# New Approximation Methods Based on Fuzzy Transform for Solving SODEs: II

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic Concepts

**Definition**

**1.**

- 1.
- (positivity and locality)—${A}_{i}\left(t\right)>0$ if $t\in \left(\right)open="("\; close=")">{t}_{i-1},{t}_{i+1}$ and ${A}_{i}\left(t\right)=0$ if $t\in \left(\right)open="["\; close="]">a,b$;
- 2.
- (continuity)—${A}_{i}$ is continuous on $\left(\right)$;
- 3.
- (covering)—for $t\in \left(\right)open="["\; close="]">a,b$.

- 4.
- ${A}_{i}\left(\right)open="("\; close=")">{t}_{i}-t$ for all $t\in \left(\right)open="["\; close="]">0,h$;
- 5.
- ${A}_{i}\left(t\right)={A}_{i-1}\left(\right)open="("\; close=")">t-h$ and ${A}_{i+1}\left(t\right)={A}_{i}\left(\right)open="("\; close=")">t-h$ for all $t\in \left(\right)open="["\; close="]">{t}_{i},\phantom{\rule{0.166667em}{0ex}}{t}_{i+1}$;

**Definition**

**2.**

**Definition**

**3.**

**Remark**

**1.**

#### New Iterative Method

## 3. New Representations for Basic Functions of FzT

#### 3.1. Generalized Uniform Fuzzy Partitions with the Generalized Normal Case

**Definition**

**4.**

**Lemma**

**1.**

**Proof.**

#### 3.2. Simpler Form of F-Transform Components Based on Generalized Uniform Fuzzy Partitions with the Generalized Normal Case

**Definition**

**5.**

**Lemma**

**2.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Corollary**

**3.**

**Proof.**

**Remark**

**2.**

**Definition**

**6.**

**Lemma**

**4.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Remark**

**3.**

- Select the generating function K which is assumed to be even, continuous and $K\left(t\right)>0$ if $\phantom{\rule{4pt}{0ex}}t\in \left(\right)open="("\; close=")">-1,1$.
- Specify the value $\beta =1/K\left(0\right)$, where $K\left(0\right)\ne 0$ to get the normal generating function K and then compute the value $\lambda =1/\left(\right)open="("\; close=")">{\int}_{-1}^{1}\beta K\left(t\right)dt$, where ${\int}_{-1}^{1}\beta K\left(t\right)dt\ne 0$.
- If conditions $\beta >0\mathrm{and}\lambda 0$ holds, then construct generalized uniform fuzzy partitions of $[-1,1]$ by $\lambda \beta K\left(t\right)$.

**Example**

**1.**

**Remark**

**4.**

## 4. New Fuzzy Numerical Methods for Solving SODEs

- Specify the number n of components and compute the step $h=\left(\right)open="("\; close=")">{t}_{n}-{t}_{1}$. If we want to obtain as best approximation of f as possible, then n should be large.
- Construct the nodes ${t}_{1}<\dots <{t}_{n}$, where ${t}_{k}={t}_{1}+h(k-1)$.
- Select the shape of basic functions. This is achieved by selecting the shape of generating function.
- Construct a h-uniform generalized fuzzy partition of $\left(\right)$ by new representations of basic functions are defined by Definition 4.

#### 4.1. Numerical Scheme I: Modified Trapezoidal Rule Based on FzT and NIM for SODEs

#### 4.2. Numerical Scheme II: Modified 2-Step Adams Moulton Method Based on FzT and NIM for SODEs

#### 4.3. Numerical Scheme III: Modified 3-Step Adams Moulton Method Based on FzT and NIM for SODEs

#### 4.4. Error Analysis of Numerical Scheme I for SODEs

**Lemma**

**5.**

**Theorem**

**3.**

## 5. Applications

**Example**

**2.**

**Example**

**3.**

- Moreover, comparison of MSE for Examples 2 and 3 shown in Table 6. It is observed that the new fuzzy approximation methods yield more accurate results in comparison with the classical Trapezoidal rule (one step) and classical Adams Moulton method (two and three steps). Hence, the new fuzzy approximation methods provide alternative techniques for solving SODEs with better results.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Algorithms

INPUT: $f(t,x,y)$; $g(t,x,y)$; endpoints $a,b$; integer N; initial condition ${y}_{1}$; m. |

Step 1 Set $h=(b-a)/N$; ${X}_{1}={x}_{1}$; ${Y}_{1}={y}_{1}$; ${t}_{1}=a$; $k=1,\dots ,N+1$; ${t}_{k}=a+(k-1)h$. |

