# Residual Series Representation Algorithm for Solving Fuzzy Duffing Oscillator Equations

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Overview of Fuzzy Calculus Theory

**Theorem**

**1.**

- ${\omega}_{1}$ and ${\omega}_{2}$ are increasing and decreasing, respectively, with ω
_{1}(1) ≤ ω_{2}(1); - ${\omega}_{1}$ and ${\omega}_{2}$ are right-hand continuous for $r=0$.
- ${\omega}_{1}$ and ${\omega}_{2}$ are left-hand continuous for $r\in \left(0\text{},\text{}1\right]$,

**Definition**

**1.**

- ${D}_{H}(\tau +\epsilon ,\delta +\epsilon )={D}_{H}(\tau ,\delta ).$
- ${D}_{H}(\lambda \tau ,\lambda \delta )=\left|\lambda \right|{D}_{H}(\tau ,\delta )$.
- ${D}_{H}(\tau ,\delta )\le {D}_{H}(\tau ,\epsilon )+{D}_{H}(\epsilon ,\delta )$.
- $({\mathbb{R}}_{\mathcal{F}},{D}_{H})$ is a complete metric space.

**Definition**

**2.**

**Definition**

**3.**

- The H-differences $p({t}_{0\text{}}+\xi )\ominus p({t}_{0}),p({t}_{0})\ominus p({t}_{0\text{}}-\xi )$ exist, for each $\xi >0$, sufficiently tends to 0, and $\underset{\xi \to {0}^{+}}{\mathrm{lim}}\frac{p({t}_{0}+\xi \text{})\ominus p({t}_{0})}{\xi}={p}^{\prime}({t}_{0})=\underset{\xi \to {0}^{+}}{\mathrm{lim}}\frac{p({t}_{0})\ominus p({t}_{0}-\xi )}{\xi},$ or
- The H-differences $p({t}_{0})\ominus p({t}_{0\text{}}+\xi ),p({t}_{0\text{}}-\xi )\ominus p({t}_{0})$ exist, for each $\xi >0$, sufficiently tends to 0, and $\underset{\xi \to {0}^{+}}{\mathrm{lim}}\frac{p({t}_{0})\text{}\ominus p({t}_{0\text{}}+\xi )}{-\xi}={p}^{\prime}({t}_{0})=\underset{\xi \to {0}^{+}}{\mathrm{lim}}\frac{p({t}_{0}-\xi )\ominus p({t}_{0})}{-\xi}$.

**Remark**

**1.**

**Theorem**

**2.**

- If $p$ is (1)-differentiable, then ${p}_{1r}$ and ${p}_{2r}$ are two differentiable functions and ${[{D}_{1}^{1}p(t)]}^{r}=\left[{{p}^{\prime}}_{1r}(t),{{p}^{\prime}}_{2r}(t)\right]$.
- If $p$ is (2)-differentiable, then ${p}_{1r}$ and ${p}_{2r}$ are two differentiable functions and ${[{D}_{2}^{1}p(t)]}^{r}=\left[{{p}^{\prime}}_{2r}(t),{{p}^{\prime}}_{1r}(t)\right]$.

**Definition**

**4.**

**Theorem**

**3.**

- If ${D}_{1}^{1}p$ is (1)-differentiable, then ${p}_{1r}^{\prime}$ and ${p}_{2r}^{\prime}$ are differentiable functions and ${\left[{D}_{1,1}^{2}p(t)\right]}^{r}=\left[{p}_{1r}^{\u2033}(t),\text{}{p}_{2r}^{\u2033}(t)\right],$
- If ${D}_{1}^{1}p$ is (2)-differentiable, then ${p}_{1r}^{\prime}$ and ${p}_{2r}^{\prime}$ are differentiable functions and ${\left[{D}_{1,2}^{2}p(t)\right]}^{r}=\left[{p}_{2r}^{\u2033}(t),\text{}{p}_{1r}^{\u2033}(t)\right],$
- If ${D}_{2}^{1}p$ is (1)-differentiable, then ${p}_{1r}^{\prime}$ and ${p}_{2r}^{\prime}$ are differentiable functions and ${\left[{D}_{2,1}^{2}p(t)\right]}^{r}=\left[{p}_{2r}^{\u2033}(t),\text{}{p}_{1r}^{\u2033}(t)\right],$
- If ${D}_{2}^{1}p$ is (2)-differentiable, then ${p}_{1r}^{\prime}$ and ${p}_{2r}^{\prime}$ are differentiable functions and ${\left[{D}_{2,2}^{2}p(t)\right]}^{r}=\left[{p}_{1r}^{\u2033}(t),\text{}{p}_{2r}^{\u2033}(t)\right]$.

