# Finite Difference Approximation Method for a Space Fractional Convection–Diffusion Equation with Variable Coefficients

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminary Remarks

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3**

**.**Let u be given on ℜ. The standard Grünwald–Letnikov estimate for $1<\alpha \le 2$ with positive order α is defined by the formula,

**Lemma**

**1**

**.**Assume that $1<\alpha \le 2$, then Grünwald–Letnikov coefficients ${\omega}_{k}^{\left(\alpha \right)}$ satisfy:

**Lemma**

**2**

**.**Let $u\in {C}^{2n}(\Re )$ that has a finite degree of smoothness with $\left({D}_{+}^{\alpha}u\right)\left(x\right)$ which is approximated by ${h}^{-\alpha}\left({\mathsf{\Delta}}_{h}^{\alpha}u\right)\left(x\right)$ possesses an asymptotic expansion in integer powers of the step-length h, then an expansion in even powers of h for the Shifted operator can be written in the form:

**Lemma**

**3**

**Theorem**

**1.**

**Proof**

**of Theorem 1.**

**Remark**

**1.**

**Remark**

**2.**

## 3. Problem Formulation of the Scheme

#### Crank–Nicolson Scheme for Time and Shifted Grünwald Difference Scheme for Space Discretization

**Theorem**

**2.**

**Proof**

**of Theorem 2.**

## 4. Theoretical Analysis of Finite Difference Scheme

#### 4.1. Boundedness of the Fractional Scheme

**Theorem**

**3.**

**Proof**

**of Theorem 3.**

#### 4.2. Stability Analysis

**Theorem**

**4.**

**Proof**

**of Theorem 4.**

#### 4.3. Convergence Analysis

**Theorem**

**5.**

**Proof**

**of Theorem 5.**

**Remark**

**3.**

## 5. Numerical Tests

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The Maximum error by C-N scheme at $(T=10,\phantom{\rule{3.33333pt}{0ex}}Max-Error=6.5276{e}^{-07})$, $(T\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}20,\phantom{\rule{3.33333pt}{0ex}}Max-Error=1.7244{e}^{-08}),\alpha =1.5$ left to right, respectively, for Example 1.

**Figure 2.**The exact (

**left**) and numerical (

**right**) solution by C-N scheme at $T=1,\alpha =1.5,\tau =0.01=h$ for Example 1.

**Figure 3.**Numerical and exact solution by C-N scheme at $\alpha =1.5,\tau =h=0.01,$ with $(T=10,T=30,T=40)$ left to right-down respectively, for Example 1.

**Figure 4.**The exact (

**left**) and numerical (

**right**) solution by C-N scheme for the FCDE at $(h=\tau =0.005,\alpha =1.5,(t=5,max-error=4.0657{e}^{-05})$ for Example 2.

**Figure 5.**The exact (

**left**) and numerical (

**right**) solution by C-N scheme for the FCDE at $(h=\tau =0.01,(t=2,max-Error=4.2158{e}^{-04}),\alpha =1.75)$ for Example 2.

**Figure 6.**The exact (

**left**) and numerical (

**right**) solution by C-N scheme at $(h=\tau =0.0025,(t=0.1,max-Error=1.4{e}^{-03},\alpha =1.1)$ for Example 4.

$\mathit{\alpha}=1.25$ | $\mathit{\alpha}=1.5$ | $\mathit{\alpha}=1.8$ | |||||
---|---|---|---|---|---|---|---|

$\mathsf{\Delta}\mathit{t}$ | $\mathsf{\Delta}\mathit{x}$ | Max-Error | Order | Max-Error | Order | Max-Error | Order |

1/50 | 1/50 | 4.9807e−04 | – | 4.0046e−04 | – | 1.4048e−04 | – |

1/100 | 1/100 | 1.0660e−04 | 2.2241 | 8.8946e−05 | 2.1707 | 3.6848e−05 | 1.9307 |

1/200 | 1/200 | 2.4413e−05 | 2.1265 | 2.0643e−05 | 2.1073 | 9.4393e−06 | 1.9648 |

1/400 | 1/400 | 5.8239e−06 | 2.0676 | 4.9592e−06 | 2.0575 | 2.3887e−06 | 1.9825 |