Step 2 Define the generalized uniform fuzzy partitions as ${B}_{k}\left(t\right)=\left(\right)open="("\; close=")">\frac{\sqrt{\pi}\mathsf{\Gamma}\left(\right)open="("\; close=")">m+1}{}2\mathsf{\Gamma}\left(\right)open="("\; close=")">m+\frac{1}{2}$. |

Step 3 for $k=1$ to N do Steps 04–15. |

$\begin{array}{ccc}\hfill \mathrm{Step}04& \hfill F\left(k\right)& =\mathrm{integral}(f(t,X\left(k\right),Y\left(k\right)){B}_{k}\left(t\right),t(k-1),t(k+1))/\mathrm{integral}({B}_{k}\left(t\right),t(k-1),t(k+1)).\hfill \\ \hfill \mathrm{Step}05& \hfill G\left(k\right)& =\mathrm{integral}(g(t,X\left(k\right),Y\left(k\right)){B}_{k}\left(t\right),t(k-1),t(k+1))/\mathrm{integral}({B}_{k}\left(t\right),t(k-1),t(k+1)).\hfill \\ \hfill \mathrm{Step}06& \hfill Xstar(k+1)& =X\left(k\right)+hF\left(k\right)/2.\hfill \\ \hfill \mathrm{Step}07& \hfill Ystar(k+1)& =Y\left(k\right)+hG\left(k\right)/2.\hfill \\ \hfill \mathrm{Step}08& \hfill Fstar(k+1)& =\mathrm{integral}(f(t,Xstar(k+1),Ystar(k+1)){B}_{k+1}\left(t\right),t(k-1),t(k+1))/\mathrm{integral}({B}_{k+1}\left(t\right),t(k-1),t(k+1)).\hfill \\ \hfill \mathrm{Step}09& \hfill Gstar(k+1)& =\mathrm{integral}(g(t,Xstar(k+1),Ystar(k+1)){B}_{k+1}\left(t\right),t(k-1),t(k+1))/\mathrm{integral}({B}_{k+1}\left(t\right),t(k-1),t(k+1)).\hfill \\ \hfill \mathrm{Step}10& \hfill Xstar2(k+1)& =Xstar(k+1)+hFstar(k+1)/2.\hfill \\ \hfill \mathrm{Step}11& \hfill Ystar2(k+1)& =Ystar(k+1)+hGstar(k+1)/2.\hfill \\ \hfill \mathrm{Step}12& \hfill Fstar2(k+1)& =\mathrm{integral}(f(t,Xstar2(k+1),Ystar2(k+1)){B}_{k+1}\left(t\right),t(k-1),t(k+1))/\mathrm{integral}({B}_{k+1}\left(t\right),t(k-1),t(k+1)).\hfill \\ \hfill \mathrm{Step}13& \hfill Gstar2(k+1)& =\mathrm{integral}(g(t,Xstar2(k+1),Ystar2(k+1)){B}_{k+1}\left(t\right),t(k-1),t(k+1))/\mathrm{integral}({B}_{k+1}\left(t\right),t(k-1),t(k+1)).\hfill \\ \hfill \mathrm{Step}14& \hfill X(k+1)& =X\left(k\right)+h\left(\right)open="("\; close=")">F\left(k\right)+Fstar2(k+1)/2.\hfill \end{array}$ |

end. |

OUTPUT: Approximation X and Y to x and y, respectively at the ($N+1$) values of t. |

INPUT: $f(t,x,y)$; $g(t,x,y)$; endpoints $a,b$; integer N; initial condition ${y}_{1}$; m. |

Step 1 Set $h=(b-a)/N$; ${X}_{1}={x}_{1}$; ${Y}_{1}={y}_{1}$; ${t}_{1}=a$; $k=1,\dots ,N+1$; ${t}_{k}=a+(k-1)h$. |

Step 2 Define the generalized uniform fuzzy partitions as ${B}_{k}\left(t\right)=\left(\right)open="("\; close=")">\frac{\sqrt{\pi}\mathsf{\Gamma}\left(\right)open="("\; close=")">m+1}{}2\mathsf{\Gamma}\left(\right)open="("\; close=")">m+\frac{1}{2}$. |

Step 3 Set ${X}_{2}={x}_{2}$; ${Y}_{2}={y}_{2}$. (In the case of no exact solutions, compute ${X}_{2}$ and ${Y}_{2}$ using Algorithm 1.) |

