# Generalized Shifted Airfoil Polynomials of the Second Kind to Solve a Class of Singular Electrohydrodynamic Fluid Model of Fractional Order

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Liouville–Caputo Fractional Derivative

**Definition**

**1.**

## 3. The Shifted Airfoil Polynomials and Their Convergence Results

#### 3.1. The Airfoil Polynomials: A Shifted Version

**Definition**

**2.**

**Lemma**

**1.**

#### 3.2. Convergent and Error Analysis

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Theorem**

**4.**

**Proof.**

## 4. The Hybrid QLM-SAPSK Procedure

**Lemma**

**3.**

**Corollary**

**1.**

**Remark**

**1.**

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Lemma**

**4.**

#### Testing Accuracy via REFs

## 5. Experimental Results and Simulations

#### The Fractional-Order Cases

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Graphics of numerical solution obtained via QLM-SAPSK approach (

**left**) and related REFs (

**right**) using $\delta ,{\mathcal{H}}^{2}=1$, $R=7$, and with integer-orders $\theta =2,\lambda =1$.

**Figure 2.**Graphics of numerical solution obtained via QLM-SAPSK approach with $\delta =0.5$ (

**left**) and $\delta =2$ (

**right**) using $R=10$, various ${\mathcal{H}}^{2}=0.5,1,2,4,10$, and with integer-orders $\theta =2,\lambda =1$.

**Figure 3.**Graphics of numerical solution obtained via the QLM-SAPSK approach with ${\mathcal{H}}^{2}=2$ (

**left**) and the associated REFs (

**right**) using $R=10$, various $\delta =0.1,0.3,0.6,1,2$, and with integer-orders $\theta =2,\lambda =1$.

**Figure 4.**Graphics of numerical solution obtained via QLM-SAPSK approach with ${\mathcal{H}}^{2}=25$ (

**left**) and ${\mathcal{H}}^{2}=100$ (

**right**) using $R=20$, various $\delta =0.1,0.3,0.6,1,2$, and with integer-orders $\theta =2,\lambda =1$.

**Figure 5.**Graphics of numerical solutions obtained via the QLM-GSAPSK technique (

**left**) and related REFs (

**right**) with $R=10$, $\alpha =1,\lambda ,\theta $, $\delta =0.5,{\mathcal{H}}^{2}=1$, and with fractional-orders $\theta =1.9,\lambda =0.9$.

**Figure 6.**Graphics of numerical solutions obtained via the QLM-GSAPSK technique (

**left**) and related REFs (

**right**) with $R=20$, $\alpha =\theta $, $\delta =0.5$, and with fractional-orders $\theta =1.9,\lambda =0.9$.

**Figure 7.**Graphics of numerical solutions obtained via the QLM-GSAPSK technique (

**left**) and related REFs (

**right**) with $R=10$, $\alpha =\theta $, $\delta =0.5$, and with fractional-orders $\theta =1.5,\lambda =0.5$.

**Figure 8.**Graphics of numerical solutions obtained via the QLM-GSAPSK technique (

**left**) and related REFs (

**right**) with $R=10$, $\alpha =\theta $, $\delta =4$, ${\mathcal{H}}^{2}=0.5$, and various fractional-orders $\theta ,\lambda $.

**Figure 9.**Graphics of numerical solutions obtained via the QLM-GSAPSK technique (

**left**) and related REFs (

**right**) with $R=30$, $\alpha =\theta $, $\delta =0.5$, ${\mathcal{H}}^{2}=50$, and various fractional-orders $\theta ,\lambda $.

**Table 1.**A comparison of numerical outcomes/REFs in QLM-SAPSK with $\theta =2,\lambda =1$, $\delta ,{\mathcal{H}}^{2}=1$, $R=7,14$, and various $p\in [0,1]$.