## 3. Fuzzy Duffing’s Equation

**Definition**

**5.**

Algorithm 1. To obtain the fuzzy solution $\mathit{p}(\mathit{t})$ for the FIVPs (3) and (4), four cases are considered according to the kinds of $(\mathit{n},\mathit{m}).$ |

1. Case I: If $\mathit{p}(\mathit{t})$ is (1,1)-differentiable, then FIVPs (3) and (4) can be converted into the crisp system given in Equation (5). Consequently, the following actions should be taken: |

2. A1: Solve the crisp system (5) using the procedures of the RPS algorithm. |

3. A2: Ensure that the solutions $\left[{\mathit{p}}_{\mathbf{1}\mathit{r}}(\mathit{t}),{\mathit{p}}_{\mathbf{2}\mathit{r}}(\mathit{t})\right],$ $\left[{\mathit{p}}_{\mathbf{1}\mathit{r}}^{\mathbf{\prime}}(\mathit{t}),{\mathit{p}}_{\mathbf{2}\mathit{r}}^{\mathbf{\prime}}(\mathit{t})\right]$ and $\left[{\mathit{p}}_{\mathbf{1}\mathit{r}}^{\mathbf{\u2033}}(\mathit{t}),{\mathit{p}}_{\mathbf{2}\mathit{r}}^{\mathbf{\u2033}}(\mathit{t})\right]$ are valid $\mathit{r}$-level sets for each $\mathit{r}\in \left[\mathbf{0},\mathbf{1}\right]$. |

4. A3: Obtain the $(\mathbf{1},\mathbf{1})$-solution $\mathit{p}(\mathit{t})$ whose $\mathit{r}$-level representation is $\left[{\mathit{p}}_{\mathbf{1}\mathit{r}}(\mathit{t}),{\mathit{p}}_{\mathbf{2}\mathit{r}}(\mathit{t})\right]$. |

5. Case II: If $\mathit{p}(\mathit{t})$ is (1,2)-differentiable, then FIVPs (3) and (4) can be converted into the crisp system given in Equation (6). Consequently, the following actions should be taken: |

6. B1: Solve the crisp system (6) using the procedures of the RPS algorithm. |

7. B2: Ensure that the solutions $\left[{\mathit{p}}_{\mathbf{1}\mathit{r}}(\mathit{t}),{\mathit{p}}_{\mathbf{2}\mathit{r}}(\mathit{t})\right],$ $\left[{\mathit{p}}_{\mathbf{1}\mathit{r}}^{\mathbf{\prime}}(\mathit{t}),{\mathit{p}}_{\mathbf{2}\mathit{r}}^{\mathbf{\prime}}(\mathit{t})\right]$ and $\left[{\mathit{p}}_{\mathbf{2}\mathit{r}}^{\mathbf{\u2033}}(\mathit{t}),{\mathit{p}}_{\mathbf{1}\mathit{r}}^{\mathbf{\u2033}}(\mathit{t})\right]$ are valid $\mathit{r}$-level sets for each $\mathit{r}\in \left[\mathbf{0},\mathbf{1}\right]$. |

8. B3: Obtain the $(\mathbf{1},\mathbf{2})$-solution $\mathit{p}(\mathit{t})$ whose $\mathit{r}$-level representation is $\left[{\mathit{p}}_{\mathbf{1}\mathit{r}}(\mathit{t}),{\mathit{p}}_{\mathbf{2}\mathit{r}}(\mathit{t})\right]$. |

9. Case III: If $\mathit{p}(\mathit{t})$ is (2,1)-differentiable, then FIVPs (3) and (4) can be converted into the crisp system given in Equation (7). Consequently, the following actions should be taken: |

10. C1: Solve the crisp system (7) using the procedures of RPS algorithm. |

11. C2: Ensure that the solutions $\left[{\mathit{p}}_{\mathbf{1}\mathit{r}}(\mathit{t}),{\mathit{p}}_{\mathbf{2}\mathit{r}}(\mathit{t})\right],$ $\left[{\mathit{p}}_{\mathbf{2}\mathit{r}}^{\mathbf{\prime}}(\mathit{t}),{\mathit{p}}_{\mathbf{1}\mathit{r}}^{\mathbf{\prime}}(\mathit{t})\right]$ and $\left[{\mathit{p}}_{\mathbf{2}\mathit{r}}^{\mathbf{\u2033}}(\mathit{t}),{\mathit{p}}_{\mathbf{1}\mathit{r}}^{\mathbf{\u2033}}(\mathit{t})\right]$ are valid $\mathit{r}$-level sets for each $\mathit{r}\in \left[\mathbf{0},\mathbf{1}\right]$. |

12. C3: Obtain the $(\mathbf{2},\mathbf{1})$-solution $\mathit{p}(\mathit{t})$ whose $\mathit{r}$-level representation is $\left[{\mathit{p}}_{\mathbf{1}\mathit{r}}(\mathit{t}),{\mathit{p}}_{\mathbf{2}\mathit{r}}(\mathit{t})\right]$. |

13. Case IV: If $\mathit{p}(\mathit{t})$ is (2,2)-differentiable, then FIVPs (3) and (4) can be converted into the crisp system given in Equation (8). Consequently, the following actions should be taken: |