Step 4 for $k=2$ to N do Steps 05–18. |

$\begin{array}{ccc}\hfill \mathrm{Step}05& \hfill F(k-1)=& \mathrm{integral}(f(t,X(k-1),Y(k-1)){B}_{k-1}\left(t\right),t(k-1),t(k+1))/\mathrm{integral}({B}_{k-1}\left(t\right),t(k-1),t(k+1)).\hfill \\ \hfill \mathrm{Step}06& \hfill G(k-1)=& \mathrm{integral}(g(t,X(k-1),Y(k-1)){B}_{k-1}\left(t\right),t(k-1),t(k+1))/\mathrm{integral}({B}_{k-1}\left(t\right),t(k-1),t(k+1)).\hfill \\ \hfill \mathrm{Step}07& \hfill F\left(k\right)=& \mathrm{integral}(f(t,X\left(k\right),Y\left(k\right)){B}_{k}\left(t\right),t(k-1),t(k+1))/\mathrm{integral}({B}_{k}\left(t\right),t(k-1),t(k+1)).\hfill \\ \hfill \mathrm{Step}08& \hfill G\left(k\right)=& \mathrm{integral}(g(t,X\left(k\right),Y\left(k\right)){B}_{k}\left(t\right),t(k-1),t(k+1))/\mathrm{integral}({B}_{k}\left(t\right),t(k-1),t(k+1)).\hfill \\ \hfill \mathrm{Step}09& \hfill Xstar(k+1)=& X\left(k\right)+h\left(8F\right(k)-F(k-1\left)\right)/12.\hfill \\ \hfill \mathrm{Step}10& \hfill Ystar(k+1)=& Y\left(k\right)+h\left(8G\right(k)-G(k-1\left)\right)/12.\hfill \\ \hfill \mathrm{Step}11& \hfill Fstar(k+1)=& \mathrm{integral}(f(t,Xstar(k+1),Ystar(k+1)){B}_{k+1}\left(t\right),t(k-1),t(k+1))/\mathrm{integral}({B}_{k+1}\left(t\right),t(k-1),t(k+1)).\hfill \\ \hfill \mathrm{Step}12& \hfill Gstar(k+1)=& \mathrm{integral}(g(t,Xstar(k+1),Ystar(k+1)){B}_{k+1}\left(t\right),t(k-1),t(k+1))/\mathrm{integral}({B}_{k+1}\left(t\right),t(k-1),t(k+1)).\hfill \\ \hfill \mathrm{Step}13& \hfill Xstar2(k+1)=& Xstar(k+1)+5hFstar(k+1)/12.\hfill \\ \hfill \mathrm{Step}14& \hfill Ystar2(k+1)=& Ystar(k+1)+5hGstar(k+1)/12.\hfill \\ \hfill \mathrm{Step}15& \hfill Fstar2(k+1)=& \mathrm{integral}(f(t,Xstar2(k+1),Ystar2(k+1)){B}_{k+1}\left(t\right),t(k-1),t(k+1))/\mathrm{integral}({B}_{k+1}\left(t\right),t(k-1),t(k+1)).\hfill \\ \hfill \mathrm{Step}16& \hfill Gstar2(k+1)=& \mathrm{integral}(g(t,Xstar2(k+1),Ystar2(k+1)){B}_{k+1}\left(t\right),t(k-1),t(k+1))/\mathrm{integral}({B}_{k+1}\left(t\right),t(k-1),t(k+1)).\hfill \\ \hfill \mathrm{Step}17& \hfill X(k+1)=& X\left(k\right)+h\left(8F\right(k)-F(k-1)+5Fstar2(k+1\left)\right)/12.\hfill \\ \hfill \mathrm{Step}18& \hfill Y(k+1)=& Y\left(k\right)+h\left(8G\right(k)-G(k-1)+5Gstar2(k+1\left)\right)/12.\hfill \end{array}$ |

end. |

OUTPUT: Approximation X and Y to x and y, respectively at the ($N+1$) values of t. |

INPUT: $f(t,x,y)$; $g(t,x,y)$; endpoints $a,b$; integer N; initial condition ${y}_{1}$; m. |

Step 1 Set $h=(b-a)/N$; ${X}_{1}={x}_{1}$; ${Y}_{1}={y}_{1}$; ${t}_{1}=a$; $k=1,\dots ,N+1$; ${t}_{k}=a+(k-1)h$. |

Step 2 Define the generalized uniform fuzzy partitions as ${B}_{k}\left(t\right)=\left(\right)open="("\; close=")">\frac{\sqrt{\pi}\mathsf{\Gamma}\left(\right)open="("\; close=")">m+1}{}2\mathsf{\Gamma}\left(\right)open="("\; close=")">m+\frac{1}{2}$. |