QLM-SAPSK | LSM [11] | GM [4] | LSCM [4] | ||||
---|---|---|---|---|---|---|---|

$\mathit{p}$ | ${\mathbf{\chi}}_{\mathbf{7}}^{\left(\mathbf{5}\right)}\left(\mathit{p}\right)$ | ${\mathcal{R}}_{\mathbf{7}}^{\left(\mathbf{5}\right)}\left(\mathit{p}\right)$ | ${\mathbf{\chi}}_{\mathbf{14}}^{\left(\mathbf{5}\right)}\left(\mathit{p}\right)$ | ${\mathcal{R}}_{\mathbf{14}}^{\left(\mathbf{5}\right)}\left(\mathit{p}\right)$ | $\mathit{N}=\mathbf{10}$ | ${\mathit{N}}_{\mathit{G}}=\mathbf{5}$ | ${\mathit{N}}_{\mathit{C}}=\mathbf{7}$ |

$0.0$ | $0.20341502$ | $0.0000\times {10}^{-0}$ | $0.203415795896659$ | $0.0000\times {10}^{-00}$ | $0.20343243$ | $0.20343574$ | $0.20342786$ |

$0.1$ | $0.20155175$ | $1.8830\times {10}^{-6}$ | $0.201552363573421$ | $2.3386\times {10}^{-13}$ | $0.20156532$ | $0.20157001$ | $0.20155941$ |

$0.2$ | $0.19593971$ | $1.5963\times {10}^{-8}$ | $0.195940129562329$ | $2.9106\times {10}^{-14}$ | $0.19594756$ | $0.19594122$ | $0.19593886$ |

$0.3$ | $0.18651339$ | $3.0982\times {10}^{-7}$ | $0.186513678715622$ | $5.3721\times {10}^{-14}$ | $0.18651760$ | $0.18651537$ | $0.18654911$ |

$0.4$ | $0.17316508$ | $1.5520\times {10}^{-7}$ | $0.173165302575071$ | $4.7990\times {10}^{-14}$ | $0.17316801$ | $0.17317053$ | $0.17316782$ |

$0.5$ | $0.15574680$ | $1.5535\times {10}^{-7}$ | $0.155746961727708$ | $4.0416\times {10}^{-14}$ | $0.15574958$ | $0.15574892$ | $0.15573556$ |

$0.6$ | $0.13407288$ | $1.6136\times {10}^{-7}$ | $0.134073001192206$ | $3.6251\times {10}^{-14}$ | $0.13407547$ | $0.13407591$ | $0.13408008$ |

$0.7$ | $0.10792349$ | $8.2611\times {10}^{-8}$ | $0.107923574051674$ | $3.9266\times {10}^{-14}$ | $0.10792535$ | $0.10792460$ | $0.10792695$ |

$0.8$ | $0.07704865$ | $2.1210\times {10}^{-7}$ | $0.077048701459288$ | $5.2063\times {10}^{-14}$ | $0.07704953$ | $0.07704912$ | $0.07705274$ |

$0.9$ | $0.04117284$ | $1.9348\times {10}^{-7}$ | $0.041172865563138$ | $3.4394\times {10}^{-14}$ | $0.04117309$ | $0.04117301$ | $0.04118162$ |

**Table 2.**The outcomes of ${\mathcal{E}}_{\infty}$ error norms, the related ${\mathrm{o}\mathrm{r}\mathrm{d}}_{R}^{\infty}$, and the spent CPU time with $\delta ,{\mathcal{H}}^{2}=1$, $\theta =2,\lambda =1$, and different R.

QLM-SAPSK | CBS [16] | ||||||
---|---|---|---|---|---|---|---|

R | ${\mathcal{E}}_{\mathit{\infty}}$ | ${\mathbf{ord}}_{\mathit{R}}^{\mathit{\infty}}$ | CPU(s) | n | MAE | ROC | CPU(s) |

2 | $5.4729\times {10}^{-1}$ | − | $0.49927$ | 16 | $1.7062\times {10}^{-11}$ | − | $0.1745$ |

4 | $8.4250\times {10}^{-3}$ | $6.0215$ | $0.62609$ | 32 | $3.5011\times {10}^{-13}$ | $5.6068$ | $0.4577$ |

8 | $1.9782\times {10}^{-7}$ | $15.378$ | $0.89085$ | 64 | $5.8009\times {10}^{-15}$ | $5.9154$ | $1.0182$ |

16 | $9.6751\times {10}^{-14}$ | $20.963$ | $1.47224$ | 128 | $9.1434\times {10}^{-17}$ | $5.9874$ | $2.8513$ |

**Table 3.**A comparison of ${\mathcal{E}}_{\infty}$ error norms utilizing $R=10$, $\delta =0.5$, $\theta =2,\lambda =1$, and different ${\mathcal{H}}^{2}$.