14. D1: Solve the crisp system (8) using the procedures of the RPS algorithm. |

15. D2: Ensure that the solutions $\left[{\mathit{p}}_{\mathbf{1}\mathit{r}}(\mathit{t}),{\mathit{p}}_{\mathbf{2}\mathit{r}}(\mathit{t})\right],$ $\left[{\mathit{p}}_{\mathbf{2}\mathit{r}}^{\mathbf{\prime}}(\mathit{t}),{\mathit{p}}_{\mathbf{1}\mathit{r}}^{\mathbf{\prime}}(\mathit{t})\right]$ and $\left[{\mathit{p}}_{\mathbf{1}\mathit{r}}^{\mathbf{\u2033}}(\mathit{t}),{\mathit{p}}_{\mathbf{2}\mathit{r}}^{\mathbf{\u2033}}(\mathit{t})\right]$ are valid $\mathit{r}$-level sets for each $\mathit{r}\in \left[\mathbf{0},\mathbf{1}\right]$. |

16. D3: Obtain the $(\mathbf{2},\mathbf{2})$-solution $\mathit{p}(\mathit{t})$ whose $\mathit{r}$-level representation is $\left[{\mathit{p}}_{\mathbf{1}\mathit{r}}(\mathit{t}),{\mathit{p}}_{\mathbf{2}\mathit{r}}(\mathit{t})\right]$. |

## 4. The RPS Method for Fuzzy Duffing Oscillator

^{th}-truncated residual functions:

^{th}-residual functions are given by

^{th}residual functions are infinitely differentiable functions about $t=0$. Moreover, $\frac{{d}^{n-2}}{d{t}^{n-2}}Re{s}_{ir}^{\infty}(0)=\frac{{d}^{n-2}}{d{t}^{n-2}}Re{s}_{n,ir}(0)=0$ for $i=1,2,$ $n=2,3,\dots $, which is a basic fact of RPS algorithm helped us to obtain the unknown parameters ${a}_{n}$ and ${b}_{n},n\ge 2$.

^{nd}unknown coefficients ${a}_{2}$ and ${b}_{2}$, substitute the 2

^{nd}-truncated series solutions of (11), ${p}_{2,1r}(t)={\alpha}_{1r}+{\beta}_{1r}t+{a}_{2}\frac{{t}^{2}}{2}$ and ${p}_{2,2r}(t)={\alpha}_{2r}+{\beta}_{2r}t+{b}_{2}\frac{{t}^{2}}{2}$, into the $2$

^{th}-residual functions of (12), such that

^{nd}-RPS approximations are ${p}_{2,1r}(t)={\alpha}_{1r}+{\beta}_{1r}t+\frac{1}{2}(f(0)-\lambda {\beta}_{1r}-\mu {\alpha}_{1r}-\gamma {\alpha}_{1r}^{3}){t}^{2}$ and ${p}_{2,2r}(t)={\alpha}_{2r}+{\beta}_{2r}t+\frac{1}{2}(f(0)-\lambda {\beta}_{2r}-\mu {\alpha}_{2r}-\gamma {\alpha}_{2r}^{3}){t}^{2}.$

^{rd}-truncated series solutions of (11), ${p}_{3,1r}(t)={\alpha}_{1r}+{\beta}_{1r}t+(f(0)-\lambda {\beta}_{1r}-\mu {\alpha}_{1r}-\gamma {\alpha}_{1r}^{3})\frac{{t}^{2}}{2}+{a}_{3}\frac{{t}^{3}}{3!}$ and ${p}_{3,2r}(t)={\alpha}_{2r}+{\beta}_{2r}t+(f(0)-\lambda {\beta}_{2r}-\mu {\alpha}_{2r}-\gamma {\alpha}_{2r}^{3})\frac{{t}^{2}}{2}+{b}_{3}\frac{{t}^{3}}{3!}$, into the $3$

^{rd}-residual functions, $Re{s}_{3,1\mathrm{r}}(t)$ and $Re{s}_{3,2r}(t)$, and then differentiate both sides of the resulting equations, $\frac{d}{dt}Re{s}_{3,1\mathrm{r}}(t)$ and $\frac{d}{dt}Re{s}_{3,2\mathrm{r}}(t)$, such that

^{rd}-RPS approximations, ${p}_{3,1r}(t)$ and ${p}_{3,2r}(t)$, are obtained.

^{th}-truncated series solutions of (11) into the 4

^{th}-residual functions $Re{s}_{4,1r}(t)$ and $Re{s}_{4,2r}(t)$ of Equation (12), and solve $\frac{{d}^{2}}{d{t}^{2}}Re{s}_{4,1r}(0)=\frac{{d}^{2}}{d{t}^{2}}Re{s}_{4,2r}(0)=0$, the 4