Step 3 Set ${X}_{2}={x}_{2}$; ${Y}_{2}={y}_{2}$; ${X}_{3}={x}_{3}$; ${Y}_{3}={y}_{3}$. (In the case of no exact solutions, compute ${X}_{2}$, ${Y}_{2}$, ${X}_{3}$ and ${Y}_{3}$ using Algorithm 1 or 2.) |

Step 4 for $k=3$ to N do Steps 05–20. |

$\begin{array}{ccc}\hfill \mathrm{Step}05& \hfill F(k-2)=& \mathrm{integral}(f(t,X(k-2),Y(k-2)){B}_{k-2}\left(t\right),t(k-1),t(k+1))/\mathrm{integral}({B}_{k-2}\left(t\right),t(k-1),t(k+1)).\hfill \\ \hfill \mathrm{Step}06& \hfill G(k-2)=& \mathrm{integral}(g(t,X(k-2),Y(k-2)){B}_{k-2}\left(t\right),t(k-1),t(k+1))/\mathrm{integral}({B}_{k-2}\left(t\right),t(k-1),t(k+1)).\hfill \\ \hfill \mathrm{Step}07& \hfill F(k-1)=& \mathrm{integral}(f(t,X(k-1),Y(k-1)){B}_{k-1}\left(t\right),t(k-1),t(k+1))/\mathrm{integral}({B}_{k-1}\left(t\right),t(k-1),t(k+1)).\hfill \\ \hfill \mathrm{Step}08& \hfill G(k-1)=& \mathrm{integral}(g(t,X(k-1),Y(k-1)){B}_{k-1}\left(t\right),t(k-1),t(k+1))/\mathrm{integral}({B}_{k-1}\left(t\right),t(k-1),t(k+1)).\hfill \\ \hfill \mathrm{Step}09& \hfill F\left(k\right)=& \mathrm{integral}(f(t,X\left(k\right),Y\left(k\right)){B}_{k}\left(t\right),t(k-1),t(k+1))/\mathrm{integral}({B}_{k}\left(t\right),t(k-1),t(k+1)).\hfill \\ \hfill \mathrm{Step}10& \hfill G\left(k\right)=& \mathrm{integral}(g(t,X\left(k\right),Y\left(k\right)){B}_{k}\left(t\right),t(k-1),t(k+1))/\mathrm{integral}({B}_{k}\left(t\right),t(k-1),t(k+1)).\hfill \\ \hfill \mathrm{Step}11& \hfill Xstar(k+1)=& X\left(k\right)+h\left(19F\right(k)-5F(k-1)+F(k-2\left)\right)/24.\hfill \\ \hfill \mathrm{Step}12& \hfill Ystar(k+1)=& Y\left(k\right)+h\left(19G\right(k)-5G(k-1)+G(k-2\left)\right)/24.\hfill \\ \hfill \mathrm{Step}13& \hfill Fstar(k+1)=& \mathrm{integral}(f(t,Xstar(k+1),Ystar(k+1)){B}_{k+1}\left(t\right),t(k-1),t(k+1))/\mathrm{integral}({B}_{k+1}\left(t\right),t(k-1),t(k+1)).\hfill \\ \hfill \mathrm{Step}14& \hfill Gstar(k+1)=& \mathrm{integral}(g(t,Xstar(k+1),Ystar(k+1)){B}_{k+1}\left(t\right),t(k-1),t(k+1))/\mathrm{integral}({B}_{k+1}\left(t\right),t(k-1),t(k+1)).\hfill \\ \hfill \mathrm{Step}15& \hfill Xstar2(k+1)=& Xstar(k+1)+9hFstar(k+1)/24.\hfill \\ \hfill \mathrm{Step}16& \hfill Ystar2(k+1)=& Ystar(k+1)+9hGstar(k+1)/24.\hfill \\ \hfill \mathrm{Step}17& \hfill Fstar2(k+1)=& \mathrm{integral}(f(t,Xstar2(k+1),Ystar2(k+1)){B}_{k+1}\left(t\right),t(k-1),t(k+1))/\mathrm{integral}({B}_{k+1}\left(t\right),t(k-1),t(k+1)).\hfill \\ \hfill \mathrm{Step}18& \hfill Gstar2(k+1)=& \mathrm{integral}(g(t,Xstar2(k+1),Ystar2(k+1)){B}_{k+1}\left(t\right),t(k-1),t(k+1))/\mathrm{integral}({B}_{k+1}\left(t\right),t(k-1),t(k+1)).\hfill \\ \hfill \mathrm{Step}19& \hfill X(k+1)=& X\left(k\right)+h\left(19F\right(k)-5F(k-1)+F(k-2)+9Fstar2(k+1\left)\right)/24.\hfill \\ \hfill \mathrm{Step}20& \hfill Y(k+1)=& Y\left(k\right)+h\left(19G\right(k)-5G(k-1)+G(k-2)+9Gstar2(k+1\left)\right)/24.\hfill \end{array}$ |

