# Numerical Inverse Laplace Transform for Solving a Class of Fractional Differential Equations

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

**Property**

**1.**

## 3. Bernstein Polynomials and Function Approximation

**Definition**

**8.**

- ${b}_{i,n}\left(t\right)=0$, if $i<0$ or $i>n$.
- ${b}_{i,n}\left(t\right)\ge 0$ for $t\in [0,1]$.
- The Bernstein polynomials form a partition of unity i.e., ${\sum}_{i=0}^{n}{b}_{i,n}\left(t\right)=1$.

## 4. The Method for Numerical Inverse Laplace Transform

#### 4.1. Application to a Class of Fractional Differential Equations

#### 4.1.1. Application to Linear Fractional Differential Equations

#### 4.1.2. Application to Linear Partial Fractional Differential Equations

#### 4.1.3. Application to Nonlinear Fractional Differential Equations

## 5. Error Estimation and Convergence Analysis

#### 5.1. Error Estimation via RK45 Method

**Theorem**

**1.**

**Proof.**

#### 5.2. Convergence Analysis

## 6. Illustrative Examples

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

**Example**

**6.**

**Example**

**7.**

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 5.**The relative errors for $k=5$ (

**left**) and $k=6$ (

**right**) for ${f}_{1}\left(t\right)$ in Example 5.

**Figure 6.**The relative errors for $k=5$ (

**left**) and $k=6$ (

**right**) for ${f}_{2}\left(t\right)$ in Example 5.

**Figure 8.**The ${L}_{\infty}-error$ for Example 1 (

**left**) and Example 2 (

**right**) for different values of k.

**Figure 9.**The ${L}_{\infty}-error$ for Example 3 (

**left**) and Example 4 (

**right**) for different values of k.

**Figure 10.**The ${L}_{\infty}-error$ for ${f}_{1}\left(t\right)$ (

**left**) and ${f}_{2}\left(t\right)$ (

**right**) at different values of k for Example 5.

t | Present Method at k = 5 | Present Method at k = 6 | FFDM [15] |
---|---|---|---|

0.1 | $8.17\times {10}^{-4}$ | $2.02\times {10}^{-7}$ | $1.16\times {10}^{-4}$ |

0.2 | $5.36\times {10}^{-4}$ | $3.09\times {10}^{-6}$ | $1.56\times {10}^{-4}$ |

0.3 | $3.64\times {10}^{-4}$ | $4.59\times {10}^{-6}$ | $1.81\times {10}^{-4}$ |

0.4 | $2.65\times {10}^{-4}$ | $4.87\times {10}^{-6}$ | $2.00\times {10}^{-4}$ |

0.5 | $2.13\times {10}^{-4}$ | $6.01\times {10}^{-6}$ | $2.15\times {10}^{-4}$ |

0.6 | $1.88\times {10}^{-4}$ | $9.61\times {10}^{-6}$ | $2.27\times {10}^{-4}$ |

0.7 | $1.77\times {10}^{-4}$ | $1.56\times {10}^{-5}$ | $2.37\times {10}^{-4}$ |

0.8 | $1.72\times {10}^{-4}$ | $2.23\times {10}^{-5}$ | $2.46\times {10}^{-4}$ |

0.9 | $1.68\times {10}^{-4}$ | $2.73\times {10}^{-5}$ | $2.54\times {10}^{-4}$ |

1 | $1.64\times {10}^{-4}$ | $2.97\times {10}^{-5}$ | $2.61\times {10}^{-4}$ |

t | Present Method at k = 6 | Cubic Spline Method [54] |
---|---|---|

0.125 | 4.49$\times {10}^{-5}$ | 1.24$\times {10}^{-3}$ |

0.250 | 1.18$\times {10}^{-6}$ | 5.12$\times {10}^{-3}$ |

0.375 | 1.80$\times {10}^{-5}$ | 1.39$\times {10}^{-2}$ |

0.500 | 1.53$\times {10}^{-5}$ | 2.61$\times {10}^{-2}$ |

0.625 | 9.29$\times {10}^{-6}$ | 4.04$\times {10}^{-2}$ |

0.75 | 7.63$\times {10}^{-6}$ | 5.58$\times {10}^{-2}$ |

0.875 | 2.47$\times {10}^{-5}$ | 7.15$\times {10}^{-2}$ |

1 | 7.43$\times {10}^{-5}$ | 8.72$\times {10}^{-2}$ |

t | Present Method at k = 5 | Present Method at k = 6 | VIM [55] |
---|---|---|---|

