# Detailed Error Analysis for a Fractional Adams Method on Caputo–Hadamard Fractional Differential Equations

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Auxiliary Results

**Theorem**

**1.**

**Theorem**

**2.**

**Corollary**

**1.**

**Theorem**

**3.**

**Proof.**

**Case 1**: $0<\alpha <1$,

**Case 2**: $\alpha >1$,

- (i)
- If $2-\alpha -p>0$, then ${x}_{p}>0$. Therefore,$$\begin{array}{cc}\hfill \sum _{j=1}^{k-1}{j}^{p-1}{(k-j)}^{\alpha -1}\le & {\int}_{0}^{{x}_{p}}{x}^{p-1}{(k+1-x)}^{\alpha -1}dx+{\int}_{{x}_{k}}^{p}{x}^{p-1}{(k+1-x)}^{\alpha -1}dx\hfill \\ \hfill \le & {(k+1-0)}^{\alpha -1}\frac{{x}_{p}^{p}}{p}+{x}_{p}^{p-1}\frac{{(k+1-{x}_{p})}^{\alpha}}{\alpha}\hfill \\ \hfill =& {(k+1)}^{\alpha -1}{C}_{1}{(k+1)}^{p}+{C}_{2}{(k+1)}^{p-1}{C}_{3}{(k+1)}^{\alpha}\hfill \\ \hfill \le & C{(k+1)}^{\alpha +p-1}\le C{k}^{\alpha +p-1}.\hfill \end{array}$$
- (ii)
- If $2-\alpha -p<0$, then ${x}_{p}<0$. This means that ${F}^{\prime}\left(x\right)<0$ and shows that $F\left(x\right)$ is a decreasing function. Therefore,$$\begin{array}{cc}\hfill \sum _{j=1}^{k-1}{j}^{p-1}{(k-j)}^{\alpha -1}\le & {\int}_{0}^{k}{x}^{p-1}{(k+1-x)}^{\alpha -1}dx\hfill \\ \hfill \le & {(k+1-0)}^{\alpha -1}{\int}_{0}^{k}{x}^{p-1}dx\le {(k+1)}^{\alpha -1}\frac{{k}^{p}}{p}\le C{k}^{\alpha +p-1},\hfill \end{array}$$

**Theorem**

**4.**

**Proof.**

**Case 1**: $0<\alpha <1$. Let

**Case 2**: $1<\alpha <2$. Let

**Remark**

**1.**

## 3. Error Analysis for the Adams Method

#### 3.1. A General Result

**Lemma**

**1.**

**Proof.**

#### 3.2. Error Estimates with Smoothness Assumptions on the Solution

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Corollary**

**2.**

**Proof of Theorem**

**7.**

#### 3.3. Error Estimates with Smoothness Assumptions on the Given Data

**Theorem**

**8.**

**Proof.**

## 4. Numerical Examples

- Give an initial value ${y}_{0}$.
- Repeat steps 2–3 to find ${y}_{2},{y}_{3},\dots ,{y}_{N}$.

**Example**

**1**

**Example**

**2**

**Example**

**3**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Graph showing the experimental order of convergence (EOC) at T = 2 in Example 1 with $\alpha =0.8$.

**Figure 2.**Graph showing the experimental order of convergence (EOC) at T = 2 in Example 1 with $\alpha =1.75$.

**Table 1.**Table showing the maximum absolute error and EOC for solving (47) using the predictor–corrector method for $0<\alpha <1$.