end. |

OUTPUT: Approximation X and Y to x and y, respectively at the ($N+1$) values of t. |

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**Figure 1.**A comparison between three fuzzy numerical methods and exact solution for two examples. (

**a**) Example 2; (

**b**) Example 3.

${\mathit{K}}_{{\mathit{C}}_{2}^{\mathit{m}}}\left(\mathit{t}\right)$ | $\mathit{\beta}$ | $\mathit{\lambda}$ | ${\mathit{B}}_{\mathit{k}}=\mathit{\lambda}\mathit{\beta}\mathit{K}\left(\right)open="("\; close=")">\frac{\mathit{t}-{\mathit{t}}_{\mathit{k}}}{\mathit{h}}$ |
---|---|---|---|

${\left(\right)}^{1}m$ | $\frac{1}{{2}^{m}}$ | $\frac{\sqrt{\pi}\mathsf{\Gamma}\left(\right)open="("\; close=")">m+1}{}$ | $\left(\right)open="("\; close=")">\frac{\sqrt{\pi}\mathsf{\Gamma}\left(\right)open="("\; close=")">m+1}{}2\mathsf{\Gamma}\left(\right)open="("\; close=")">m+\frac{1}{2}$ |

${\mathit{t}}_{\mathit{i}}$ | Solution $\mathit{x}\left(\mathit{t}\right)$ | Proposed Scheme I | Proposed Scheme II | Proposed Scheme III | Trap ${}^{1}$ | 2-Step Adams ${}^{2}$ | 3-Step Adams ${}^{3}$ |
---|---|---|---|---|---|---|---|

0.00 | −4.00000 | −4.00000 | −4.00000 | −4.00000 | −4.00000 | −4.00000 | −4.00000 |

0.05 | −4.00501 | −4.00506 | −4.00501 | −4.00501 | −4.00714 | −4.00501 | −4.00501 |

0.10 | −4.02008 | −4.02059 | −4.02011 | −4.02008 | −4.02627 | −4.02187 | −4.02008 |

0.15 | −4.04543 | −4.04589 | −4.04536 | −4.04536 | −4.05752 | −4.04875 | −4.04703 |

0.20 | −4.08136 | −4.08128 | −4.08121 | −4.08119 | −4.10134 | −4.08610 | −4.08434 |

0.25 | −4.12834 | −4.12721 | −4.12813 | −4.12811 | −4.15850 | −4.13442 | −4.13261 |

0.30 | −4.18701 | −4.18423 | −4.18673 | −4.18670 | −4.23015 | −4.19440 | −4.19250 |

0.35 | −4.25816 | −4.25304 | −4.25783 | −4.25779 | −4.31783 | −4.26691 | −4.26487 |

0.40 | −4.34282 | −4.33453 | −4.34243 | −4.34236 | −4.42361 | −4.35300 | −4.35077 |

0.45 | −4.44224 | −4.42976 | −4.44178 | −4.44170 | −4.55014 | −4.45399 | −4.45151 |

0.50 | −4.55798 | −4.54001 | −4.55745 | −4.55733 | −4.70086 | −4.57151 | −4.56871 |

0.55 | −4.69195 | −4.66686 | −4.69134 | −4.69117 | −4.88023 | −4.70755 | −4.70433 |

0.60 | −4.84651 | −4.81220 | −4.84581 | −4.84558 | −5.09399 | −4.86455 | −4.86079 |

0.65 | −5.02460 | −4.97833 | −5.02378 | −5.02346 | −5.34964 | −5.04556 | −5.04112 |

0.70 | −5.22984 | −5.16805 | −5.22887 | −5.22845 | −5.65700 | −5.25435 | −5.24903 |

0.75 | −5.46680 | −5.38480 | −5.46565 | −5.46508 | −6.02912 | −5.49569 | −5.48924 |

0.80 | −5.74130 | −5.63286 | −5.73990 | −5.73914 | −6.48349 | −5.77559 | −5.76768 |

