# Numerical Solution of Fractional Diffusion Wave Equation and Fractional Klein–Gordon Equation via Two-Dimensional Genocchi Polynomials with a Ritz–Galerkin Method

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## Abstract

**:**

## 1. Introduction

## 2. Basic Concept for Fractional Calculus

## 3. Genocchi Polynomials and Function Approximation

#### 3.1. Definition and Properties of Genocchi Polynomials

#### 3.2. Function Approximation of Genocchi Polynomials

## 4. Ritz–Galerkin Method with the Two-Dimensional Genocchi Polynomials Basis

#### 4.1. Ritz–Galerkin Method

#### 4.2. Satisfier Function

#### 4.3. Transformation of Nonhomogeneous Initial and Boundary Conditions into Homogeneous Conditions

## 5. Error Bound

**Theorem**

**1.**

**Proof.**

**Lemma**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

## 6. Numerical Results

**Problem**

**1.**

**Problem**

**2.**

**Problem**

**3.**

**Problem**

**4.**

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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M | Ref. [3] | Ref. [21] | Ref. [22] | Our Method |
---|---|---|---|---|

4 | 5.46 $\times {10}^{-5}$ | 3.38 $\times {10}^{-3}$ | 1.09 $\times {10}^{-1}$ | 4.39377 $\times {10}^{-6}$ |

8 | 5.50 $\times {10}^{-6}$ | 9.69 $\times {10}^{-4}$ | 2.76 $\times {10}^{-2}$ | 4.39381 $\times {10}^{-6}$ |

**Table 2.**Comparison of the absolute errors with [4] of Problem 1 for different values of $\alpha $ and $M=3$.

$(\mathit{x},\mathit{t})$ | $\mathit{\alpha}=1.1$ | $\mathit{\alpha}=1.5$ | $\mathit{\alpha}=1.9$ | |||
---|---|---|---|---|---|---|

Our Method | Ref. [4] | Our Method | Ref. [4] | Our Method | Ref. [4] | |

(0.1,0.1) | 1.15559 $\times {10}^{-8}$ | 6.7028 $\times {10}^{-5}$ | 4.97870 $\times {10}^{-8}$ | 6.0407 $\times {10}^{-5}$ | 1.15539 $\times {10}^{-7}$ | 2.0243 $\times {10}^{-5}$ |

(0.2, 0.2) | 5.15080 $\times {10}^{-7}$ | 1.8718 $\times {10}^{-4}$ | 3.22420 $\times {10}^{-7}$ | 1.5683 $\times {10}^{-4}$ | 1.02961 $\times {10}^{-8}$ | 5.9155 $\times {10}^{-5}$ |

(0.3, 0.3) | 2.63293 $\times {10}^{-6}$ | 3.0913 $\times {10}^{-4}$ | 2.27559 $\times {10}^{-6}$ | 2.7985 $\times {10}^{-4}$ | 1.66006 $\times {10}^{-6}$ | 1.0947 $\times {10}^{-4}$ |

(0.4, 0.4) | 4.60055 $\times {10}^{-6}$ | 4.0221 $\times {10}^{-4}$ | 4.25541 $\times {10}^{-6}$ | 3.7035 $\times {10}^{-4}$ | 3.66878 $\times {10}^{-6}$ | 1.6790 $\times {10}^{-4}$ |

(0.5, 0.5) | 5.41170 $\times {10}^{-7}$ | 4.5801 $\times {10}^{-4}$ | 5.14200 $\times {10}^{-7}$ | 3.8089 $\times {10}^{-4}$ | 4.80654 $\times {10}^{-7}$ | 1.6277 $\times {10}^{-4}$ |

(0.6, 0.6) | 1.50610 $\times {10}^{-5}$ | 4.5260 $\times {10}^{-4}$ | 1.44824 $\times {10}^{-5}$ | 3.6309 $\times {10}^{-4}$ | 1.35084 $\times {10}^{-5}$ | 1.9284 $\times {10}^{-4}$ |

(0.7, 0.7) | 3.84166 $\times {10}^{-5}$ | 4.0597 $\times {10}^{-4}$ | 3.71459 $\times {10}^{-5}$ | 3.2603 $\times {10}^{-4}$ | 3.51060 $\times {10}^{-5}$ | 6.2825 $\times {10}^{-5}$ |

(0.8, 0.8) | 5.05410 $\times {10}^{-5}$ | 3.1039 $\times {10}^{-4}$ | 4.88530 $\times {10}^{-5}$ | 6.5594 $\times {10}^{-4}$ | 4.63092 $\times {10}^{-5}$ | 1.0181 $\times {10}^{-5}$ |

