= & {} \frac{(s-a)^2}{2}\odot f_{ii.gH}^{\prime \prime \prime }(a)\\&\ominus (-1)\int _{a}^{s}\bigg (\int _{a}^{s_1}\bigg (\int _{a}^{s_2} f_{ii.gH}^{(4)}(s_3){\text {d}}s_3){\text {d}}s_2\bigg )\bigg ){\text {d}}s_1, \end{aligned}$$

thus, we conclude:

$$\begin{aligned} f_{ii.gH}^{\prime }(s)= & {} f_{ii.gH}^{\prime }(a)\oplus (-1)f_{i.gH}^{\prime \prime }(a)\odot (s-a)\ominus (-1)f_{ii.gH}^{\prime \prime \prime }(a)\odot \frac{(s-a)^2}{2}\\&\oplus \int _{a}^{s}\bigg (\int _{a}^{s_1} \bigg (\int _{a}^{s_2}f_{i.gH}^{(4)}(s_3){\text {d}}s_3){\text {d}}s_2\bigg )\bigg ){\text {d}}s_1; \end{aligned}$$

now, if $(s-a)\rightarrow 0$, we get

$$\begin{aligned} D(f_{ii.gH}^{\prime }(s) \ominus f_{ii.gH}^{\prime }(a),(-1) f_{i.gH}^{\prime \prime }(a)\odot (s-a))\rightarrow 0; \end{aligned}$$

this means

$$\begin{aligned} (-1)f_{i.gH}^{\prime \prime }(a) \approx \frac{f_{ii.gH}^{'}(s)\ominus f_{ii.gH}^{'}(a)}{s-a}. \end{aligned}$$

(iv):

Suppose that $f^{(i)},~i=1,3$, are $[(ii)-gH]$-differentiable and $f^{(i)},~i=2,4$, are $[(i)-gH]$-differentiable, so base on the above procedures,

$$\begin{aligned} f_{i.gH}^{\prime }(s)=f_{i.gH}^{\prime }(a)\ominus (-1) \odot \int _{a}^{s} f_{ii.gH}^{\prime \prime }(s_1){\text {d}}s_{1}, \end{aligned}$$

where

$$\begin{aligned} f_{ii.gH}^{\prime \prime }(s_1)=f_{ii.gH}^{\prime \prime }(a) \ominus (-1)\odot \int _{a}^{s_{1}}f_{i.gH}^{\prime \prime \prime }(s_{2}){\text {d}}s_{2}. \end{aligned}$$

also

$$\begin{aligned} \int _{a}^{s}f_{ii.gH}^{\prime \prime }(s_1){\text {d}}s_{1}=f_{ii.gH}^{\prime \prime }(a) \odot (s-a)\ominus (-1)\odot \int _{a}^{s}\bigg (\int _{a}^{s_1} f_{i.gH}^{\prime \prime \prime }(s_2){\text {d}}s_2\bigg ) d s_1. \end{aligned}$$

By a similar way, it is easy to see that

$$\begin{aligned} f_{i.gH}^{\prime }(s)= & {} f_{i.gH}^{\prime }(a)\oplus (-1)f_{ii.gH}^{\prime \prime }(a)\odot (s-a)\ominus (-1)f_{i.gH}^{\prime \prime \prime }(a)\odot \frac{(s-a)^2}{2}\\&\oplus \int _{a}^{s}\bigg (\int _{a}^{s_1}\bigg (\int _{a}^{s_2} f_{ii.gH}^{(4)}(s_3){\text {d}}s_3){\text {d}}s_2\bigg )\bigg ){\text {d}}s_1, \end{aligned}$$

and finally if $(s-a)\rightarrow 0,$ we can conclude

$$\begin{aligned} f_{i.gH}^{\prime \prime }(a)\approx & {} \frac{f_{i.gH}^{'}(s) \ominus f_{i.gH}^{'}(a)}{s-a}. \end{aligned}$$

