# Application of Fractional Residual Power Series Algorithm to Solve Newell–Whitehead–Segel Equation of Fractional Order

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries and Notations

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

- ${\mathcal{D}}_{t}^{\beta}c=0$, $c\in \mathbb{R}$.
- ${\mathcal{D}}_{t}^{\beta}{t}^{q}=\frac{\Gamma \left(q+1\right)}{\Gamma \left(q+1-\beta \right)}{t}^{q-\beta}$, $n-1<\beta \le n,qn-1,n\in \mathbb{N},q\in \mathbb{R}$.
- ${\mathcal{D}}_{t}^{\beta}\left(\lambda {\phi}_{1}\left(t\right)+\mu {\phi}_{2}\left(t\right)\right)=\lambda {\mathcal{D}}_{t}^{\beta}{\phi}_{1}\left(t\right)+\mu {\mathcal{D}}_{t}^{\beta}{\phi}_{2}\left(t\right)$, that is, ${\mathcal{D}}_{t}^{\beta}$ is a linear, $\lambda ,\mu \in \mathbb{R}$.

**Definition**

**4.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

## 3. Solution Methodology of the FRPS Algorithm

**Algorithm**

**1.**

- Step 1: Assume that the solution of FNWSEs (1) and (2) has the MFPS about ${t}_{0}=0$:$$\phi \left(x,t\right)={\displaystyle \sum}_{n=0}^{\infty}{h}_{n}\left(x\right)\frac{{t}^{n\beta}}{\Gamma \left(n\beta +1\right)},$$
- Step 2: Define the $k$th truncated series ${\phi}_{k}\left(x,t\right)$ of $\phi \left(x,t\right)$ such that$${\phi}_{k}\left(x,t\right)={\displaystyle \sum}_{n=0}^{k}{h}_{n}\left(x\right)\frac{{t}^{n\beta}}{\Gamma \left(n\beta +1\right)}.$$
- Step 3: Consider the initial condition $\phi \left(x,0\right)={h}_{0}\left(x\right)$, then the zeroth MFPS approximate solution of $\phi \left(x,t\right)$ is ${\phi}_{k}\left(x,t\right)={h}_{0}\left(x\right)+{{\displaystyle \sum}}_{n=1}^{k}{h}_{n}\left(x\right)\frac{{t}^{n\beta}}{\Gamma \left(n\beta +1\right)}$.
- Step 4: Define the $k$th residual function $Re{s}_{\phi}^{k}\left(x,t\right)$ such that$$Re{s}_{\phi}^{k}\left(x,t\right)={\mathcal{D}}_{t}^{\beta}{\phi}_{k}\left(x,t\right)-a{\mathcal{D}}_{x}^{2}{\phi}_{k}\left(x,t\right)-b{\phi}_{k}\left(x,t\right)+c{\phi}_{k}^{p}\left(t,x\right).$$
- Step 5: Substitute the $k$th truncated series ${\phi}_{k}\left(x,t\right)$ into the $k$th residual function $Re{s}_{\phi}^{k}\left(x,t\right)$ such that$$\begin{array}{l}Re{s}_{\phi}^{k}\left(x,t\right)={\mathcal{D}}_{t}^{\beta}\left({h}_{0}\left(x\right)+{{\displaystyle \sum}}_{n=1}^{k}{h}_{n}\left(x\right)\frac{{t}^{n\beta}}{\Gamma \left(n\beta +1\right)}\right)-a{\mathcal{D}}_{x}^{2}\left({h}_{0}\left(x\right)+{{\displaystyle \sum}}_{n=1}^{k}{h}_{n}\left(x\right)\frac{{t}^{n\beta}}{\Gamma \left(n\beta +1\right)}\right)-\\ \phantom{\rule{2cm}{0ex}}b\left({h}_{0}\left(x\right)+{{\displaystyle \sum}}_{n=1}^{k}{h}_{n}\left(x\right)\frac{{t}^{n\beta}}{\Gamma \left(n\beta +1\right)}\right)+c{\left({h}_{0}\left(x\right)+{{\displaystyle \sum}}_{n=1}^{k}{h}_{n}\left(x\right)\frac{{t}^{n\beta}}{\Gamma \left(n\beta +1\right)}\right)}^{p}.\end{array}$$
- Step 6: Set $k=1$ in Step 5, then by using $Re{s}_{\phi}^{1}{\left(x,t\right)}_{\u2502t={t}_{0}}=0$, the first unknown coefficient ${\phi}_{1}\left(x\right)$ is obtained. Therefore, the first approximate PS solution ${\phi}_{1}\left(x,t\right)$ is also obtained.
- Step 7: For $k=2,3,\dots ,m$, do the following subroutine:
- (A)
- Apply the operator ${\mathcal{D}}_{t}^{\beta}$, ($k-1$) times, on both sides of the $k$th residual function $Re{s}_{\phi}^{k}\left(x,t\right)$ in Step 4 such that ${D}_{t}^{\left(n-1\right)\beta}Re{s}_{\phi}^{k}\left(x,t\right)$.
- (B)
- Compute the resulting equation at $t=0$ with equality to zero such that ${\mathcal{D}}_{t}^{\left(k-1\right)\beta}Re{s}_{\phi}^{k}\left(x,0\right)=0$, with the help of ${\mathcal{D}}_{t}^{\beta}{t}^{q}=0$ for $q>\beta $ at $t=0$.
- (C)
- Find the $k$th unknown coefficient ${\phi}_{k}\left(x\right)$ and do Step 7 for $k=k+1$ until the arbitrary $m$.

