# A Novel Analytical Approach for the Solution of Fractional-Order Diffusion-Wave Equations

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

## 3. ADM Implementation

## 4. New Idea Based on ADM

## 5. Numerical Results

#### 5.1. Example 1

#### 5.2. Example 2

#### 5.3. Example 3

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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$|\mathit{E}\mathit{x}\mathit{a}\mathit{c}\mathit{t}-\mathit{A}\mathit{D}\mathit{M}|$ | $|\mathit{E}\mathit{x}\mathit{a}\mathit{c}\mathit{t}-\mathit{A}\mathit{D}\mathit{M}|$ | $|\mathit{E}\mathit{x}\mathit{a}\mathit{c}\mathit{t}-\mathit{A}\mathit{D}\mathit{M}|$ | |
---|---|---|---|

$\mathbf{\xi}$ | $\mathbf{\alpha}=\mathbf{1}.\mathbf{3}$ | $\mathbf{\alpha}=\mathbf{1}.\mathbf{5}$ | $\mathbf{\alpha}=\mathbf{1}.\mathbf{8}$ |

0.1 | 2.469772 × 10${}^{-3}$ | 4.68002 × 10${}^{-4}$ | 3.2855 × 10${}^{-5}$ |

0.2 | 2.729520 × 10${}^{-3}$ | 5.17222 × 10${}^{-4}$ | 3.6311 × 10${}^{-5}$ |

0.3 | 3.016585 × 10${}^{-3}$ | 5.71619 × 10${}^{-4}$ | 4.0129 × 10${}^{-5}$ |

0.4 | 3.333843 × 10${}^{-3}$ | 6.31737 × 10${}^{-4}$ | 4.4351 × 10${}^{-5}$ |

0.5 | 3.684465 × 10${}^{-3}$ | 6.98177 × 10${}^{-4}$ | 4.9015 × 10${}^{-5}$ |

0.6 | 4.071964 × 10${}^{-3}$ | 7.71604 × 10${}^{-4}$ | 5.4170 × 10${}^{-5}$ |

0.7 | 4.500216 × 10${}^{-3}$ | 8.52755 × 10${}^{-4}$ | 5.9866 × 10${}^{-5}$ |

0.8 | 4.973509 × 10${}^{-3}$ | 9.42440 × 10${}^{-4}$ | 6.6163 × 10${}^{-5}$ |

0.9 | 5.496578 × 10${}^{-3}$ | 1.041558 × 10${}^{-3}$ | 7.3121 × 10${}^{-5}$ |

1 | 6.074659 × 10${}^{-3}$ | 1.151100 × 10${}^{-3}$ | 8.0812 × 10${}^{-5}$ |

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**MDPI and ACS Style**

Mustafa, S.; Hajira; Khan, H.; Shah, R.; Masood, S.
A Novel Analytical Approach for the Solution of Fractional-Order Diffusion-Wave Equations. *Fractal Fract.* **2021**, *5*, 206.
https://doi.org/10.3390/fractalfract5040206

**AMA Style**

Mustafa S, Hajira, Khan H, Shah R, Masood S.
A Novel Analytical Approach for the Solution of Fractional-Order Diffusion-Wave Equations. *Fractal and Fractional*. 2021; 5(4):206.
https://doi.org/10.3390/fractalfract5040206

**Chicago/Turabian Style**

Mustafa, Saima, Hajira, Hassan Khan, Rasool Shah, and Saadia Masood.
2021. "A Novel Analytical Approach for the Solution of Fractional-Order Diffusion-Wave Equations" *Fractal and Fractional* 5, no. 4: 206.
https://doi.org/10.3390/fractalfract5040206