# Predictor Laplace Fractional Power Series Method for Finding Multiple Solutions of Fractional Boundary Value Problems

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## Abstract

**:**

## 1. Introduction

## 2. Preliminary and Notations

- $L\left[ax\right(t)+by(t\left)\right]=aX\left(s\right)+bY\left(s\right)$.
- ${L}^{-1}[aX\left(s\right)+bY\left(s\right)]=ax\left(t\right)+by\left(t\right)$.
- $\underset{s\to \infty}{lim}\left[sY\left(s\right)\right]=y\left(0\right)$.
- $L\left[{D}^{k\alpha}y\left(t\right)\right]={s}^{k\alpha}Y\left(s\right)-{\displaystyle \sum _{j=0}^{k-1}}{s}^{(k-j)\alpha -1}\left({D}^{j\alpha}y\right)\left(0\right)$, for $\alpha \in (0,1]$.

## 3. Predictor Laplace FPSM

- Assume the initial conditions ${y}^{\left(i\right)}\left(0\right)={A}_{i}$.
- Apply the LT to Equation (4) to obtain$$Y\left(s\right)-\sum _{i=0}^{n-1}\left({\displaystyle \frac{{y}^{\left(i\right)}\left(0\right)}{{s}^{i+1}}}\right)={\displaystyle \frac{1}{{s}^{\alpha}}}L\left[N(t,y)\right].$$
- Assume that $Y\left(s\right)$ and the LT of the nonlinear term can be formulated as$$\begin{array}{ccc}\hfill Y\left(s\right)& =& \sum _{i=0}^{J}\left({\displaystyle \frac{{c}_{i}}{{s}^{qi+1}}}\right),\hfill \end{array}$$$$\begin{array}{ccc}\hfill L\left[N\right(t,y]& =& \sum _{i={s}_{1}}^{{s}_{2}}\left({\displaystyle \frac{{w}_{i}}{{s}^{qi+1}}}\right),\hfill \end{array}$$
- Set the ${c}_{i}$ for $i=0,1,2,\dots (n-1)/q+{\alpha}_{1}$ as ${c}_{0}={A}_{0}=y\left(0\right),{c}_{1/q}={A}_{1}={y}^{\prime}\left(0\right),$${c}_{2/q}={A}_{2}={y}^{\u2033}\left(0\right),\dots {c}_{(n-1)/q}={y}^{(n-1)}\left(0\right)={A}_{n-1}$, and ${c}_{i}=0$ for the other values.
- Multiply (9) by ${s}^{jq+1}$ and take the limit $\underset{s\u27f6\infty}{lim}{s}^{jq+1}{R}_{j}=0$ to obtain$$\underset{s\u27f6\infty}{lim}{s}^{jq+1}{R}_{j}=\underset{s\u27f6\infty}{lim}\sum _{i=0}^{j}\left({\displaystyle \frac{{c}_{i}}{{s}^{q(i-j)}}}\right)-\sum \left({\displaystyle \frac{{w}_{i}}{{s}^{q[(n-1)/q+{\alpha}_{1}+i-j]}}}\right)=0.$$This gives ${c}_{i}={w}_{i-(n-1)/q-{\alpha}_{1}}$ for $i>(n-1)/q+{\alpha}_{1}$.
- Apply the inverse Laplace transform to Equation (8); the N-th order of the approximate solution becomes$$y(t,{\mathbf{A}}_{n-1})=\sum _{i=0}^{N}{\displaystyle \frac{{c}_{i}{t}^{qi}}{\Gamma (qi+1)}}.$$
- Substitute the solution $y(t,{\mathbf{A}}_{n-1})$ into the conditions in Equation (5) to generate n-nonlinear algebraic equations.
- Solve the equations generated in Step 8 using an accurate method such as the Newton–Raphson method or the new optimal numerical root-solver [29] to determine all solutions for the unknown ${A}_{i}$.
- Substitute the solutions for ${A}_{i}$ from Step 9 into $y(t,{\mathbf{A}}_{n-1})$ to obtain the complete solution to Equation (4).

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

## 4. Applications

**Example**

**1.**

**Example**

**2.**

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The solutions of Example 1 using the 70th order of approximation for $\alpha =1.9,1.8,1.7,1.68$. The first and the second solution are represent in the orange and blue curves respectively.