- Step 8: Collect the obtained coefficients ${\phi}_{k}\left(x\right)$ for each $k=0,1,2,\dots ,m$ in terms of expanded MFPS ${\phi}_{k}\left(x,t\right)$ and try to find a general pattern with the term of infinite series so that the exact solution $\phi \left(x,t\right)$ of FNWSEs (1) and (2) is obtained; otherwise, the pattern obtained in the sense of the series coefficients will be the $m$th approximate MFPS solution of FNWSEs (1) and (2).
- Step 9: Stop.

**Theorem**

**3.**

**Proof.**

## 4. Numerical Results and Discussion

**Example**

**1.**

**Example**

**2.**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The solution behavior of the approximate solutions ${\phi}_{8}\left(x,t\right)$ for $\beta \in \left\{0.75,0.85,0.95,1\right\}$ compared with the exact solution. Red, $\beta =0.75$; dashed blue, $\beta =0.85$; dashed green, $\beta =0.95$; dashed darker red, $\beta =1$; and blue, exact solution.

**Figure 2.**Solution behavior of the exact $\phi \left(x,t\right)$ and approximate solution ${\phi}_{4}\left(x,t\right)$ at $\beta =1$, for nonlinear fractional Newell–Whitehead–Segel equations (FNWSEs) (16) and (17): (

**a**) Exact solution; (

**b**) FRPS solution.

**Figure 3.**Plots of the solutions for Example 2 at $\beta =1$ and $t\in \left[0,1\right]$. Red for the exact solution; blue for the fractional residual power series (FRPS) solution; green for the FCT-HP solution [12]: (

**a**) Solutions at $\gamma =0.05$; (

**b**) Solution at $\gamma =0.005$.

**Figure 4.**Plots of the FRPS and FCT-HP solutions for Example 2 at $x=3$ and $t\in \left[0,2\right]$ for different values of fractional order $\beta $. Red, $\beta =1$; blue, $\beta =0.85$; green, $\beta =0.75$; orange, $\beta =0.55$; gray, $\beta =0.45$: (

**a**) FRPS solutions; (

**b**) FCT-HP solutions.

**Figure 5.**Surface plots of the fifth-order FRPS solutions for Example 2 at $x\in \left[-4,4\right]$ and $t\in \left[0,2\right]$ for different values of fractional order $\beta $ such that $\beta =\left\{1,0.75,0.5,0.25\right\}$: (