Now, based on the above theorem, we introduce the Modified Euler method for solving FDE (3.1). First to integrate the system given Eq. (3.1), we replace the interval [0, T] by a set of discrete equally spaced grid points, $0=t_0< t_1< \cdots < t_N = T$, where $t_{n}=nh$, $h=\frac{T}{N}$.$\square$

Now in order to obtain the proposed method, we consider three cases:

Case 1

Let $y(t)\in {\mathcal {C}}^{4}_{gH} ([0,T], ~{\mathbb {R}}_{\mathcal {F}})$ is $[(i)-gH]$-differentiable solution of problem (3.1) and $y^{(i)}(t),~~i=1,\ldots ,4$ are $[(i)-gH]$-differentiable. Now by using the Taylor series expansion of $y(t_{k+1})$ at the point $t_k$ , for each $k=0,1,\ldots ,N$, we get

$$\begin{aligned} y(t_{k+1})= & {} y(t_k)\oplus (t_{k+1}-t_k )\odot y^{\prime }_{i.gH}(t_k) \\&\oplus \frac{(t_{k+1}-t_k)^2}{2}\odot y^{\prime \prime }_{i.gH}(t_k) \oplus \frac{(t_{k+1}-t_k)^3}{3!}\odot y^{\prime \prime \prime }_{i.gH}(\eta _k); \end{aligned}$$

for some points $\eta _k$ lie between $t_k$ and $t_{k+1}$. Since $h=t_{k+1}- t_k$, gives

$$\begin{aligned} y(t_{k+1})= y(t_k)\oplus h\odot y^{\prime }_{i.gH}(t_k) \oplus \frac{h^2}{2}\odot y^{\prime \prime }_{i.gH}(t _k) \oplus \frac{h^3}{3!}\odot y^{\prime \prime \prime }_{i.gH}(\eta _k). \end{aligned}$$

(3.7)

Consequently, by taking Eq. (3.2) into Eq. (3.7) we obtain:

$$\begin{aligned} y(t_{k+1})= y(t_k) \oplus \frac{h}{2}\odot \bigg (\frac{y^{\prime }_{i.gH}(t_{k+1}) \ominus y^{\prime }_{i.gH}(t_k)}{h}\bigg ) \oplus \frac{h^3}{3!} \odot y^{\prime \prime \prime }_{i.gH}(\eta _k). \end{aligned}$$

Fuzzy differential equation (3.1) implies that $y^{\prime }_{gH}(t)=f(t,y(t))$; so we have

$$\begin{aligned} y(t_{k+1})= y(t_k) \oplus \frac{h}{2}\odot \bigg (\frac{f(t_{k+1} , y(t_{k+1}))\oplus f(t_{k} , y(t_{k}))}{h}\bigg ) \oplus \frac{h^3}{3!}\odot y^{\prime \prime \prime }_{i.gH}(\eta _k). \end{aligned}$$

(3.8)

In order to obtain a numerical method, the value of $y(t_{k+1})$ appearing on the RHS is not known. To handel this, the value of $y(t_{k+1})$ is first predicted by Euler method (Allahviranloo et al. 2015), and then the predicted value is used in Eq. (3.8).

Thus, the Modified Euler method can be written as follows:

$$\begin{aligned} \left\{ \begin{array}{l} y_{k+1}^{*}=y_k \oplus h\odot f(t_k, y_k),\\ y_{k+1}=y_k \oplus \frac{h}{2}\odot \bigg (f(t_{k+1} , y^{*}(t_{k+1}))\oplus f(t_{k} , y(t_{k}))\bigg ), \quad k=0,1,\ldots ,N-1. \end{array}\right. \end{aligned}$$

(3.9)