0.85 | −6.06076 | −5.91759 | −6.05906 | −6.05803 | −7.04390 | −6.10182 | −6.09202 |

0.90 | −6.43490 | −6.24582 | −6.43281 | −6.43141 | −7.74310 | −6.48450 | −6.47222 |

0.95 | −6.87660 | −6.62636 | −6.87400 | −6.87209 | −8.62699 | −6.93706 | −6.92148 |

1.00 | −7.40326 | −7.07221 | −7.40106 | −7.39831 | −9.76103 | −7.47766 | −7.45769 |

${\mathit{t}}_{\mathit{i}}$ | Solution $\mathit{y}\left(\mathit{t}\right)$ | Proposed Scheme I | Proposed Scheme II | Proposed Scheme III | Trap ${}^{1}$ | 2-Step Adams ${}^{2}$ | 3-Step Adams ${}^{3}$ |
---|---|---|---|---|---|---|---|

0.00 | 4.00000 | 4.00000 | 4.00000 | 4.00000 | 4.00000 | 4.00000 | 4.00000 |

0.05 | 3.61935 | 3.62135 | 3.61935 | 3.61935 | 3.62045 | 3.61935 | 3.61935 |

0.10 | 3.27492 | 3.27848 | 3.27485 | 3.27492 | 3.27766 | 3.27601 | 3.27492 |

0.15 | 2.96327 | 2.96802 | 2.96314 | 2.96327 | 2.96757 | 2.96496 | 2.96415 |

0.20 | 2.68128 | 2.68698 | 2.68112 | 2.68124 | 2.68671 | 2.68326 | 2.68261 |

0.25 | 2.42612 | 2.43262 | 2.42595 | 2.42607 | 2.43201 | 2.42819 | 2.42769 |

0.30 | 2.19525 | 2.20247 | 2.19508 | 2.19520 | 2.20079 | 2.19724 | 2.19688 |

0.35 | 1.98634 | 1.99425 | 1.98619 | 1.98630 | 1.99067 | 1.98815 | 1.98792 |

0.40 | 1.79732 | 1.80593 | 1.79718 | 1.79729 | 1.79951 | 1.79888 | 1.79875 |

0.45 | 1.62628 | 1.63564 | 1.62616 | 1.62627 | 1.62541 | 1.62754 | 1.62752 |

0.50 | 1.47152 | 1.48170 | 1.47143 | 1.47153 | 1.46664 | 1.47245 | 1.47253 |

0.55 | 1.33148 | 1.34257 | 1.33142 | 1.33153 | 1.32166 | 1.33207 | 1.33224 |

0.60 | 1.20478 | 1.21688 | 1.20474 | 1.20485 | 1.18908 | 1.20500 | 1.20526 |

0.65 | 1.09013 | 1.10337 | 1.09012 | 1.09023 | 1.06762 | 1.08998 | 1.09033 |

0.70 | 0.98639 | 1.00091 | 0.98641 | 0.98652 | 0.95615 | 0.98587 | 0.98632 |

0.75 | 0.89252 | 0.90848 | 0.89257 | 0.89269 | 0.85366 | 0.89163 | 0.89217 |

0.80 | 0.80759 | 0.82515 | 0.80766 | 0.80779 | 0.75923 | 0.80632 | 0.80696 |

0.85 | 0.73073 | 0.75011 | 0.73084 | 0.73097 | 0.67208 | 0.72910 | 0.72984 |

0.90 | 0.66120 | 0.68260 | 0.66133 | 0.66148 | 0.59150 | 0.65919 | 0.66004 |

0.95 | 0.59827 | 0.62195 | 0.59844 | 0.59860 | 0.51692 | 0.59590 | 0.59686 |

1.00 | 0.54134 | 0.56745 | 0.54145 | 0.54163 | 0.44787 | 0.53860 | 0.53969 |

${\mathit{t}}_{\mathit{i}}$ | Solution $\mathit{x}\left(\mathit{t}\right)$ | Proposed Scheme I | Proposed Scheme II | Proposed Scheme III | Trap ${}^{1}$ | 2-Step Adams ${}^{2}$ | 3-Step Adams ${}^{3}$ |
---|---|---|---|---|---|---|---|