0.1 | 1.59$\times {10}^{-3}$ | 5.99$\times {10}^{-6}$ | 1.66$\times {10}^{-4}$ |

0.2 | 9.84$\times {10}^{-4}$ | 1.40$\times {10}^{-5}$ | 1.32$\times {10}^{-3}$ |

0.3 | 6.08$\times {10}^{-4}$ | 1.32$\times {10}^{-5}$ | 4.39$\times {10}^{-3}$ |

0.4 | 3.80$\times {10}^{-4}$ | 1.02$\times {10}^{-5}$ | 1.02$\times {10}^{-2}$ |

0.5 | 2.34$\times {10}^{-4}$ | 1.53$\times {10}^{-5}$ | 1.93$\times {10}^{-2}$ |

0.6 | 1.22$\times {10}^{-4}$ | 3.16$\times {10}^{-5}$ | 3.21$\times {10}^{-2}$ |

0.7 | 2.24$\times {10}^{-5}$ | 5.15$\times {10}^{-5}$ | 4.86$\times {10}^{-2}$ |

0.8 | 7.21$\times {10}^{-5}$ | 6.18$\times {10}^{-5}$ | 6.81$\times {10}^{-2}$ |

0.9 | 1.73$\times {10}^{-4}$ | 5.60$\times {10}^{-5}$ | 8.95$\times {10}^{-2}$ |

1 | 3.29$\times {10}^{-4}$ | 5.56$\times {10}^{-5}$ | 1.11$\times {10}^{-1}$ |

t | Present Method at k = 5 | Present Method at k = 6 | VIM [55] |
---|---|---|---|

0.1 | 1.33$\times {10}^{-4}$ | 4.81$\times {10}^{-5}$ | 1.79$\times {10}^{-4}$ |

0.2 | 2.89$\times {10}^{-4}$ | 1.31$\times {10}^{-5}$ | 1.54$\times {10}^{-4}$ |

0.3 | 4.07$\times {10}^{-4}$ | 1.74$\times {10}^{-5}$ | 5.57$\times {10}^{-3}$ |

0.4 | 5.09$\times {10}^{-4}$ | 3.42$\times {10}^{-7}$ | 1.41$\times {10}^{-2}$ |

0.5 | 6.02$\times {10}^{-4}$ | 7.26$\times {10}^{-5}$ | 2.94$\times {10}^{-2}$ |

0.6 | 6.85$\times {10}^{-4}$ | 1.45$\times {10}^{-4}$ | 5.40$\times {10}^{-2}$ |

0.7 | 7.64$\times {10}^{-4}$ | 1.25$\times {10}^{-4}$ | 9.11$\times {10}^{-2}$ |

0.8 | 8.44$\times {10}^{-4}$ | 2.68$\times {10}^{-5}$ | 1.44$\times {10}^{-1}$ |

0.9 | 9.28$\times {10}^{-4}$ | 1.53$\times {10}^{-4}$ | 2.17$\times {10}^{-1}$ |

1 | 9.96$\times {10}^{-4}$ | 2.98$\times {10}^{-4}$ | 3.13$\times {10}^{-1}$ |

x | t = 0 | t = 0.06 | t = 0.13 | t = 0.29 | t = 0.50 |
---|---|---|---|---|---|

0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

0.11 | 0.11 | 0.110729 | 0.111595 | 0.113677 | 0.116726 |

0.31 | 0.31 | 0.315791 | 0.322671 | 0.339207 | 0.363420 |

0.88 | 0.88 | 0.926665 | 0.982110 | 1.115365 | 1.310479 |

1.00 | 1.00 | 1.060260 | 1.131857 | 1.303932 | 1.555887 |

x | t = 0 | t = 0.06 | t = 0.13 | t = 0.29 | t = 0.50 |
---|---|---|---|---|---|

0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

0.11 | 0.11 | 0.110729 | 0.111595 | 0.113677 | 0.116726 |

0.31 | 0.31 | 0.315791 | 0.322670 | 0.339207 | 0.363419 |

0.88 | 0.88 | 0.926669 | 0.982101 | 1.115360 | 1.310469 |

1.00 | 1.00 | 1.060265 | 1.131845 | 1.303926 | 1.555874 |

x | t = 0 | t = 0.06 | t = 0.13 | t = 0.29 | t = 0.50 |
---|---|---|---|---|---|

0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

0.11 | 0.11 | 0.110729 | 0.111595 | 0.113679 | 0.116710 |

0.31 | 0.31 | 0.315794 | 0.322675 | 0.339223 | 0.363298 |

0.88 | 0.88 | 0.926696 | 0.982140 | 1.115490 | 1.309495 |

1.00 | 1.00 | 1.060300 | 1.131896 | 1.304094 | 1.554617 |

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**MDPI and ACS Style**

Rani, D.; Mishra, V.; Cattani, C.
Numerical Inverse Laplace Transform for Solving a Class of Fractional Differential Equations. *Symmetry* **2019**, *11*, 530.
https://doi.org/10.3390/sym11040530

**AMA Style**

Rani D, Mishra V, Cattani C.
Numerical Inverse Laplace Transform for Solving a Class of Fractional Differential Equations. *Symmetry*. 2019; 11(4):530.
https://doi.org/10.3390/sym11040530

**Chicago/Turabian Style**

Rani, Dimple, Vinod Mishra, and Carlo Cattani.
2019. "Numerical Inverse Laplace Transform for Solving a Class of Fractional Differential Equations" *Symmetry* 11, no. 4: 530.
https://doi.org/10.3390/sym11040530