Case 2

Now, consider y(t) is $[(ii)-gH]$-differentiable and belongs to ${\mathcal {C}}^{3}_{gH} ([0,T], {\mathbb {R}}_{\mathcal {F}})$ and $y^{(i)}(t),~~i=1,\ldots ,4$ are $[(ii)-gH]$-differentiable. So the Taylor’s series expansion of y(t) about the point $t_k$ at $t_{k+1}$ is

$$\begin{aligned} y(t_{k+1})= & {} y(t_k)\ominus (-1) h\odot y^{\prime }_{ii.gH}(t_k) \nonumber \\&\ominus (-1) \frac{h^2}{2}\odot y^{\prime \prime }_{ii.gH} \nonumber \\&\ominus (-1)\frac{h^3}{3!}\odot y^{\prime \prime \prime }_{ii.gH}(\eta _k), \end{aligned}$$

(3.10)

Now, by taking Eq. (3.3) into Eq. (3.10) we obtain

$$\begin{aligned} y(t_{k+1})= y(t_k)\ominus (-1) \frac{h}{2}\odot \bigg ( y^{\prime }_{ii.gH}(t_{k}) \oplus y^{\prime }_{ii.gH}(t_{k+1})\bigg ), \end{aligned}$$

As described in Case 1, the Modified Euler method takes the following form:

$$\begin{aligned} \left\{ \begin{array}{l} y_{k+1}^{*}=y_k \ominus (-1) h\odot f(t_k, y_k),\\ y_{k+1}= y_k \ominus (-1)\frac{h}{2}\odot \bigg (f(t_{k+1} , y^{*}(t_{k+1}))\oplus f(t_{k} , y(t_{k}))\bigg ), k=0,1,\ldots ,N-1. \end{array}\right. \nonumber \\ \end{aligned}$$

(3.11)

Case 3

We Consider partition of [0, T] as follows:

$$\begin{aligned} t_0=0, t_1, \ldots ,t_j,\gamma , t_{j+1},\ldots ,t_N=T. \end{aligned}$$

(3.12)

If the assumptions of case 1 are true for $y(t),~t \in [0,t_{j}]$ and case 2 are true for $y(t),~t \in [t_{j+1},T]$ ($\gamma \in [0,T]$ is a switching point type I), the Modified Euler method can be written as follows:

$$\begin{aligned} \left\{ \begin{array}{l} y_{k+1}^{*}=y_k \oplus h\odot f(t_k, y_k),\\ y_{k+1}= y_k \oplus \frac{h}{2}\odot \bigg (f(t_{k+1} , y^{*}(t_{k+1})) \oplus f(t_{k} , y(t_{k}))\bigg ),\qquad k=0,1,\ldots ,j.\\ \\ y_{k+1}^{*}=y_k \ominus (-1) h\odot f(t_k, y_k),\\ y_{k+1}= y_k \ominus (-1)\frac{h}{2}\odot \bigg (f(t_{k+1} , y^{*}(t_{k+1})) \oplus f(t_{k}, y(t_{k}))\bigg ),\quad k=j+1,1,\ldots ,N-1.\\ \end{array}\right. \nonumber \\ \end{aligned}$$

(3.13)

Case 4

If the assumptions of case 2 are true for $y(t),~t \in [0,t_{j}]$ and case 1 are true for $y(t),~t \in [t_{j+1},T]$ ($\gamma \in [0,T]$ is a switching point type II); hence the Modified Euler method can be written as follows:

$$\begin{aligned} \left\{ \begin{array}{l} y_{k+1}^{*}=y_k \ominus (-1) h\odot f(t_k, y_k),\\ y_{k+1}= y_k \ominus (-1)\frac{h}{2}\odot \bigg (f(t_{k+1} , y^{*}(t_{k+1}))\oplus f(t_{k} , y(t_{k}))\bigg ),\quad k=0,1,\ldots ,j, \\ \\ y_{k+1}^{*}=y_k \oplus h\odot f(t_k, y_k),\\ y_{k+1}= y_k \oplus \frac{h}{2}\odot \bigg (f(t_{k+1} , y^{*}(t_{k+1}))\oplus f(t_{k} , y(t_{k}))\bigg ),\qquad k=j+1,1,\ldots ,N-1.\\ \end{array}\right. \nonumber \\ \end{aligned}$$

(3.14)

3.1 Consistency, convergence and stability

In this section, consistency, convergence and stability of the Modified Euler Method are discussed in detail.