0.00 | 2.00000 | 2.00000 | 2.00000 | 2.00000 | 2.00000 | 2.00000 | 2.00000 |

0.05 | 2.00250 | 2.00257 | 2.00250 | 2.00250 | 2.00250 | 2.00250 | 2.00250 |

0.10 | 2.01008 | 2.01016 | 2.01007 | 2.01008 | 2.01005 | 2.01006 | 2.01008 |

0.15 | 2.02289 | 2.02299 | 2.02289 | 2.02289 | 2.02279 | 2.02285 | 2.02286 |

0.20 | 2.04124 | 2.04138 | 2.04124 | 2.04124 | 2.04101 | 2.04117 | 2.04118 |

0.25 | 2.06557 | 2.06577 | 2.06558 | 2.06558 | 2.06511 | 2.06546 | 2.06547 |

0.30 | 2.09650 | 2.09676 | 2.09652 | 2.09651 | 2.09567 | 2.09633 | 2.09633 |

0.35 | 2.13485 | 2.13520 | 2.13488 | 2.13486 | 2.13344 | 2.13459 | 2.13459 |

0.40 | 2.18171 | 2.18218 | 2.18176 | 2.18172 | 2.17942 | 2.18132 | 2.18132 |

0.45 | 2.23852 | 2.23915 | 2.23860 | 2.23854 | 2.23493 | 2.23795 | 2.23796 |

0.50 | 2.30720 | 2.30803 | 2.30731 | 2.30722 | 2.30167 | 2.30637 | 2.30638 |

0.55 | 2.39031 | 2.39140 | 2.39047 | 2.39034 | 2.38192 | 2.38910 | 2.38911 |

0.60 | 2.49133 | 2.49279 | 2.49157 | 2.49138 | 2.47868 | 2.48956 | 2.48957 |

0.65 | 2.61513 | 2.61708 | 2.61549 | 2.61520 | 2.59605 | 2.61251 | 2.61251 |

0.70 | 2.76863 | 2.77126 | 2.76917 | 2.76873 | 2.73967 | 2.76465 | 2.76466 |

0.75 | 2.96202 | 2.96563 | 2.96288 | 2.96218 | 2.91754 | 2.95585 | 2.95584 |

0.80 | 3.21093 | 3.21600 | 3.21235 | 3.21120 | 3.14130 | 3.20103 | 3.20099 |

0.85 | 3.54059 | 3.54791 | 3.54308 | 3.54108 | 3.42845 | 3.52398 | 3.52386 |

0.90 | 3.99443 | 4.00539 | 3.99914 | 3.99541 | 3.80653 | 3.96482 | 3.96449 |

0.95 | 4.65413 | 4.67130 | 4.66407 | 4.65640 | 4.32100 | 4.59671 | 4.59578 |

1.00 | 5.69348 | 5.72071 | 5.71719 | 5.69904 | 5.05197 | 5.56751 | 5.56473 |

${\mathit{t}}_{\mathit{i}}$ | Solution $\mathit{y}\left(\mathit{t}\right)$ | Proposed Scheme I | Proposed Scheme II | Proposed Scheme III | Trap ^{1} | 2-Step Adams ^{2} | 3-Step Adams ^{3} |
---|---|---|---|---|---|---|---|