^{th}coefficients ${a}_{4}$ and ${b}_{4}$ can be obtained, such that ${a}_{4}={f}^{\u2033}(0)-\lambda {f}^{\prime}(0)-{\lambda}^{2}f(0)-\mu f(0)-{\lambda}^{2}\mu {\alpha}_{1r}+{\mu}^{2}{\alpha}_{1r}-3\gamma f(0){\alpha}_{1r}^{2}-\gamma {\lambda}^{2}{\alpha}_{1r}^{3}+4\gamma \mu {\alpha}_{1r}^{3}+3{\gamma}^{2}{\alpha}_{1r}^{5}-{\lambda}^{3}{\beta}_{1r}+2\lambda \mu {\beta}_{1r}+6\gamma \lambda {\alpha}_{1r}^{2}{\beta}_{1r}-6\gamma {\alpha}_{1r}{\beta}_{1r}^{2}$ and ${b}_{4}={f}^{\u2033}(0)-\lambda {f}^{\prime}(0)-{\lambda}^{2}f(0)-\mu f(0)-{\lambda}^{2}\mu {\alpha}_{2r}+{\mu}^{2}{\alpha}_{2r}-3\gamma f(0){\alpha}_{2r}^{2}-\gamma {\lambda}^{2}{\alpha}_{2r}^{3}+4\gamma \mu {\alpha}_{2r}^{3}+3{\gamma}^{2}{\alpha}_{2r}^{5}-{\lambda}^{3}{\beta}_{2r}+2\lambda \mu {\beta}_{2r}+6\gamma \lambda {\alpha}_{2r}^{2}{\beta}_{2r}-6\gamma {\alpha}_{2r}{\beta}_{2r}^{2}$.

## 5. Numerical Applications

**Application**

**1.**

**Case 1:**If $p(t)$ is (1,1)- solution, then the corresponding (1,1)-system is

**Case 2:**If $p(t)$ is (1,2)-solution, then the corresponding (1,2)-system is

**Case 3:**If $p(t)$ is (2,1)-solution, then the correspondence (2,1)-system is

**Case 4:**If $p(t)$ is (2,2)-solution, then the correspondence (2,2)-system is

^{th}-RPS solutions for Equations (15) and (16) at $r=1$ and different values of $t$ in $\left[0,1\right]$, such that $t\in \left\{0.1,0.3,0.5,0.7,0.9\right\}$. While Table 3 and Table 4 display the fuzzy approximate solutions of (1,1)-system and (1,2)-system, respectively, at various values of $r\in \left[0,1\right]$, with step size $0.25$ and various values of $t$, such that $t\in \left\{0.2,0.4,0.6,0.8\right\}$. From these results, we conclude that the results are in good agreement with each other in our two cases and are closer to solutions at $r=1$, as $r$ values increase. In any case, to demonstrate the 5

^{th}-order RPS solutions behavior of Equations (15) and (16), the coupled surface has been plotted in a 3-dim graph for all possible $(n,m)$-systems of Equations (15) and (16) for each $t\in \left[0,1\right]$ and $r\in \left[0,1\right],$ as shown in Figure 1, whereas blue and yellow correspond to upper and lower bounds of the fuzzy solutions, respectively.

**Application**

**2.**

**Case 1:**If $p(t)$ is (1,1)- solution, then the corresponding (1,1)-system is

**Case 2:**If $p(t)$ is (1,2)-solution, then the corresponding (1,2)-system is

**Case 3:**If $p(t)$ is (2,1)-solution, then the correspondence (2,1)-system is

**Case 4:**If $p(t)$ is (2,2)-solution, then the correspondence (2,2)-system is

^{th}-RPS approximate solutions of (1,1)-system, with their residual errors, are computed and listed in Table 5 for $t=0.1$ and $r\in \left[0,1\right]$ with step size $0.25$, which can illustrate the efficiency of the proposed method.

^{th}-RPS approximate solutions for each possible $(n,m)$-systems of Equations (21) and (22) at $t\in \left[0,1\right]$ and $r\in \left\{0,0.25,0.5,0.75,1\right\}$, which guides us to a deep understanding of the pattern behavior of the fuzzy Duffing oscillator, in which gray, blue, green, orange and red refer to the $r$-level solutions at $r=0$, $r=0.25$, $r=0.5$, $r=0.75$ and $r=1$, respectively. The graphs of the $r$-level solutions of the fuzzy Duffing equation are implemented and presented in patterns similar to the interference patterns. The solutions to each case of the fuzzy Duffing equations are compatible with each other in some very simple way and slip into each other in the same direction without any intersections, where the behavior of these solution curves is harmonic in a smooth way around central symmetry at $r=1$ (the red line in the middle, Figure 4).