Theorem 3.2

The Modified Euler method is consistent.

Proof

First, let y(t) is $[(i)-gH]$-differentiable; by Eq. (3.9) we can write

$$\begin{aligned} \phi _k = y_{k+1}\ominus \bigg (y_k \oplus \frac{h}{2}\odot \bigg (f(t_{k+1} , y_{k} \oplus f(t_{k} , y_{k}) )\oplus f(t_{k} , y_{k})\bigg )\bigg ). \end{aligned}$$

The local truncation error is defined as

$$\begin{aligned} \tau _{k}=\frac{1}{h}\odot \phi _{k}; \end{aligned}$$

now, it is sufficient to show

$$\begin{aligned} \lim _{h\rightarrow 0}\max D(\tau _{k},0)=0, \end{aligned}$$

where $\tau _{k}=\frac{h^2}{3!}y_{i.gH}^{\prime \prime \prime } (\xi _{k})$ and $D(y_{i.gH}^{\prime \prime \prime },0)\le M_{1}$.

Consequently,

$$\begin{aligned} \lim _{h\rightarrow 0}\max D(\tau _{k},0)=\lim _{h\rightarrow 0} \frac{h^2}{3!}\odot \max D(y_{i.gH}^{\prime \prime \prime }(\xi _{k}),0) \le \frac{h^2}{3!}M_{1}=0. \end{aligned}$$

If y(t) is $[(ii)-gH]$-differentiable, then

$$\begin{aligned} \tau _{k}=\ominus (-1)\frac{h^2}{3!}\odot y_{ii.gH}^{\prime \prime \prime }(\xi _{k}), \end{aligned}$$

and $D(y_{ii.gH}^{\prime \prime \prime },0)\le M_{2}$;

therefore,

$$\begin{aligned} \lim _{h\rightarrow 0}\max D(\tau _{k},0)=\lim _{h\rightarrow 0}\left| \frac{-h^2}{3!}\right| \odot \max D(\ominus y_{i.gH}^{\prime \prime \prime }(\xi _{k}),0) \le \frac{h^2}{3!}M_{2}=0; \end{aligned}$$

so, the proof of the theorem is complete. $\square$

Lemma 3.1

(see Nieto et al. 2009) For all real z,

$$\begin{aligned} 1+z\le e^{z}. \end{aligned}$$

(3.15)

Theorem 3.3

Let $y_{gH}^{\prime \prime \prime }(t)$ exist and f(t, y(t)) satisfy in Lipschitz condition on the $\{(t,y(t))| t \in [0,p], y\in \overline{B}(y_{0},q),~p,q>0\}$; then the Modified Euler Method converges to the solution of fuzzy differential Eq. (3.1).

Proof

Let y(t) is $[(i)-gH]$-differentiable; then the Modified Euler method may be written in the following form:

$$\begin{aligned} y_{k+1}=y_{k}\oplus h \odot \phi (t_{n},y_{n};h), \end{aligned}$$

(3.16)

where

$$\begin{aligned} \phi (t_{n},y_{n};h)=\frac{1}{2}[f(t_{n},y_{n}) \oplus f(t_{n}+h,y_{n}\oplus h \odot f(t_{n},y_{n}))], \end{aligned}$$

and $\phi (.,.;.)$ is continuous function of its variables. First we want to verify the Lipschitz condition of the function $\phi$ for the Modified Euler method.

From Lipschitz condition, we obtain

$$\begin{aligned} D(\phi (t,u;h),\phi (t,v;h))\le & {} \frac{1}{2} D(f(t,u),f(t,v))\\&+\frac{1}{2}D(f(t+h,u\oplus h \odot f(t,u)),f(t+h,v \oplus h \odot f(t,v))),\\\le & {} \frac{L}{2}D(u,v)+\frac{L}{2}D(u\oplus h \odot f(t,u),v\oplus h \odot f(t,v)),\\\le & {} L\left( 1+\frac{hL}{2}\right) D(u,v)=L_{\phi }D(u,v), \end{aligned}$$

i.e. $\phi$ satisfies a Lipschitz condition with constant $(L+\frac{1}{2}hL^2)$.