**a**) $\beta =1$; (

**b**) $\beta =0.75$; (

**c**) $\beta =0.5$; (

**d**) $\beta =0.25$.

$\mathit{x}$ | $\mathit{t}$ | Exact Solution | Approximation | Absolute Error |
---|---|---|---|---|

$2$ | $0.16$ | $6.29653826102665$ | $6.29653826102803$ | $1.377564\times {10}^{-12}$ |

$0.32$ | $5.36555597112197$ | $5.36555597181613$ | $6.941567\times {10}^{-10}$ | |

$0.48$ | $4.57222519514215$ | $4.57222522141770$ | $2.627554\times {10}^{-8}$ | |

$0.64$ | $3.89619330179521$ | $3.89619364642941$ | $3.446342\times {10}^{-7}$ | |

$0.80$ | $3.32011692273654$ | $3.32011945197702$ | $2.529240\times {10}^{-6}$ | |

$0.96$ | $2.82921701435156$ | $2.82922987150470$ | $1.285715\times {10}^{-5}$ | |

$-2$ | $0.16$ | $0.115325121038062$ | $0.115325121038087$ | $2.524369\times {10}^{-14}$ |

$0.32$ | $0.098273585604361$ | $0.098273585617075$ | $1.271391\times {10}^{-11}$ | |

$0.48$ | $0.083743225592195$ | $0.083743226073449$ | $4.812533\times {10}^{-10}$ | |

$0.64$ | $0.071361269556386$ | $0.071361275868581$ | $6.312195\times {10}^{-9}$ | |

$0.80$ | $0.060810062625217$ | $0.060810108949873$ | $4.632465\times {10}^{-8}$ | |

$0.96$ | $0.051818917172725$ | $0.051819152659700$ | $2.354869\times {10}^{-7}$ |

**Table 2.**Error analysis of ${\phi}_{n}\left(x,t\right)$, $n=4,8,12,16,$ at $\beta =1$ for Example 1.

Iteration | Errors | $\mathit{t}=0.2$ | $\mathit{t}=0.4$ | $\mathit{t}=0.6$ | $\mathit{t}=0.8$ |
---|---|---|---|---|---|

$\mathit{n}\mathbf{=}\mathbf{4}$ | Absolute | $7.013861\times {10}^{-6}$ | $2.173374\times {10}^{-4}$ | $1.599339\times {10}^{-3}$ | $6.535753\times {10}^{-3}$ |

Relative | $3.151531\times {10}^{-6}$ | $1.192773\times {10}^{-4}$ | $1.072069\times {10}^{-3}$ | $5.351022\times {10}^{-3}$ | |

$\mathit{n}\mathbf{=}\mathbf{8}$ | Absolute | $3.759659\times {10}^{-12}$ | $1.887899\times {10}^{-9}$ | $7.119639\times {10}^{-8}$ | $9.304556\times {10}^{-7}$ |

Relative | $1.689324\times {10}^{-12}$ | $1.036101\times {10}^{-9}$ | $4.772437\times {10}^{-8}$ | $7.617926\times {10}^{-7}$ | |

$\mathit{n}\mathbf{=}\mathbf{12}$ | Absolute | $0$ | $2.88658\times {10}^{-15}$ | $5.46452\times {10}^{-13}$ | $2.26967\times {10}^{-11}$ |

Relative | $0$ | $1.58419\times {10}^{-15}$ | $3.66298\times {10}^{-13}$ | $1.85825\times {10}^{-11}$ | |

$\mathit{n}\mathbf{=}\mathbf{16}$ | Absolute | $0$ | $0$ | $8.88178\times {10}^{-16}$ | $5.95364\times {10}^{-16}$ |

Relative | $0$ | $0$ | $6.94999\times {10}^{-14}$ | $1.95599\times {10}^{-12}$ |

${\mathit{t}}_{\mathit{i}}$ | $\mathit{\beta}=1$ | $\mathit{\beta}=0.95$ | $\mathit{\beta}=0.85$ | $\mathit{\beta}=0.75$ |
---|---|---|---|---|