N | $\mathit{\alpha}=0.4$ | EOC | $\mathit{\alpha}=0.6$ | EOC | $\mathit{\alpha}=0.8$ | EOC |
---|---|---|---|---|---|---|

10 | 6.839 × 10${}^{-2}$ | 2.607 × 10${}^{-2}$ | 1.350 × 10${}^{-2}$ | |||

20 | 2.064 × 10${}^{-2}$ | 1.728 | 7.433 × 10${}^{-3}$ | 1.810 | 3.542 × 10${}^{-3}$ | 1.930 |

40 | 6.576 × 10${}^{-3}$ | 1.650 | 2.209 × 10${}^{-3}$ | 1.751 | 9.534 × 10${}^{-4}$ | 1.894 |

80 | 2.193 × 10${}^{-3}$ | 1.584 | 6.768 × 10${}^{-4}$ | 1.706 | 2.610 × 10${}^{-4}$ | 1.869 |

160 | 7.561 × 10${}^{-4}$ | 1.536 | 2.120 × 10${}^{-4}$ | 1.675 | 7.225 × 10${}^{-5}$ | 1.853 |

320 | 2.671 × 10${}^{-4}$ | 1.502 | 6.744 × 10${}^{-5}$ | 1.653 | 2.015 × 10${}^{-5}$ | 1.842 |

640 | 9.600 × 10${}^{-5}$ | 1.476 | 2.169 × 10${}^{-5}$ | 1.637 | 5.652 × 10${}^{-6}$ | 1.834 |

1280 | 3.497 × 10${}^{-5}$ | 1.457 | 7.026 × 10${}^{-6}$ | 1.626 | 1.591 × 10${}^{-6}$ | 1.829 |

**Table 2.**Table showing the maximum absolute error and EOC for solving (47) using the predictor–corrector method for $\alpha >1$.

N | $\mathit{\alpha}=1.25$ | EOC | $\mathit{\alpha}=1.50$ | EOC | $\mathit{\alpha}=1.75$ | EOC |
---|---|---|---|---|---|---|

10 | 6.166 × 10${}^{-3}$ | 5.475 × 10${}^{-3}$ | 5.369 × 10${}^{-3}$ | |||

20 | 1.463 × 10${}^{-3}$ | 2.076 | 1.319 × 10${}^{-3}$ | 2.053 | 1.319 × 10${}^{-3}$ | 2.026 |

40 | 3.497 × 10${}^{-4}$ | 2.065 | 3.207 × 10${}^{-4}$ | 2.040 | 3.259 × 10${}^{-4}$ | 2.017 |

80 | 8.408 × 10${}^{-5}$ | 2.056 | 7.853 × 10${}^{-5}$ | 2.030 | 8.091 × 10${}^{-5}$ | 2.010 |

160 | 2.033 × 10${}^{-5}$ | 2.049 | 1.934 × 10${}^{-5}$ | 2.022 | 2.014 × 10${}^{-5}$ | 2.006 |

320 | 4.935 × 10${}^{-6}$ | 2.042 | 4.783 × 10${}^{-6}$ | 2.016 | 5.022 × 10${}^{-6}$ | 2.004 |

640 | 1.203 × 10${}^{-6}$ | 2.036 | 1.187 × 10${}^{-6}$ | 2.011 | 1.253 × 10${}^{-6}$ | 2.002 |

1280 | 2.943 × 10${}^{-7}$ | 2.031 | 2.951 × 10${}^{-7}$ | 2.008 | 3.131 × 10${}^{-7}$ | 2.001 |

**Table 3.**Table showing the maximum absolute error and EOC for solving (53) using the predictor–corrector method for $0<\alpha \le 0.5$.

N | $\mathit{\alpha}=0.1$ | EOC | $\mathit{\alpha}=0.3$ | EOC | $\mathit{\alpha}=0.5$ | EOC |
---|---|---|---|---|---|---|

10 | 2.225 × 10${}^{-2}$ | 9.375 × 10${}^{-3}$ | 5.123 × 10${}^{-3}$ | |||

20 | 1.261 × 10${}^{-2}$ | 0.819 | 3.345 × 10${}^{-3}$ | 1.487 | 1.622 × 10${}^{-3}$ | 1.660 |

40 | 5.625 × 10${}^{-3}$ | 1.164 | 1.196 × 10${}^{-3}$ | 1.484 | 5.255 × 10${}^{-4}$ | 1.626 |