Now the exact solution of Eq. (3.1) is satisfied:

$$\begin{aligned} y(t_{k+1})= y(t_{k}) \oplus {h}\odot \phi (t_{k},y(t_{k});h) \oplus R_{k}, \end{aligned}$$

(3.17)

where $R_{k}=\frac{h^3}{3!}\odot y_{i.gH}^{\prime \prime \prime }(t_{k})$.

Subtracting Eq. (3.17) from Eq. (3.16) and by using Lipschitz condition, the following equation is obtained:

$$\begin{aligned} D(y(t_{k+1}),y_{k+1})\le (1+hL_{\phi _{k}})D(y(t_{k}),y_{k}) \oplus D(R_{k},0); \end{aligned}$$

(3.18)

we put

$$\begin{aligned} L_{\phi }=\max _{0\le k \le N} \{L_{\phi _{k}}\},\quad R =\max _{0\le k \le N}D(R_{k},0), \end{aligned}$$

and by substituting in Eq. (3.18), get

$$\begin{aligned} D(y(t_{k+1}),y_{k+1})\le (1+hL_{\phi })D(y(t_{k}),y_{k})\oplus R, \end{aligned}$$

it is easy to see that

$$\begin{aligned} D(y(t_{k+1}),y_{k+1})\le (1+hL_{\phi })^{k+1}D(y(t_{0}),y_{0}) \oplus [1+(1+hL_{\phi })+\cdots + (1+hL_{\phi })^{k}]\odot R. \end{aligned}$$

On the other hand,

$$\begin{aligned} \sum _{i=0}^{k}(1+hL_{\phi })^{i}=\frac{(1+hL_{\phi })^{k+1}-1}{hL_{\phi }}, \end{aligned}$$

Now for $0\le (k+1)h\le T$, $(k+1)\le (N-1)$, and by using Eq. (3.15), we obtain

$$\begin{aligned} D(y(t_{k+1}),y_{k+1}) \le e^{TL_{\phi _{k}}} D(y(t_{0}),y_{0})+\frac{R}{hL_{\phi }}[e^{TL_{\phi }}-1], \end{aligned}$$

where $R=\max _{0\le k \le N}D(\frac{h^3}{3!} \odot y_{i.gH}^{\prime \prime \prime }(\xi _{k}),0)$.

It is clear that $D(y(t_{0}),y_{0})=0$; so

$$\begin{aligned} D(y(t_{k+1}),y_{k+1}) \le \frac{h^{2}}{6L_{\phi }}[e^{TL_{\phi }}-1] \max _{0\le t\le T} D(y^{\prime \prime \prime }_{i.gH}(t),0). \end{aligned}$$

Thus, $\lim _{h\rightarrow 0}D(y(t_{k+1}),y_{k+1}) \rightarrow 0$ and in this case, Modified Euler method converges.

Consider y(t) is $[(ii)-gH]$-differentiable, the Modified Euler method is written as follows:

$$\begin{aligned} y_{k+1}=y_{k}\ominus (-1) h \odot \varphi (t_{n},y_{n};h), \end{aligned}$$

(3.19)

where

$$\begin{aligned} \varphi (t_{n},y_{n};h)=\frac{1}{2}[f(t_{n},y_{n})\oplus f(t_{n}+h,y_{n}\ominus (-1) h \odot f(t_{n},y_{n}))], \end{aligned}$$

and $\varphi (.,.;.)$ is continuous function of its variables. The Lipschitz condition of the function $\varphi$ for the Modified Euler method is as follows:

$$\begin{aligned} D(\varphi (t,u;h),\varphi (t,v;h))\le & {} \frac{1}{2} D(f(t,u),f(t,v))\\&+\frac{1}{2}D(f(t+h,u\ominus (-1)h f(t,u)),f(t+h,v\ominus (-1)h f(t,v))),\\\le & {} \frac{L}{2}D(u,v)+\frac{L}{2}D(u\ominus (-1) h \odot f(t,u),v\ominus (-1)h \odot f(t,v)),\\\le & {} L\left( 1-\frac{hL}{2}\right) D(u,v)=L_{\varphi }D(u,v), \end{aligned}$$

i.e. $\varphi$ satisfies a Lipschitz condition with constant $(L-\frac{1}{2}hL^2)$.