$0.01$ | $0.99019625$ | $0.98749024$ | $0.97988049$ | $0.96832475$ |

$0.02$ | $0.98076933$ | $0.97640839$ | $0.96501163$ | $0.94926497$ |

$0.03$ | $0.97169425$ | $0.96609889$ | $0.95210517$ | $0.93362989$ |

$0.04$ | $0.96294399$ | $0.95638186$ | $0.94043021$ | $0.91968384$ |

$0.05$ | $0.95448958$ | $0.94714379$ | $0.92958436$ | $0.90653732$ |

$0.06$ | $0.94629999$ | $0.93829533$ | $0.91928645$ | $0.89360624$ |

$0.07$ | $0.93834225$ | $0.92975813$ | $0.90931428$ | $0.88045413$ |

$0.08$ | $0.93058133$ | $0.92145933$ | $0.89947874$ | $0.86672793$ |

$0.09$ | $0.92298025$ | $0.91332880$ | $0.88961102$ | $0.85212675$ |

$0.10$ | $0.91550000$ | $0.90529762$ | $0.87955555$ | $0.83638473$ |

**Table 4.**Numerical comparison of ${\phi}_{5}\left(x,t\right)$ at $\beta =1$ and $\gamma =0.001$ for Example 2. FPS, fractional power series; FCT-HP, fractional complex transform coupled with He’s polynomials.

$\mathit{t}$ | Exact | FPS Method | FCT-HP Method [12] | ||
---|---|---|---|---|---|

Fifth Appr. Sol. | Absolute Error | Fifth Appr. Sol. | Absolute Error | ||

$0.16$ | $0.0013764$ | $0.00137634$ | $5.54268\times {10}^{-9}$ | $0.00137396$ | $2.39074\times {10}^{-6}$ |

$0.32$ | $0.0018939$ | $0.00189377$ | $1.64202\times {10}^{-7}$ | $0.00187473$ | $1.92088\times {10}^{-5}$ |

$0.48$ | $0.0026054$ | $0.00260390$ | $1.49387\times {10}^{-6}$ | $0.00253977$ | $6.56309\times {10}^{-5}$ |

$0.64$ | $0.0035827$ | $0.00357487$ | $7.81441\times {10}^{-6}$ | $0.00342320$ | $1.59483\times {10}^{-4}$ |

$0.80$ | $0.0049238$ | $0.00489447$ | $2.93697\times {10}^{-5}$ | $0.00459900$ | $3.24833\times {10}^{-4}$ |

$0.96$ | $0.0067619$ | $0.00667333$ | $8.85884\times {10}^{-5}$ | $0.00616420$ | $5.97721\times {10}^{-4}$ |

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**MDPI and ACS Style**

Saadeh, R.; Alaroud, M.; Al-Smadi, M.; Ahmad, R.R.; Salma Din, U.K.
Application of Fractional Residual Power Series Algorithm to Solve Newell–Whitehead–Segel Equation of Fractional Order. *Symmetry* **2019**, *11*, 1431.
https://doi.org/10.3390/sym11121431

**AMA Style**

Saadeh R, Alaroud M, Al-Smadi M, Ahmad RR, Salma Din UK.
Application of Fractional Residual Power Series Algorithm to Solve Newell–Whitehead–Segel Equation of Fractional Order. *Symmetry*. 2019; 11(12):1431.
https://doi.org/10.3390/sym11121431

**Chicago/Turabian Style**

Saadeh, Rania, Mohammad Alaroud, Mohammed Al-Smadi, Rokiah Rozita Ahmad, and Ummul Khair Salma Din.
2019. "Application of Fractional Residual Power Series Algorithm to Solve Newell–Whitehead–Segel Equation of Fractional Order" *Symmetry* 11, no. 12: 1431.
https://doi.org/10.3390/sym11121431