0.00 | 2.00000 | 2.00000 | 2.00000 | 2.00000 | 2.00000 | 2.00000 | 2.00000 |

0.05 | 2.00250 | 2.00257 | 2.00250 | 2.00250 | 2.00250 | 2.00250 | 2.00250 |

0.10 | 2.01008 | 2.01016 | 2.01007 | 2.01008 | 2.01005 | 2.01006 | 2.01008 |

0.15 | 2.02289 | 2.02299 | 2.02289 | 2.02289 | 2.02279 | 2.02285 | 2.02286 |

0.20 | 2.04124 | 2.04138 | 2.04124 | 2.04124 | 2.04101 | 2.04117 | 2.04118 |

0.25 | 2.06557 | 2.06577 | 2.06558 | 2.06558 | 2.06511 | 2.06546 | 2.06547 |

0.30 | 2.09650 | 2.09676 | 2.09652 | 2.09651 | 2.09567 | 2.09633 | 2.09633 |

0.35 | 2.13485 | 2.13520 | 2.13488 | 2.13486 | 2.13344 | 2.13459 | 2.13459 |

0.40 | 2.18171 | 2.18218 | 2.18176 | 2.18172 | 2.17942 | 2.18132 | 2.18132 |

0.45 | 2.23852 | 2.23915 | 2.23860 | 2.23854 | 2.23493 | 2.23795 | 2.23796 |

0.50 | 2.30720 | 2.30803 | 2.30731 | 2.30722 | 2.30167 | 2.30637 | 2.30638 |

0.55 | 2.39031 | 2.39140 | 2.39047 | 2.39034 | 2.38192 | 2.38910 | 2.38911 |

0.60 | 2.49133 | 2.49279 | 2.49157 | 2.49138 | 2.47868 | 2.48956 | 2.48957 |

0.65 | 2.61513 | 2.61708 | 2.61549 | 2.61520 | 2.59605 | 2.61251 | 2.61251 |

0.70 | 2.76863 | 2.77126 | 2.76917 | 2.76873 | 2.73967 | 2.76465 | 2.76466 |

0.75 | 2.96202 | 2.96563 | 2.96288 | 2.96218 | 2.91754 | 2.95585 | 2.95584 |

0.80 | 3.21093 | 3.21600 | 3.21235 | 3.21120 | 3.14130 | 3.20103 | 3.20099 |

0.85 | 3.54059 | 3.54791 | 3.54308 | 3.54108 | 3.42845 | 3.52398 | 3.52386 |

0.90 | 3.99443 | 4.00539 | 3.99914 | 3.99541 | 3.80653 | 3.96482 | 3.96449 |

0.95 | 4.65413 | 4.67130 | 4.66407 | 4.65640 | 4.32100 | 4.59671 | 4.59578 |

1.00 | 5.69348 | 5.72071 | 5.71719 | 5.69904 | 5.05197 | 5.56751 | 5.56473 |

(a) The Values of MSE of$\mathit{x}\left(\mathit{t}\right)$for SODEs | |||||||

Case | Proposed Scheme for$\mathit{x}\left(\mathit{t}\right)$ | Classical Method for$\mathit{x}\left(\mathit{t}\right)$ | |||||

I | II | III | Trapezoidal Rule | 2-Step Adams Moulton | 3-Step Adams Moulton | ||

Ex.1 | ${K}_{{C}_{2}^{201}}$ | 1.21569 × ${10}^{-2}$ | 1.21112 × ${10}^{-6}$ | 3.71739 × ${10}^{-6}$ | 5.99915 × ${10}^{-1}$ | 8.37463 × ${10}^{-4}$ | 4.72086 × ${10}^{-4}$ |

Ex.2 | ${K}_{{C}_{2}^{201}}$ | 6.02153 × ${10}^{-5}$ | 3.29699 × ${10}^{-5}$ | 1.77640 × ${10}^{-6}$ | 2.75574 × ${10}^{-2}$ | 9.75470 × ${10}^{-4}$ | 1.01541 × ${10}^{-3}$ |

(b) The Values of MSE of $\mathit{y}\left(\mathit{t}\right)$ for SODEs | |||||||

Case | Proposed Scheme for $\mathit{y}\left(\mathit{t}\right)$ | Classical Method for $\mathit{y}\left(\mathit{t}\right)$ | |||||

I | II | III | Trapezoidal Rule | 2-Step Adams Moulton | 3-Step Adams Moulton | ||

Ex.1 | ${K}_{{C}_{2}^{201}}$ | 1.80417 × ${10}^{-4}$ | 1.20902 × ${10}^{-8}$ | 2.08739 × ${10}^{-8}$ | 1.40165 × ${10}^{-3}$ | 2.25155 × ${10}^{-6}$ | 1.09674 × ${10}^{-6}$ |

Ex.2 | ${K}_{{C}_{2}^{201}}$ | 6.02153 × ${10}^{-5}$ | 3.29699 × ${10}^{-5}$ | 1.77640 × ${10}^{-6}$ | 2.75574 × ${10}^{-2}$ | 9.75470 × ${10}^{-4}$ | 1.01541 × ${10}^{-3}$ |

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**MDPI and ACS Style**

ALKasasbeh, H.; Perfilieva, I.; Ahmad, M.Z.; Yahya, Z.R.
New Approximation Methods Based on Fuzzy Transform for Solving SODEs: II. *Appl. Syst. Innov.* **2018**, *1*, 30.
https://doi.org/10.3390/asi1030030

**AMA Style**

ALKasasbeh H, Perfilieva I, Ahmad MZ, Yahya ZR.
New Approximation Methods Based on Fuzzy Transform for Solving SODEs: II. *Applied System Innovation*. 2018; 1(3):30.
https://doi.org/10.3390/asi1030030

**Chicago/Turabian Style**

ALKasasbeh, Hussein, Irina Perfilieva, Muhammad Zaini Ahmad, and Zainor Ridzuan Yahya.
2018. "New Approximation Methods Based on Fuzzy Transform for Solving SODEs: II" *Applied System Innovation* 1, no. 3: 30.
https://doi.org/10.3390/asi1030030