Now the exact solution of Eq. (3.1) is satisfied:

$$\begin{aligned} y(t_{k+1}) = y(t_{k}) \ominus (-1) {h}\odot \varphi (t_{k},y(t_{k});h) \oplus J_{k}, \end{aligned}$$

(3.20)

where $J_{k}=-\frac{h^3}{3!}\odot y_{ii.gH}^{\prime \prime \prime }(t_{k})$.

Subtracting Eq. (3.20) from Eq. (3.19) and by using Lipschitz condition, the following equation is obtained:

$$\begin{aligned} D(y(t_{k+1}),y_{k+1})\le (1-hL_{\phi _{k}})D(y(t_{k}),y_{k})\oplus D(J_{k},0); \end{aligned}$$

(3.21)

now by putting

$$\begin{aligned} L_{\varphi }=\max _{0\le k \le N} \{L_{\varphi _{k}}\}, \quad J=\max _{0\le k \le N}D(J_{k},0), \end{aligned}$$

and by similar procedure, we obtain

$$\begin{aligned} D(y(t_{k+1}),y_{k+1}) \le \frac{-h^{2}}{6L_{\phi }} \left[ \frac{1}{e^{TL_{\phi }}}-1\right] \max _{0\le t\le T} D(y^{\prime \prime \prime }_{ii.gH}(t),0). \end{aligned}$$

Thus, $\lim _{h\rightarrow 0}D(y(t_{k+1}),y_{k+1}) \rightarrow 0$. $\square$

Theorem 3.4

The Modified Euler method is stable.

Proof

Let $y_{k+1}$, $k+1\ge 0$ be the solution of the Modified Euler method with initial condition $y_{0}\in {\mathbb {R}}_{F}$ and let $\tau _{k+1}$ be the solution of the Modified Euler method with perturbed fuzzy initial condition $\tau _{0}=y_{0}\oplus \epsilon \in {\mathbb {R}}_{F}$; therefore, if y(t) is $[(i)-gH]$-differentiable

$$\begin{aligned} \tau _{k+1}=\tau _{k}\oplus \frac{h}{2}\odot \bigg (f(t_{k+1} , \tau _{k}\oplus h \odot f(t_{k},\tau _{k}))\oplus f(t_{k} , \tau _{k})\bigg ),\quad \tau _{0}=y_{0}\oplus \epsilon , \end{aligned}$$

then by using Lipschitz condition and properties of Hausdorff distance, we have

$$\begin{aligned} D(y_{k+1},\tau _{k+1})\le D(y_{k},\tau _{k})+\frac{h}{2}L\bigg (D(y_{k},\tau _{k}) + hL D(y_{k},\tau _{k})\bigg )+\frac{h}{2}LD(y_{k},\tau _{k}), \end{aligned}$$

Thus

$$\begin{aligned} D(y_{k+1},\tau _{k+1})\le \left( 1+hL+\frac{(hL)^2}{2}\right) D(y_{k},\tau _{k}). \end{aligned}$$

Now by iterating the above inequality, we can write

$$\begin{aligned} D(y_{k+1},\tau _{k+1})\le & {} \left( 1+hL+\frac{(hL)^2}{2}\right) ^{k}D(y_{0},\tau _{0}) \le (e^{hL})^{k}D(y_{0},\tau _{0})\\\le & {} e^{LT}D(y_{0},\tau _{0}) \le \kappa D(y_{0},\tau _{0}), \end{aligned}$$

where $\kappa =e^{TL}$, and $kh\le (k+1)h \le T$. So, if y(t) is $[(i)-gH]$-differentiable, this method is stable. $\square$