1/800 | 1/800 | 1.4211e−06 | 2.0350 | 1.2146e−06 | 2.0296 | 6.0078e−07 | 1.9913 |

$\mathit{T}=1$ | $\mathit{T}=5$ | ||||
---|---|---|---|---|---|

$\mathsf{\Delta}\mathit{t}$ | $\mathsf{\Delta}\mathit{x}$ | Max-Error | Order | Max-Error | Order |

1/50 | 1/50 | 1.4048e−04 | – | 2.5297e−05 | – |

1/100 | 1/100 | 3.6848e−05 | 1.9307 | 7.4748e−06 | 1.7589 |

1/200 | 1/200 | 9.4393e−06 | 1.9648 | 2.0122e−06 | 1.8933 |

1/400 | 1/400 | 2.3887e−06 | 1.9825 | 4.9017e−07 | 2.0374 |

1/800 | 1/800 | 6.0078e−07 | 1.9913 | 1.0620e−07 | 2.2065 |

**Table 3.**The maximum error and convergence order by C-N for SFCDE in example 2 at $T=1,\alpha =1.55$.

$\mathsf{\Delta}\mathit{t}$ | $\mathsf{\Delta}\mathit{x}$ | Max-Error | Order |
---|---|---|---|

1/50 | 1/50 | 2.6e−03 | – |

1/100 | 1/100 | 7.695e−04 | 1.7563 |

1/150 | 1/150 | 2.144e−04 | 1.8436 |

1/200 | 1/200 | 5.688e−05 | 1.9143 |

**Table 4.**The Maximum error and convergence order produced by C-N scheme for example 3 at $T=1,{N}_{t}=100$.

$\mathit{\alpha}=1.35$ | $\mathit{\alpha}=1.5$ | $\mathit{\alpha}=1.75$ | ||||
---|---|---|---|---|---|---|

$\mathsf{\Delta}\mathit{x}$ | Max-Error | Order | Max-Error | Order | Max-Error | Order |

1/50 | 4.5e−03 | – | 2.8e−03 | – | 1.7e−03 | – |

1/100 | 2.7e−03 | 0.7370 | 1.6e−03 | 0.8074 | 8.9641–04 | 0.97224 |

1/200 | 1.6e−03 | 0.7549 | 8.6405e−04 | 0.8889 | 4.6491e−04 | 0.8981 |

1/400 | 9.5896e−04 | 0.7385 | 4.7955e−04 | 0.8494 | 2.4086e−04 | 0.9488 |

1/800 | 5.7034e−04 | 0.7496 | 2.6609e−04 | 0.8498 | 1.2473e−04 | 0.9494 |

$\mathit{\alpha}=1.25$ | $\mathit{\alpha}=1.55$ | ||||
---|---|---|---|---|---|

$\mathsf{\Delta}\mathit{t}$ | $\mathsf{\Delta}\mathit{x}$ | Max-Error | Error-Rate | Max-Error | Error-Rate |

1/50 | 1/50 | 1.91e−02 | – | 1.52e−02 | – |

1/100 | 1/100 | 9.9e−03 | 1.93 | 7.9e−03 | 1.9 |

1/200 | 1/200 | 5.2e−03 | 1.90 | 4.3e−03 | 1.84 |

1/400 | 1/400 | 2.8e−03 | 1.86 | 2.4e−03 | 1.79 |

1/800 | 1/800 | 1.6e−03 | 1.75 | 1.4e−03 | 1.7 |

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**MDPI and ACS Style**

Anley, E.F.; Zheng, Z.
Finite Difference Approximation Method for a Space Fractional Convection–Diffusion Equation with Variable Coefficients. *Symmetry* **2020**, *12*, 485.
https://doi.org/10.3390/sym12030485

**AMA Style**

Anley EF, Zheng Z.
Finite Difference Approximation Method for a Space Fractional Convection–Diffusion Equation with Variable Coefficients. *Symmetry*. 2020; 12(3):485.
https://doi.org/10.3390/sym12030485

**Chicago/Turabian Style**

Anley, Eyaya Fekadie, and Zhoushun Zheng.
2020. "Finite Difference Approximation Method for a Space Fractional Convection–Diffusion Equation with Variable Coefficients" *Symmetry* 12, no. 3: 485.
https://doi.org/10.3390/sym12030485