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Case 1:**Starting with ${p}_{0,1r}(0)=r-1,{p}_{0,2r}(0)=1-r$ and ${p}_{0,1r}^{\prime}(0)=r,{p}_{0,2r}^{\prime}(0)=2-r$, the first-RPS solutions of (1,1)-system are ${p}_{1,1r}(t)=(r-1)+rt$ and ${p}_{1,2r}(t)=(1-r)+(2-r)t$. Apparently, the $n$

^{th}-residual functions of (1,1)-system can be defined as follows

${a}_{2}=1+3r-6{r}^{2}+2{r}^{3}$, | ${b}_{2}=-1-3r+6{r}^{2}-2{r}^{3}$, |

${a}_{3}=2+3r-12{r}^{2}+6{r}^{3}$, | ${b}_{3}=8-27r+24{r}^{2}-6{r}^{3}$, |

${a}_{4}=3(1-r-20{r}^{2}+36{r}^{3}-20{r}^{4}+4{r}^{5})$, | ${b}_{4}=3(1-r-20{r}^{2}+36{r}^{3}-20{r}^{4}+4{r}^{5})$, |

${a}_{5}=-8-51r-132{r}^{2}+516{r}^{3}-432{r}^{4}+108{r}^{5}$, | ${b}_{5}=34-429r+1236{r}^{2}-1380{r}^{3}+648{r}^{4}-108{r}^{5}$, |

^{th}approximate solutions of (1,1)-system can be written as

**Case 2:**The $n$

^{th}-residual functions of (1,2)-system can be defined as follows

${a}_{2}=-1-3r+6{r}^{2}-2{r}^{3}$, | ${b}_{2}=1+3r-6{r}^{2}+2{r}^{3}$, |

${a}_{3}=8-27r+24{r}^{2}-6{r}^{3}$, | ${b}_{3}=2+3r-12{r}^{2}+6{r}^{3}$, |

${a}_{4}=-3(-15+31r-36{r}^{2}+36{r}^{3}-20{r}^{4}+4{r}^{5})$, | ${b}_{4}=3(1-r-20{r}^{2}+36{r}^{3}-20{r}^{4}+4{r}^{5})$, |

${a}_{5}=16-267r+732{r}^{2}-732{r}^{3}+288{r}^{4}-36{r}^{5}$, | ${b}_{5}=10-213r+372{r}^{2}-132{r}^{3}-72{r}^{4}+36{r}^{5}$, |

^{th}approximate solutions of (1,2)-system can be written as

**Case 3:**The $n$

^{th}-residual functions of (2,1)-system can be defined as follows

${a}_{2}=5-9r+6{r}^{2}-2{r}^{3}$, | ${b}_{2}=-5+9r-6{r}^{2}+2{r}^{3}$, |

${a}_{3}=14-33r+24{r}^{2}-6{r}^{3}$, | ${b}_{3}=-4+9r-12{r}^{2}+6{r}^{3}$, |

${a}_{4}=3(1+15r-44{r}^{2}+44{r}^{3}-20{r}^{4}+4{r}^{5})$, | ${b}_{4}=3(15-47r+60{r}^{2}-44{r}^{3}+20{r}^{4}-4{r}^{5})$, |

${a}_{5}=-344+1245r-1788{r}^{2}+1356{r}^{3}-576{r}^{4}+108{r}^{5}$, | ${b}_{5}=82-573r+1164{r}^{2}-1068{r}^{3}+504{r}^{4}-108{r}^{5}$, |

^{th}approximate solutions of (2,1)-system can be written as

**Case 4:**Following the RPS method, the $n$

^{th}-residual functions of system (23) can be defined as follows

${a}_{2}=-5+9r-6{r}^{2}+2{r}^{3}$, | ${b}_{2}=5-9r+6{r}^{2}-2{r}^{3}$, |

${a}_{3}=-4+9r-12{r}^{2}+6{r}^{3}$, | ${b}_{3}=14-33r+24{r}^{2}-6{r}^{3}$, |

${a}_{4}=3(-15+47r-68{r}^{2}+52{r}^{3}-20{r}^{4}+4{r}^{5})$, | ${b}_{4}=3(31-79r+84{r}^{2}-52{r}^{3}+20{r}^{4}-4{r}^{5})$, |

${a}_{5}=-80+381r-780{r}^{2}+804{r}^{3}-432{r}^{4}+108{r}^{5}$, | ${b}_{5}=538-1725r+2316{r}^{2}-1668{r}^{3}+648{r}^{4}-108{r}^{5}$, |

^{th}approximate solutions of the (2,2)-system can be written as

**Case 1:**The $n$

^{th}-residual functions of (1,1)-system can be defined as follows

${a}_{2}=-1.629-0.343r-0.027{r}^{2}-0.001{r}^{3}$, | ${b}_{2}=-2.431+0.463r-0.033{r}^{2}+0.001{r}^{3}$, |

${a}_{3}=-3.087-0.829r-0.081{r}^{2}-0.003{r}^{3}$, | ${b}_{3}=-5.093+1.189r-0.099{r}^{2}+0.003{r}^{3}$, |

${a}_{4}=1.21347+0.59815r+0.1647{r}^{2}+0.0223{r}^{3}+0.00135{r}^{4}+0.00003{r}^{5}$, | ${b}_{4}=3.26953-1.57015r+0.3333{r}^{2}-0.0343{r}^{3}+0.00165{r}^{4}-0.00003{r}^{5}$, |

${a}_{5}=29.96523+13.33135r+2.4543{r}^{2}+0.2367{r}^{3}+0.01215{r}^{4}+0.00027{r}^{5}$, | ${b}_{5}=68.54177\text{}-\text{}26.39935r+4.1877{r}^{2}-0.3447{r}^{3}+0.01485{r}^{4}-0.00027{r}^{5}$, |