Now, we suppose that y(t) is $[(ii)-gH]$-differentiable; then it is easy to see that

$$\begin{aligned} D(y_{k+1},\tau _{k+1})\le D(y_{k},\tau _{k})-\frac{h}{2}L\bigg (D(y_{k},\tau _{k}) - hL D(y_{k},\tau _{k})\bigg )-\frac{h}{2}LD(y_{k},\tau _{k}); \end{aligned}$$

therefore

$$\begin{aligned} D(y_{k+1},\tau _{k+1})\le \left( 1-hL+\frac{(hL)^2}{2}\right) D(y_{k},\tau _{k}); \end{aligned}$$

by repeating the earlier procedure we obtain

$$\begin{aligned} D(y_{k+1},\tau _{k+1})\le & {} \left( 1-hL+\frac{(hL)^2}{2}\right) ^{k}D(y_{0},\tau _{0}) \le (e^{-hL})^{k}D(y_{0},\tau _{0})\\\le & {} e^{-LT}D(y_{0},\tau _{0}) \le \kappa D(y_{0},\tau _{0}), \end{aligned}$$

where $\kappa =e^{-TL}$, and $kh\le (k+1)h \le T$; therefore, the Modified Euler method is stable.

4 Numerical examples

In this section we will provide three examples to emphasize acceptable accuracy of Modified Euler Method. All numerical computations were performed using Maple 13 software package.

Example 4.1

(see. Allahviranloo et al. 2015) Consider the first-order fuzzy initial value problem as follows:

$$\begin{aligned} \left\{ \begin{array}{l} y^{\prime }_{i.gH}(t)= y(t)~ \oplus ~ (1.3, 2, 2.1), \quad 0\le t \le 1; \\ y(0)= (0.82, 1, 1.2), \end{array}\right. \end{aligned}$$

such that the exact $[(i)-gH]$-differentiable solution is obtained by solving the following system:

$$\begin{aligned} \left\{ \begin{array}{ll} (y_{-})^{\prime }(t;{\alpha } ) = y_{-}(t; \alpha ) + 1.3 + 0.7 \alpha , &{} 0\le t \le 1; \\ (y^{+})^{\prime }(t;{\alpha } ) = y^{+}(t; \alpha ) + 2.1 - 0.1 \alpha , &{} 0\le t \le 1; \\ y (0; \alpha ) = [0.82 + 0.18 \alpha , 1.2 - 0.2 \alpha ] , &{} . \end{array}\right. \end{aligned}$$

Now we use the Modified Euler method to obtain approximate solution and we compare our solution with Euler method (Allahviranloo et al. 2015). The global truncation errors of the Modified Euler and Euler method have been reported for $h=0.025$ and 0.005 in Table 1, and the approximated solution by the Modified Euler method is shown in Fig. 1.

Table 1 The global truncation errors for Example 4.1

Fig. 1

Approximated solution for Example 4.1. Red lines: $y^{+}(t,r)$; black lines:$y^{-}(t,r)$

Example 4.2

(see Allahviranloo et al. 2015) Let us consider the following initial value problem:

$$\begin{aligned} \left\{ \begin{array}{ll} y'_{ii.gH}(t)= -y(t)\oplus t \odot (0.7, 1, 1.8), &{} \quad 0\le t \le 1, \\ y(0) = (0, 1, 2.2) &{} \end{array}\right. \end{aligned}$$

(4.1)

Since y(t) has $[(ii)-gH]$-derivative, the $[(ii)-gH]$-differentiable solution is obtained by solving

$$\begin{aligned} \left\{ \begin{array}{ll} (y_{-})^{\prime }(t;{\alpha } ) = - y_{-}(t; \alpha ) + t ~[1.8 - 0.8\alpha ], &{} \ 0\le t \le 1,\\ (y^{+})^{\prime }(t;{\alpha } ) = - y^{+}(t; \alpha ) + t ~[0.7+ 0.3 \alpha ], &{} \ 0\le t \le 1,\\ y (0; \alpha ) = [\alpha , 2.2 - 1.2 \alpha ] &{} \end{array}\right. \end{aligned}$$

The global truncation errors of Modified Euler and Euler method have been reported for $h=0.025$ and 0.005 in Table 2, and the approximated solution by the Modified Euler method is shown in Fig. 2.