(0.9, 0.9) | 2.89942 $\times {10}^{-5}$ | 1.7283 $\times {10}^{-4}$ | 2.76030 $\times {10}^{-5}$ | 7.1269 $\times {10}^{-3}$ | 2.57007 $\times {10}^{-5}$ | 2.0918 $\times {10}^{-5}$ |

**Table 3.**Comparison of the maximum absolute errors (MAEs) of Problem 2 with [3] for different values of $\alpha $.

$\mathit{\alpha}$ | Our Method | Ref. [3] |
---|---|---|

1.2 | 1.11 $\times {10}^{-16}$ | 5.35 $\times {10}^{-6}$ |

1.4 | 1.11 $\times {10}^{-16}$ | 1.01 $\times {10}^{-6}$ |

1.6 | 1.11 $\times {10}^{-16}$ | 8.77 $\times {10}^{-6}$ |

1.8 | 1.11 $\times {10}^{-16}$ | 2.27 $\times {10}^{-6}$ |

**Table 4.**Comparison of the maximum absolute errors (MAEs) with [10] of Problem 3 for $\alpha =1.25$ and for different values of M.

M | $\mathit{\tau}$ | Our Method | Ref. [10] |
---|---|---|---|

3 | $\frac{1}{10}$ | 8.2146 $\times {10}^{-4}$ | 1.3189 $\times {10}^{-3}$ |

4 | $\frac{1}{20}$ | 2.9523 $\times {10}^{-5}$ | 3.3035 $\times {10}^{-4}$ |

5 | $\frac{1}{40}$ | 7.9725 $\times {10}^{-7}$ | 8.2925 $\times {10}^{-5}$ |

6 | $\frac{1}{80}$ | 6.1421 $\times {10}^{-7}$ | 2.0856 $\times {10}^{-5}$ |

7 | $\frac{1}{160}$ | 9.9173 $\times {10}^{-6}$ | 5.2649 $\times {10}^{-6}$ |

**Table 5.**Comparison of the maximum absolute errors (MAEs) with [10] of Problem 3 for $\alpha =1.75$ and for different values of M.

M | $\mathit{\tau}$ | Our Method | Ref. [10] |
---|---|---|---|

3 | $\frac{1}{10}$ | 8.2677 $\times {10}^{-4}$ | 3.7014 $\times {10}^{-3}$ |

4 | $\frac{1}{20}$ | 2.9040 $\times {10}^{-5}$ | 1.5270 $\times {10}^{-3}$ |

5 | $\frac{1}{40}$ | 2.4291 $\times {10}^{-7}$ | 6.2904 $\times {10}^{-4}$ |

6 | $\frac{1}{80}$ | 5.6104 $\times {10}^{-7}$ | 2.5916 $\times {10}^{-4}$ |

7 | $\frac{1}{160}$ | 7.7767 $\times {10}^{-6}$ | 1.0715 $\times {10}^{-4}$ |

**Table 6.**Comparison of the maximum absolute errors (MAEs) of Problem 4 with [2] for $\alpha =1.3$ and different values of M.

M | Our Method | Ref. [2] |
---|---|---|

4 | 1.8485 $\times {10}^{-3}$ | 6.9789 $\times {10}^{-3}$ |

6 | 3.0582 $\times {10}^{-4}$ | 2.7048 $\times {10}^{-3}$ |

8 | 8.3497 $\times {10}^{-5}$ | 1.1765 $\times {10}^{-3}$ |

10 | 3.4893 $\times {10}^{-4}$ | 5.7402 $\times {10}^{-4}$ |

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**MDPI and ACS Style**

Kanwal, A.; Phang, C.; Iqbal, U.
Numerical Solution of Fractional Diffusion Wave Equation and Fractional Klein–Gordon Equation via Two-Dimensional Genocchi Polynomials with a Ritz–Galerkin Method. *Computation* **2018**, *6*, 40.
https://doi.org/10.3390/computation6030040

**AMA Style**

Kanwal A, Phang C, Iqbal U.
Numerical Solution of Fractional Diffusion Wave Equation and Fractional Klein–Gordon Equation via Two-Dimensional Genocchi Polynomials with a Ritz–Galerkin Method. *Computation*. 2018; 6(3):40.
https://doi.org/10.3390/computation6030040

**Chicago/Turabian Style**

Kanwal, Afshan, Chang Phang, and Umer Iqbal.
2018. "Numerical Solution of Fractional Diffusion Wave Equation and Fractional Klein–Gordon Equation via Two-Dimensional Genocchi Polynomials with a Ritz–Galerkin Method" *Computation* 6, no. 3: 40.
https://doi.org/10.3390/computation6030040