# Solution of Multi-Term Time-Fractional PDE Models Arising in Mathematical Biology and Physics by Local Meshless Method

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## Abstract

**:**

## 1. Introduction

## 2. Proposed Methodology

#### 2.1. Time Discretization

#### 2.2. A $\theta $-Weighted Technique

## 3. Numerical Results and Discussions

#### 3.1. Test Problem

#### 3.2. Test Problem

#### 3.3. Test Problem

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Simulation results using $N=30$ (

**left**), $t=1$ (

**right**) and $\tau =0.001$, $\beta =0.3$, $\gamma =0.2$ for Test Problem in Section 3.1.

**Figure 2.**Simulation results using $N=30$, $\tau =0.001$, $\beta =0.3$, $\gamma =0.2$ and $\alpha =0.75$ (

**left**), $\alpha =0.95$ (

**right**) for Test Problem in Section 3.1.

**Figure 3.**Simulation results of the LMM and the MSRPIM [35] using $N=25$, $\tau =0.001$, $t=1$, $\beta =0.3$ and $\gamma =0.2$ for Test Problem in Section 3.1.

**Figure 4.**Simulation results of the LMM and the MSRPIM [35] using $[0,1]$, ${N}^{2}=20$, $\tau =0.001$, $t=0.1$, $\beta =0.3$ and $\gamma =0.2$ for Test Problem in Section 3.2.

**Figure 5.**Simulation results of the LMM and the MSRPIM [35] using $[0,1]$, ${N}^{2}=20$, $t=0.1$, $\alpha =0.7$, $\beta =0.3$ and $\gamma =0.2$ for Test Problem in Section 3.2.

**Figure 6.**Simulation results using ${N}^{2}=20$, $\tau =0.001$, $t=0.1$, $\alpha =0.75$, $\beta =0.3$ and $\gamma =0.2$ for Test Problem in Section 3.2.

**Figure 7.**Computational domain (

**left**) and numerical results in term of different errors norms (

**right**) using ${N}^{2}=11$, $\tau =0.001$, $t=0.5$, $\beta =0.3$ and $\gamma =0.2$ for Test Problem in Section 3.2.

**Figure 8.**Computational domain (

**left**) and ${L}_{absolute}$ (

**right**) using $N=112$, $\tau =0.001$, $t=1$, $\alpha =0.75$, $\beta =0.3$ and $\gamma =0.2$ for Test Problem in Section 3.2.

**Figure 9.**Simulation results using $\tau =0.01$, $t=1$, $\beta =0.3$ and $\gamma =0.2$ (

**left**) and condition number (

**right**) for Test Problem in Section 3.3.

**Figure 10.**Computational domain (

**left**) and absolute error (

**right**) using ${N}^{3}=11$, $\tau =0.001$, $\beta =0.3$ and $\gamma =0.2$ for Test Problem in Section 3.3.

**Figure 11.**Computational domain (

**left**) and absolute error (

**right**) using ${N}^{3}=11$, $\tau =0.001$, $\beta =0.3$ and $\gamma =0.2$ for Test Problem in Section 3.3.

**Table 1.**Numerical results using $\alpha =0.75$, $\beta =0.3$, $\gamma =0.2$ and $t=0.5$ for Test Problem Section 3.1.

$\mathit{N}=10$ | $\mathit{N}=20$ | $\mathit{N}=30$ | ||||
---|---|---|---|---|---|---|

$\mathbf{\tau}$ | $\mathbf{Max}\left(\mathbf{\epsilon}\right)$ | ${\mathbf{L}}_{\mathbf{2}}$ | $\mathbf{Max}\left(\mathbf{\epsilon}\right)$ | ${\mathbf{L}}_{\mathbf{2}}$ | $\mathbf{Max}\left(\mathbf{\epsilon}\right)$ | ${\mathbf{L}}_{\mathbf{2}}$ |

1.0000 × 10${}^{-3}$ | 1.9384 × 10${}^{-4}$ | 1.2747 × 10${}^{-4}$ | 1.6001 × 10${}^{-4}$ | 7.7998 × 10${}^{-5}$ | 1.1545 × 10${}^{-4}$ | 6.2711 × 10${}^{-5}$ |

5.0000 × 10${}^{-4}$ | 2.6002 × 10${}^{-4}$ | 1.6562 × 10${}^{-4}$ | 8.7511 × 10${}^{-5}$ | 4.0421 × 10${}^{-5}$ | 6.6581 × 10${}^{-5}$ | 3.0006 × 10${}^{-5}$ |

2.5000 × 10${}^{-4}$ | 3.1467 × 10${}^{-4}$ | 2.1193 × 10${}^{-4}$ | 6.2482 × 10${}^{-5}$ | 4.1427 × 10${}^{-5}$ | 3.7293 × 10${}^{-5}$ | 1.7774 × 10${}^{-5}$ |

Condition No. | 4.6506 × 10${}^{4}$ | 7.2601 × 10${}^{5}$ | 3.6585 × 10${}^{6}$ |

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**MDPI and ACS Style**

Ahmad, I.; Ahmad, H.; Thounthong, P.; Chu, Y.-M.; Cesarano, C.
Solution of Multi-Term Time-Fractional PDE Models Arising in Mathematical Biology and Physics by Local Meshless Method. *Symmetry* **2020**, *12*, 1195.
https://doi.org/10.3390/sym12071195

**AMA Style**

Ahmad I, Ahmad H, Thounthong P, Chu Y-M, Cesarano C.
Solution of Multi-Term Time-Fractional PDE Models Arising in Mathematical Biology and Physics by Local Meshless Method. *Symmetry*. 2020; 12(7):1195.
https://doi.org/10.3390/sym12071195

**Chicago/Turabian Style**

Ahmad, Imtiaz, Hijaz Ahmad, Phatiphat Thounthong, Yu-Ming Chu, and Clemente Cesarano.
2020. "Solution of Multi-Term Time-Fractional PDE Models Arising in Mathematical Biology and Physics by Local Meshless Method" *Symmetry* 12, no. 7: 1195.
https://doi.org/10.3390/sym12071195