$\vdots $ | $\vdots $ |

**Case 2:**The $n$

^{th}-residual functions of (1,2)-system can be defined as follows

${a}_{2}=-2.431+0.463r-0.033{r}^{2}+0.001{r}^{3}$, | ${b}_{2}=-1.629-0.343r-0.027{r}^{2}-0.001{r}^{3}$, |

${a}_{3}=-5.093+1.189r-0.099{r}^{2}+0.003{r}^{3}$, | ${b}_{3}=-3.087-0.829r-0.081{r}^{2}-0.003{r}^{3}$, |

${a}_{4}=-0.44373+2.69095r-0.2505{r}^{2}+0.0031{r}^{3}+0.00015{r}^{4}+0.00003{r}^{5}$, | ${b}_{4}=3.96433-1.73335r-0.2259{r}^{2}-0.0055{r}^{3}+0.00045{r}^{4}-0.00003{r}^{5}$, |

${a}_{5}=41.78643+4.99855r-0.7545{r}^{2}-0.0321{r}^{3}+0.00135{r}^{4}+0.00027{r}^{5}$, | ${b}_{5}=48.53897\text{}-\text{}1.66015r-0.8931{r}^{2}+0.0105{r}^{3}+0.00405{r}^{4}-0.00027{r}^{5}$, |

$\vdots $ | $\vdots $ |

**Case 3:**The $n$

^{th}-residual functions of (2,1)-system can be defined as follows

${a}_{2}=-2.231+0.263r-0.033{r}^{2}+0.001{r}^{3}$, | ${b}_{2}=-1.829-0.143r-0.027{r}^{2}-0.001{r}^{3}$, |

${a}_{3}=-4.893+0.989r-0.099{r}^{2}+0.003{r}^{3}$, | ${b}_{3}=-3.287-0.629r-0.081{r}^{2}-0.003{r}^{3}$, |

${a}_{4}=0.88427+1.22695r-0.1065{r}^{2}-0.0049{r}^{3}+0.00015{r}^{4}+0.00003{r}^{5}$, | ${b}_{4}=2.87633\text{}-\text{}0.74935r-0.1299{r}^{2}+0.0025{r}^{3}+0.00045{r}^{4}-0.00003{r}^{5}$, |

${a}_{5}=48.67443-2.82545r+0.2295{r}^{2}-0.0801{r}^{3}+0.00135{r}^{4}+0.00027{r}^{5}$, | ${b}_{5}=43.33097+2.80385r-0.1971{r}^{2}+0.0585{r}^{3}+0.00405{r}^{4}-0.00027{r}^{5}$, |

$\vdots $ | $\vdots $ |

**Case 4:**The $n$

^{th}-residual functions of (2,2)-system can be defined as follows

${a}_{2}=-1.829-0.143r-0.027{r}^{2}-0.001{r}^{3}$, | ${b}_{2}=-2.231+0.263r-0.033{r}^{2}+0.001{r}^{3}$, |

${a}_{3}=-3.287-0.629r-0.081{r}^{2}-0.003{r}^{3}$, | ${b}_{3}=-4.893+0.989r-0.099{r}^{2}+0.003{r}^{3}$, |

${a}_{4}=2.30147-0.38585r+0.0687{r}^{2}+0.0143{r}^{3}+0.00135{r}^{4}+0.00003{r}^{5}$, | ${b}_{4}=1.94153-0.10615r+0.1893{r}^{2}-0.0263{r}^{3}+0.00165{r}^{4}-0.00003{r}^{5}$, |

${a}_{5}=35.17323+8.86735r+1.7583{r}^{2}+0.1887{r}^{3}+0.01215{r}^{4}+0.00027{r}^{5}$, | ${b}_{5}=61.65377-18.57535r+3.2037{r}^{2}-0.2967{r}^{3}+0.01485{r}^{4}-0.00027{r}^{5}$, |

$\vdots $ | $\vdots $ |

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**Figure 1.**Surface plots of Equations (15) and (16) at $t\in \left[0,1\right]$ and $r\in \left[0,1\right]$: (

**a**) (1,1)-system; (

**b**) (1,2)-system; (

**c**) (2,1)-system; (

**d**) (2,2)-system, where blue and yellow represent the lower and upper boundaries, respectively, of 5th-RPS fuzzy solutions.

**Figure 2.**Graphs of triangular fuzzy solutions of Equations (15) and (16) for each $r$ in $\left[0,1\right]$, n = 5 and t = 0.5: (

**a**) (1,1)-solutions; (

**b**) (1,2)-solutions; (

**c**) (2,1)-solutions; (

**d**) (2,2)-solutions, where green, blue and orange represent p

_{1r}(0.5), p

_{2r}(0.5) and p(0.5) at r = 1, respectively.