Table 2 The global truncation errors for Example 4.2

Fig. 2

Approximated solution for Example 4.2. Red lines: $y^{+}(t,r)$; black lines:$y^{-}(t,r)$

Example 4.3

(see Allahviranloo et al. 2015) Consider the initial value problem

$$\begin{aligned} \left\{ \begin{array}{ll} y'_{gH}(t)= (1-t)~y(t), &{} \quad 0\le t \le 2, \\ y(0) = (0, 1, 2) &{} \end{array}\right. \end{aligned}$$

Obviously that initial value problem on [0, 1] is $[(i)-gH]$-differentiable and at $t=1$ the problem is switched to $[(ii)-gH]$-differentiable. So, the point $t=1$ is a switching point and the obtained solution on [0,  1] is $[(i)-gH]$-differentiable and $[(ii)-gH]$-differentiable on (1,  2]. The $[(i)-gH]$-differentiable solution can be obtained by solving

$$\begin{aligned} \left\{ \begin{array}{l} (y_{-})^{\prime }(t;{\alpha } ) = \left\{ \begin{array}{ll} (1-t) y_{-}(t; \alpha ), &{} \quad 0\le t \le 1, \\ (1-t) y^{+}(t; \alpha ), &{} \quad 1< t \le 2, \end{array}\right. \\ (y^{+})^{\prime }(t;{\alpha } ) =\left\{ \begin{array}{ll} (1-t) y^{+}(t; \alpha ), &{} \quad 0\le t \le 1, \\ (1-t) y_{-}(t; \alpha ), &{} \quad 1 < t \le 2, \end{array}\right. \\ y (0; \alpha ) = [\alpha , 2 - \alpha ] \end{array}\right. \end{aligned}$$

The global truncation errors of Modified Euler and Euler method have been reported for $h=0.025$ and 0.005 in Table 3, and the approximated solution by the Modified Euler method is plotted in Fig. 3.

Table 3 The global truncation errors for Example 4.3

Fig. 3

Approximated solution for Example 4.3. Red lines: $y^{+}(t,r)$; black lines:$y^{-}(t,r)$

Example 4.4

Jafari and Razvarz (2018) A tank with a heating system is displayed in Fig. 4, where $\tilde{R} = 0.5$, the thermal capacitance is $\tilde{C} = 2$, and the temperature is $\psi$. The model is formulated as follows:

$$\begin{aligned} \left\{ \begin{array}{ll} \phi ^{\prime }(t)=-\frac{1}{\tilde{R}\tilde{C}}\phi (t), \quad 0 \le t \le T, \\ \phi (0)=(\phi ^{-}(0,\alpha ),\quad \phi ^{+}(0,\alpha )), \end{array}\right. \end{aligned}$$

(4.2)

where the initial condition is a symmetric triangular fuzzy number as $\phi (0)=(-a(1-\alpha ),a(1-\alpha )).$

The solution of FDE (4.2), by using Modified Euler method, for $a=2, h=0.005$ is shown in Fig. 5.

Fig. 4

Thermal system

Fig. 5

Approximated and Real solution for Example 4.4

5 Conclusion

In this paper, we provided the numerical method for solving fuzzy differential equations under generalized differentiability. By improving Euler method (Allahviranloo et al. 2015), the Modified Euler method without losing its simplicity was introduced, and it was shown that the global error for the Modified Euler method was $o(h^2)$, while the global error of Euler method was o(h). Finally, some examples were provided to emphasize acceptable accuracy of proposed method.