**Figure 2.**The solutions of Example 2 using the 700th order of approximation for $\alpha =3.9,3.8,3.7,3.26$. The first and the second solution are represent in the blue and orange curves respectively.

n | $\mathit{\alpha}=1.9$ | $\mathit{\alpha}=1.8$ | $\mathit{\alpha}=1.7$ | |||
---|---|---|---|---|---|---|

40 | 0.6052120962 | 0.8666280497 | 0.6454299587 | 0.8409846122 | 0.7040690899 | 0.7968988326 |

50 | 0.6052120962 | 0.8666280497 | 0.6454299587 | 0.8409846122 | 0.7040690899 | 0.7968988326 |

60 | 0.6052065265 | 0.8666339906 | 0.6454123301 | 0.8410056456 | 0.7039799571 | 0.7970001618 |

70 | 0.6052065265 | 0.8666339906 | 0.6454123301 | 0.8410056456 | 0.7039795681 | 0.7970000599 |

80 | 0.6052064987 | 0.8666339632 | 0.6454122363 | 0.8410055710 | 0.7039795681 | 0.7970000599 |

90 | 0.6052064987 | 0.8666339632 | 0.6454122364 | 0.8410055708 | 0.7039795696 | 0.7970000577 |

100 | 0.6052064987 | 0.8666339632 | 0.6454122364 | 0.8410055708 | 0.7039795696 | 0.7970000577 |

**Table 2.**The values of ${A}_{2}$ for different $\alpha $ and n-th orders using $\sigma =-20$ for Example 2.

n | $\mathit{\alpha}=3.9$ | $\mathit{\alpha}=3.8$ | $\mathit{\alpha}=3.7$ | |||
---|---|---|---|---|---|---|

300 | −2.908126478 | 145.8780662 | −2.890668959 | 125.3769389 | −2.870259296 | 106.3892558 |

400 | −2.908126478 | 147.7398896 | −2.890668959 | 131.8532334 | −2.870259296 | 113.2663820 |

500 | −2.908126478 | 148.4348950 | −2.890668959 | 130.8731202 | −2.870259296 | 118.8103639 |

600 | −2.908126478 | 148.3813672 | −2.890668959 | 130.4388597 | −2.870259296 | 115.3101551 |

700 | −2.908126478 | 148.3654474 | −2.890668959 | 130.4794925 | −2.870259296 | 115.9764097 |

800 | −2.908126478 | 148.3669774 | −2.890668959 | 130.4985713 | −2.870259296 | 115.7959115 |

900 | −2.908126478 | 148.3668728 | −2.890668959 | 130.4956720 | −2.870259296 | 115.8171922 |

1000 | −2.908126478 | 148.3668434 | −2.890668959 | 130.4959435 | −2.870259296 | 115.8344660 |

**Table 3.**The values of ${A}_{2}$ for different $\alpha $ and n-th orders using $\sigma =20$ for Example 2.

n | $\mathit{\alpha}=3.9$ | $\mathit{\alpha}=3.8$ | $\mathit{\alpha}=3.7$ | |||
---|---|---|---|---|---|---|

300 | −138.0711661 | −3.101976104 | −118.1546084 | −3.123617677 | −100.1782084 | −3.149851401 |

400 | −139.3947405 | −3.101976104 | −122.1853108 | −3.123617677 | −109.4983522 | −3.149851401 |

500 | −139.7981285 | −3.101976104 | −121.7141473 | −3.123617677 | −107.3604086 | −3.149851401 |

600 | −139.7714048 | −3.101976104 | −121.5374888 | −3.123617677 | −106.3390332 | −3.149851401 |

700 | −139.7646425 | −3.101976104 | −121.5522702 | −3.123617677 | −106.524706 | −3.149851401 |

800 | −139.7651863 | −3.101976104 | −121.5578166 | −3.123617677 | −106.4873698 | −3.149851401 |

900 | −139.7651539 | −3.101976104 | −121.5571362 | −3.123617677 | −106.4913908 | −3.149851401 |

1000 | −139.7651461 | −3.101976104 | −121.5571918 | −3.123617677 | −106.4937748 | −3.149851401 |