**Figure 3.**Graphs of triangular fuzzy solutions of Equations (15) and (16) for each $r$ in [0, 1], n = 5 and t = 0.95: (

**a**) (1,1)-solutions; (

**b**) (1,2)-solutions; (

**c**) (2,1)-solutions; (

**d**) (2,2)-solutions, where green, blue and orange represent p

_{1r}(0.95), p

_{2r}(0.95) and p(0.95) at r = 1, respectively.

**Figure 4.**Solution behavior for each case of Equations (21) and (22) based on r-level solutions at $t\in \left[0,1\right]$ and n = 6: (

**a**) (1,1)-system; (

**b**) (1,2)-system; (

**c**) (2,1)-system; (

**d**) (2,2)-system, where gray, blue, green, orange and red refer to the 6th-RPS solutions at r = 0, r = 0.25, r = 0.5, r = 0.75 and r = 1, respectively.

Lower ${\mathit{p}}_{\mathbf{5}\mathbf{,}\mathbf{1}\mathit{r}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$ | $\mathit{r}$ | RPSM | VIM [18] | Exact Solution | MNS [18] |

$0$ | $-0.9946548333$ | $-0.9946548745$ | $-0.9946548739$ | $-0.9946548688$ | |

$0.25$ | $-0.7176218148$ | $-0.7176219076$ | $-0.7176219071$ | $-0.7176219141$ | |

$0.5$ | $-0.4435578646$ | $-0.4435578806$ | $-0.4435578799$ | $-0.4435578164$ | |

$0.75$ | $-0.1714170902$ | $-0.1714170389$ | $-0.1714170382$ | $-0.1714169861$ | |

$1$ | $0.09983341667$ | $0.09983341601$ | $0.09983341664$ | $0.09983341519$ | |

Upper ${\mathit{p}}_{\mathbf{5}\mathbf{,}\mathbf{2}\mathit{r}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$ | $\mathit{r}$ | RPSM | VIM [18] | Exact Solution | MNS [18] |

$0$ | $1.19652366667$ | $1.19652475384$ | $1.19652475400$ | $1.19652479981$ | |

$0.25$ | $0.91852284342$ | $0.91852280319$ | $0.91852280370$ | $0.91852292721$ | |

$0.5$ | $0.64377182292$ | $0.64377156027$ | $0.64377156090$ | $0.64377160488$ | |

$0.75$ | $0.37122049382$ | $0.37122034006$ | $0.37122034070$ | $0.37122034586$ | |

$1$ | $0.09983341667$ | $0.09983341601$ | $0.09983341664$ | $0.09983341519$ |

${\mathit{t}}_{\mathit{i}}$ | Exact | RPS | Absolute Error | Relative Error |
---|---|---|---|---|

$0.1$ | $0.0998334166$ | $0.0998334167$ | $1.98385\times {10}^{-11}$ | $1.98716\times {10}^{-10}$ |

$0.3$ | $0.2955202067$ | $0.2955202500$ | $4.33387\times {10}^{-8}$ | $1.46652\times {10}^{-7}$ |

$0.5$ | $0.4794255386$ | $0.4794270833$ | $1.54473\times {10}^{-6}$ | $3.22204\times {10}^{-6}$ |

$0.7$ | $0.6442176872$ | $0.6442339166$ | $1.62294\times {10}^{-5}$ | $2.51925\times {10}^{-5}$ |

$0.9$ | $0.7833269096$ | $0.7834207500$ | $9.38404\times {10}^{-5}$ | $1.19797\times {10}^{-4}$ |

${\mathit{p}}_{\mathbf{5}\mathbf{,}\mathbf{1}\mathit{r}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$ | ${\mathit{t}}_{\mathit{i}}$ | $\mathit{r}\mathbf{=}\mathbf{0}$ | $\mathit{r}\mathbf{=}\mathbf{0.25}$ | $\mathit{r}\mathbf{=}\mathbf{0.5}$ | $\mathit{r}\mathbf{=}\mathbf{0.75}$ | $\mathit{r}\mathbf{=}\mathbf{1}$ |

$0.2$ | $-0.9771546$ | $-0.6691458$ | $-0.3736266$ | $-0.0858586$ | $0.1986693$ | |

$0.4$ | $-0.8961493$ | $-0.5171258$ | $-0.1924533$ | $0.1028372$ | $0.3894187$ | |

$0.6$ | $-0.7369840$ | $-0.2862825$ | $0.0351700$ | $0.3022108$ | $0.5646480$ | |

$0.8$ | $-0.4799786$ | $0.0165733$ | $0.2790933$ | $0.4837893$ | $0.7173973$ | |

${\mathit{p}}_{\mathbf{5}\mathbf{,}\mathbf{2}\mathit{r}}(\mathit{t})$ | ${\mathit{t}}_{\mathit{i}}$ | $\mathit{r}\mathbf{=}\mathbf{0}$ | $\mathit{r}\mathbf{=}\mathbf{0.25}$ | $\mathit{r}\mathbf{=}\mathbf{0.5}$ | $\mathit{r}\mathbf{=}\mathbf{0.75}$ | $\mathit{r}\mathbf{=}\mathbf{1}$ |