QLM-SAPSK | Jacobi [24] | Legendre [24] | Chebyshev [24] | HW [18] | DADS [14] | |
---|---|---|---|---|---|---|

${\mathcal{H}}^{\mathbf{2}}$ | $\mathit{R}=\mathbf{10}$ | $\mathit{N}=\mathbf{10}$ | $\mathit{N}=\mathbf{10}$ | $\mathit{N}=\mathbf{10}$ | $\mathit{K}=\mathbf{16}$ | $\mathit{N},\mathit{n}=\mathbf{5}$ |

$0.5$ | $2.2246\times {10}^{-11}$ | $2.4881\times {10}^{-9}$ | $1.3623\times {10}^{-10}$ | $6.4622\times {10}^{-10}$ | − | $1.6287\times {10}^{-8}$ |

$1.0$ | $9.8648\times {10}^{-10}$ | $6.0006\times {10}^{-9}$ | $3.1226\times {10}^{-9}$ | $3.3842\times {10}^{-9}$ | $4.189\times {10}^{-5}$ | $9.4029\times {10}^{-7}$ |

$2.0$ | $3.4171\times {10}^{-8}$ | $3.3174\times {10}^{-7}$ | $1.7631\times {10}^{-7}$ | $1.1158\times {10}^{-7}$ | $2.421\times {10}^{-7}$ | $4.5114\times {10}^{-5}$ |

$4.0$ | $9.9502\times {10}^{-7}$ | $8.2384\times {10}^{-6}$ | $4.5498\times {10}^{-6}$ | $2.9458\times {10}^{-6}$ | $6.733\times {10}^{-6}$ | $1.4724\times {10}^{-5}$ |

**Table 4.**A comparison of numerical outcomes/REFs in QLM-GSAPSK with $\theta =1.9,\lambda =0.9$, $\delta =0.5,{\mathcal{H}}^{2}=2$, $R=10$, $\alpha =1,1.9$, and various $p\in [0,1]$.

QLM-SAPSK | ||||||
---|---|---|---|---|---|---|

p | $\mathit{\alpha}=1.9$ | REFs | $\mathit{\alpha}=1$ | REFs | RKHSM [1] | RKHSM [23] |

$0.0$ | $0.368955980597229$ | $0.0000\times {10}^{-00}$ | $0.3687331653$ | $0.0000\times {10}^{-0}$ | $0.381236310$ | $0.3825201644$ |

$0.1$ | $0.365172617552478$ | $2.6995\times {10}^{-10}$ | $0.3650687781$ | $5.7028\times {10}^{-5}$ | $0.374950080$ | $0.3754454339$ |

$0.2$ | $0.354724901744587$ | $1.9371\times {10}^{-10}$ | $0.3546658303$ | $9.9371\times {10}^{-6}$ | $0.359968470$ | $0.3600500656$ |

$0.3$ | $0.337827276925944$ | $1.5404\times {10}^{-11}$ | $0.3377883382$ | $1.0603\times {10}^{-5}$ | $0.342105510$ | $0.3434211029$ |

$0.4$ | $0.314292044298967$ | $1.2527\times {10}^{-11}$ | $0.3142653985$ | $9.2000\times {10}^{-6}$ | $0.315723980$ | $0.3158370163$ |

$0.5$ | $0.283712502798752$ | $7.4387\times {10}^{-12}$ | $0.2836938281$ | $4.7247\times {10}^{-6}$ | $0.284128540$ | $0.2874089253$ |