**Table 4.**The solutions $y\left(t\right)$ of Example 2 with $\sigma =-20$ using the 700th order of approximation at several values of t.

t | $\mathit{y}\left(\mathit{t}\right)$ for $\mathit{\alpha}=3.9$ | $\mathit{y}\left(\mathit{t}\right)$ for $\mathit{\alpha}=3.8$ | $\mathit{y}\left(\mathit{t}\right)$ for $\mathit{\alpha}=3.7$ | |||
---|---|---|---|---|---|---|

0.00 | 1.48985 | −13.7152 | 1.48802 | −11.6233 | 1.48590 | −9.89024 |

0.25 | 1.39896 | −9.10445 | 1.39768 | −7.57271 | 1.39619 | −6.30149 |

0.50 | 1.12574 | 3.34540 | 1.12593 | 3.21957 | 1.12616 | 3.11011 |

0.75 | 0.66544 | 13.7643 | 0.667053 | 11.7956 | 0.668914 | 10.1577 |

1.00 | 3.04497 $\times \phantom{\rule{4pt}{0ex}}{10}^{-18}$ | 4.16334 $\times \phantom{\rule{4pt}{0ex}}{10}^{-16}$ | 9.49464$\times \phantom{\rule{4pt}{0ex}}{10}^{-17}$ | 1.03251 $\times \phantom{\rule{4pt}{0ex}}{10}^{-14}$ | −5.60716 $\times \phantom{\rule{4pt}{0ex}}{10}^{-17}$ | 1.66533 $\times \phantom{\rule{4pt}{0ex}}{10}^{-14}$ |

**Table 5.**The solutions $y\left(t\right)$ of Example 2 with $\sigma =20$ using the 700th order of approximation at several values of t.

t | $\mathit{y}\left(\mathit{t}\right)$ for $\mathit{\alpha}=3.9$ | $\mathit{y}\left(\mathit{t}\right)$ for $\mathit{\alpha}=3.8$ | $\mathit{y}\left(\mathit{t}\right)$ for $\mathit{\alpha}=3.7$ | |||
---|---|---|---|---|---|---|

0.00 | 15.2927 | 1.51121 | 13.205 | 1.51345 | 11.4798 | 1.51614 |

0.25 | 10.9479 | 1.41428 | 9.43037 | 1.41585 | 8.17621 | 1.41773 |

0.50 | −0.857015 | 1.12413 | −0.710199 | 1.12388 | −0.582739 | 1.12356 |

0.75 | −11.2594 | 0.646121 | −9.31746 | 0.644145 | −7.71311 | 0.6418 |

1.00 | 8.88178 $\times \phantom{\rule{4pt}{0ex}}{10}^{-16}$ | 2.93434 $\times \phantom{\rule{4pt}{0ex}}{10}^{-16}$ | −2.9976 $\times \phantom{\rule{4pt}{0ex}}{10}^{-15}$ | 1.78424 $\times \phantom{\rule{4pt}{0ex}}{10}^{-16}$ | −5.9952 $\times \phantom{\rule{4pt}{0ex}}{10}^{-15}$ | −1.09784 $\times \phantom{\rule{4pt}{0ex}}{10}^{-16}$ |

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## Share and Cite

**MDPI and ACS Style**

Alomari, A.-K.; Salameh, W.M.M.; Alaroud, M.; Tahat, N.
Predictor Laplace Fractional Power Series Method for Finding Multiple Solutions of Fractional Boundary Value Problems. *Symmetry* **2024**, *16*, 1152.
https://doi.org/10.3390/sym16091152

**AMA Style**

Alomari A-K, Salameh WMM, Alaroud M, Tahat N.
Predictor Laplace Fractional Power Series Method for Finding Multiple Solutions of Fractional Boundary Value Problems. *Symmetry*. 2024; 16(9):1152.
https://doi.org/10.3390/sym16091152

**Chicago/Turabian Style**

Alomari, Abedel-Karrem, Wael Mahmoud Mohammad Salameh, Mohammad Alaroud, and Nedal Tahat.
2024. "Predictor Laplace Fractional Power Series Method for Finding Multiple Solutions of Fractional Boundary Value Problems" *Symmetry* 16, no. 9: 1152.
https://doi.org/10.3390/sym16091152