$0.2$ | $-0.9771547$ | $1.0753739$ | $0.7743423$ | $0.4836909$ | $0.1986693$ | |

$0.4$ | $-0.8961493$ | $1.3645042$ | $0.9931466$ | $0.6754232$ | $0.3894187$ | |

$0.6$ | $-0.7369840$ | $1.6325016$ | $1.1433200$ | $0.8082544$ | $0.5646480$ | |

$0.8$ | $-0.4799787$ | $1.8847333$ | $1.2022933$ | $0.8625413$ | $0.7173973$ |

${\mathit{p}}_{\mathbf{5}\mathbf{,}\mathbf{1}\mathit{r}}(\mathit{t})$ | ${\mathit{t}}_{\mathit{i}}$ | $\mathit{r}\mathbf{=}\mathbf{0}$ | $\mathit{r}\mathbf{=}\mathbf{0.25}$ | $\mathit{r}\mathbf{=}\mathbf{0.5}$ | $\mathit{r}\mathbf{=}\mathbf{0.75}$ | $\mathit{r}\mathbf{=}\mathbf{1}$ |

$0.2$ | $-0.9662910$ | $-0.6665720$ | $-0.3743330$ | $-0.0868917$ | $0.1986693$ | |

$0.4$ | $-0.7853010$ | $-0.4816390$ | $-0.1870450$ | $0.1009540$ | $0.3894187$ | |

$0.6$ | $-0.2786320$ | $-0.1152050$ | $0.0931120$ | $0.3195730$ | $0.5646480$ | |

$0.8$ | $0.8143570$ | $0.5461570$ | $0.5161490$ | $0.5863333$ | $0.7173973$ | |

${\mathit{p}}_{\mathbf{5}\mathbf{,}\mathbf{2}\mathit{r}}(\mathit{t})$ | ${\mathit{t}}_{\mathit{i}}$ | $\mathit{r}\mathbf{=}\mathbf{0}$ | $\mathit{r}\mathbf{=}\mathbf{0.25}$ | $\mathit{r}\mathbf{=}\mathbf{0.5}$ | $\mathit{r}\mathbf{=}\mathbf{0.75}$ | $\mathit{r}\mathbf{=}\mathbf{1}$ |

$0.2$ | $-0.9662910$ | $1.0746048$ | $0.7763790$ | $0.4854010$ | $0.1986693$ | |

$0.4$ | $-0.7853010$ | $1.3578921$ | $1.0077386$ | $0.6881310$ | $0.3894187$ | |

$0.6$ | $-0.2786320$ | $1.6076042$ | $1.1866280$ | $0.8456930$ | $0.5646480$ | |

$0.8$ | $0.8143570$ | $1.8171493$ | $1.2852373$ | $0.9331970$ | $0.7173973$ |

$\mathit{r}$ | Lower Bound | Upper Bound | ||
---|---|---|---|---|

${\mathit{p}}_{1\mathit{r}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$ | ${\mathit{E}}_{1}\mathbf{\left(}\mathit{t}\mathbf{;}\mathit{r}\mathbf{\right)}$ | ${\mathit{p}}_{\mathbf{2}\mathit{r}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$ | ${\mathit{E}}_{\mathbf{2}}\mathbf{\left(}\mathit{t}\mathbf{;}\mathit{r}\mathbf{\right)}$ | |

$0$ | $0.9813481271$ | $6.32899\times {10}^{-10}$ | $1.1970156393$ | $7.95815\times {10}^{-10}$ |

$0.25$ | $1.0083764337$ | $7.17582\times {10}^{-10}$ | $1.1701305851$ | $9.18593\times {10}^{-10}$ |

0.5 | $1.0353857847$ | $8.00644\times {10}^{-10}$ | $1.1432235606$ | $9.68646\times {10}^{-10}$ |

0.75 | $1.0623756761$ | $8.76832\times {10}^{-10}$ | $1.1162950665$ | $9.67723\times {10}^{-10}$ |

1 | $1.0893456043$ | $9.38968\times {10}^{-10}$ | $1.0893456043$ | $9.38968\times {10}^{-10}$ |

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**MDPI and ACS Style**

Alshammari, M.; Al-Smadi, M.; Arqub, O.A.; Hashim, I.; Alias, M.A.
Residual Series Representation Algorithm for Solving Fuzzy Duffing Oscillator Equations. *Symmetry* **2020**, *12*, 572.
https://doi.org/10.3390/sym12040572

**AMA Style**

Alshammari M, Al-Smadi M, Arqub OA, Hashim I, Alias MA.
Residual Series Representation Algorithm for Solving Fuzzy Duffing Oscillator Equations. *Symmetry*. 2020; 12(4):572.
https://doi.org/10.3390/sym12040572

**Chicago/Turabian Style**

Alshammari, Mohammad, Mohammed Al-Smadi, Omar Abu Arqub, Ishak Hashim, and Mohd Almie Alias.
2020. "Residual Series Representation Algorithm for Solving Fuzzy Duffing Oscillator Equations" *Symmetry* 12, no. 4: 572.
https://doi.org/10.3390/sym12040572