$0.6$ | $0.245509789495880$ | $4.2012\times {10}^{-12}$ | $0.2454968764$ | $2.1833\times {10}^{-7}$ | $0.244546771$ | $0.2475168041$ |

$0.7$ | $0.198955842096497$ | $2.7814\times {10}^{-12}$ | $0.1989472632$ | $3.2885\times {10}^{-6}$ | $0.197105564$ | $0.2007161568$ |

$0.8$ | $0.143189521840292$ | $2.6542\times {10}^{-12}$ | $0.1431843235$ | $4.3863\times {10}^{-6}$ | $0.141053509$ | $0.1443651864$ |

$0.9$ | $0.077231456051811$ | $1.7987\times {10}^{-12}$ | $0.0772290694$ | $4.8701\times {10}^{-6}$ | $0.075613972$ | $0.0777826865$ |

**Table 5.**A comparison of numerical outcomes/REFs in QLM-GSAPSK with $\theta =1.9,\lambda =0.9$, $\delta =0.5,{\mathcal{H}}^{2}=2$, $R=5,10,15,20$, $\alpha =1.9$, and various $p\in [0,1]$.

p | $\mathit{R}=5$ | $\mathit{R}=10$ | $\mathit{R}=15$ | $\mathit{R}=20$ |
---|---|---|---|---|

$0.0$ | $0.368913634324475$ | $0.368955980597229$ | $0.368955980703425$ | $0.368955980703425$ |

$0.1$ | $0.365132468934994$ | $0.365172617552478$ | $0.365172617645275$ | $0.365172617645275$ |

$0.2$ | $0.354690183822644$ | $0.354724901744587$ | $0.354724901811129$ | $0.354724901811129$ |

$0.3$ | $0.337799646966645$ | $0.337827276925944$ | $0.337827276969954$ | $0.337827276969954$ |

$0.4$ | $0.314271598052510$ | $0.314292044298967$ | $0.314292044328889$ | $0.314292044328889$ |

$0.5$ | $0.283698182319713$ | $0.283712502798752$ | $0.283712502819713$ | $0.283712502819713$ |

$0.6$ | $0.245500072203232$ | $0.245509789495880$ | $0.245509789510364$ | $0.245509789510364$ |

$0.7$ | $0.198949413418262$ | $0.198955842096497$ | $0.198955842106158$ | $0.198955842106158$ |

$0.8$ | $0.143185607077043$ | $0.143189521840292$ | $0.143189521846127$ | $0.143189521846127$ |

$0.9$ | $0.077229663096623$ | $0.077231456051811$ | $0.077231456054508$ | $0.077231456054508$ |

**Table 6.**The outcomes of ${\mathcal{E}}_{\infty}$ error norms and the related ${\mathrm{o}\mathrm{r}\mathrm{d}}_{R}^{\infty}$ with $\delta =0.5,{\mathcal{H}}^{2}=1,2,5,10$, $\theta =1.5,\lambda =0.5$, $\alpha =\theta $, and different R.

${\mathcal{H}}^{2}=1$ | ${\mathcal{H}}^{2}=2$ | ${\mathcal{H}}^{2}=5$ | ${\mathcal{H}}^{2}=10$ | |||||
---|---|---|---|---|---|---|---|---|

$\mathit{R}$ | ${\mathcal{E}}_{\infty}$ | ${\mathbf{ord}}_{\mathit{R}}^{\infty}$ | ${\mathcal{E}}_{\infty}$ | ${\mathbf{ord}}_{\mathit{R}}^{\infty}$ | ${\mathcal{E}}_{\infty}$ | ${\mathbf{ord}}_{\mathit{R}}^{\infty}$ | ${\mathcal{E}}_{\infty}$ | ${\mathbf{ord}}_{\mathit{R}}^{\infty}$ |

2 | $8.2045\times {10}^{-01}$ | − | $3.1534\times {10}^{+00}$ | − | $1.7334\times {10}^{+01}$ | − | $5.6053\times {10}^{+01}$ | − |

4 | $2.6454\times {10}^{-03}$ | $8.2768$ | $3.4686\times {10}^{-02}$ | $6.5064$ | $3.0398\times {10}^{-01}$ | $5.8335$ | $3.6669\times {10}^{+00}$ | $3.9345$ |

8 | $2.5453\times {10}^{-08}$ | $16.665$ | $7.9286\times {10}^{-06}$ | $12.095$ | $1.5185\times {10}^{-03}$ | $7.6452$ | $7.4638\times {10}^{-02}$ | $5.6181$ |

16 | $8.3563\times {10}^{-16}$ | $24.860$ | $1.9781\times {10}^{-13}$ | $25.256$ | $2.7494\times {10}^{-08}$ | $15.753$ | $5.5263\times {10}^{-06}$ | $13.721$ |

**Table 7.**The outcomes of ${\mathcal{E}}_{\infty}$ error norms and the related ${\mathrm{o}\mathrm{r}\mathrm{d}}_{R}^{\infty}$ with ${\mathcal{H}}^{2}=0.5,\delta =0.5,1,2,5$, $\theta =1.5,\lambda =0.5$, $\alpha =\theta $, and different R.

$\mathit{\delta}=0.5$ | $\mathit{\delta}=1$ | $\mathit{\delta}=2$ | $\mathit{\delta}=5$ | |||||
---|---|---|---|---|---|---|---|---|

$\mathit{R}$ | ${\mathcal{E}}_{\infty}$ | ${\mathbf{ord}}_{\mathit{R}}^{\infty}$ | ${\mathcal{E}}_{\infty}$ | ${\mathbf{ord}}_{\mathit{R}}^{\infty}$ | ${\mathcal{E}}_{\infty}$ | ${\mathbf{ord}}_{\mathit{R}}^{\infty}$ | ${\mathcal{E}}_{\infty}$ | ${\mathbf{ord}}_{\mathit{R}}^{\infty}$ |

2 | $2.0904\times {10}^{-01}$ | − | $2.1260\times {10}^{-01}$ | − | $2.2009\times {10}^{-01}$ | − | $2.4592\times {10}^{-01}$ | − |

4 | $1.7041\times {10}^{-04}$ | $10.2606$ | $1.3556\times {10}^{-04}$ | $10.6150$ | $1.0364\times {10}^{-03}$ | $7.7304$ | $1.8625\times {10}^{-02}$ | $3.7229$ |

8 | $1.8749\times {10}^{-11}$ | $23.1157$ | $3.4860\times {10}^{-09}$ | $15.2470$ | $1.3152\times {10}^{-07}$ | $12.944$ | $1.8081\times {10}^{-04}$ | $6.6866$ |

16 | $2.7613\times {10}^{-16}$ | $16.0511$ | $2.7841\times {10}^{-16}$ | $23.5779$ | $1.7266\times {10}^{-15}$ | $26.183$ | $7.9406\times {10}^{-09}$ | $14.475$ |

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## Share and Cite

**MDPI and ACS Style**

Srivastava, H.M.; Izadi, M.
Generalized Shifted Airfoil Polynomials of the Second Kind to Solve a Class of Singular Electrohydrodynamic Fluid Model of Fractional Order. *Fractal Fract.* **2023**, *7*, 94.
https://doi.org/10.3390/fractalfract7010094

**AMA Style**

Srivastava HM, Izadi M.
Generalized Shifted Airfoil Polynomials of the Second Kind to Solve a Class of Singular Electrohydrodynamic Fluid Model of Fractional Order. *Fractal and Fractional*. 2023; 7(1):94.
https://doi.org/10.3390/fractalfract7010094

**Chicago/Turabian Style**

Srivastava, Hari M., and Mohammad Izadi.
2023. "Generalized Shifted Airfoil Polynomials of the Second Kind to Solve a Class of Singular Electrohydrodynamic Fluid Model of Fractional Order" *Fractal and Fractional* 7, no. 1: 94.
https://doi.org/10.3390/fractalfract7010094