80 | 2.387 × 10${}^{-3}$ | 1.237 | 4.343 × 10${}^{-4}$ | 1.461 | 1.739 × 10${}^{-4}$ | 1.595 |

160 | 1.004 × 10${}^{-3}$ | 1.249 | 1.606 × 10${}^{-4}$ | 1.436 | 5.856 × 10${}^{-5}$ | 1.571 |

320 | 4.242 × 10${}^{-4}$ | 1.243 | 6.033 × 10${}^{-5}$ | 1.412 | 1.998 × 10${}^{-5}$ | 1.552 |

640 | 1.805 × 10${}^{-4}$ | 1.233 | 2.299 × 10${}^{-5}$ | 1.392 | 6.884 × 10${}^{-6}$ | 1.537 |

1280 | 7.742 × 10${}^{-5}$ | 1.221 | 8.864 × 10${}^{-6}$ | 1.375 | 2.389 × 10${}^{-6}$ | 1.527 |

**Table 4.**Table showing the maximum absolute error and EOC for solving (53) using the predictor–corrector method for $0.5<\alpha <1$.

N | $\mathit{\alpha}=0.7$ | EOC | $\mathit{\alpha}=0.9$ | EOC |
---|---|---|---|---|

10 | 5.507 × 10${}^{-3}$ | 1.162 × 10${}^{-2}$ | ||

20 | 1.931 × 10${}^{-3}$ | 1.512 | 5.100 × 10${}^{-3}$ | 1.188 |

40 | 7.031 × 10${}^{-4}$ | 1.457 | 2.300 × 10${}^{-3}$ | 1.151 |

80 | 2.635 × 10${}^{-4}$ | 1.416 | 1.051 × 10${}^{-3}$ | 1.129 |

160 | 1.008 × 10${}^{-4}$ | 1.386 | 4.847 × 10${}^{-4}$ | 1.116 |

320 | 3.920 × 10${}^{-5}$ | 1.363 | 2.247 × 10${}^{-4}$ | 1.109 |

640 | 1.541 × 10${}^{-5}$ | 1.347 | 1.045 × 10${}^{-4}$ | 1.105 |

1280 | 6.110 × 10${}^{-6}$ | 1.335 | 4.863 × 10${}^{-5}$ | 1.103 |

**Table 5.**Table showing the maximum absolute error and EOC for solving (53) using the predictor–corrector method for $\alpha >1$.

N | $\mathit{\alpha}=1.25$ | EOC | $\mathit{\alpha}=1.50$ | EOC | $\mathit{\alpha}=1.75$ | EOC |
---|---|---|---|---|---|---|

10 | 6.166 × 10${}^{-3}$ | 5.475 × 10${}^{-3}$ | 5.369 × 10${}^{-3}$ | |||

20 | 1.463 × 10${}^{-3}$ | 2.076 | 1.319 × 10${}^{-3}$ | 2.053 | 1.319 × 10${}^{-3}$ | 2.026 |

40 | 3.497 × 10${}^{-4}$ | 2.065 | 3.207 × 10${}^{-4}$ | 2.040 | 3.259 × 10${}^{-4}$ | 2.017 |

80 | 8.408 × 10${}^{-5}$ | 2.056 | 7.853 × 10${}^{-5}$ | 2.030 | 8.091 × 10${}^{-5}$ | 2.010 |

160 | 2.033 × 10${}^{-5}$ | 2.049 | 1.934 × 10${}^{-5}$ | 2.022 | 2.014 × 10${}^{-5}$ | 2.006 |

320 | 4.935 × 10${}^{-6}$ | 2.042 | 4.783 × 10${}^{-6}$ | 2.016 | 5.022 × 10${}^{-6}$ | 2.004 |

640 | 1.203 × 10${}^{-6}$ | 2.036 | 1.187 × 10${}^{-6}$ | 2.011 | 1.253 × 10${}^{-6}$ | 2.002 |

1280 | 2.943 × 10${}^{-7}$ | 2.031 | 2.951 × 10${}^{-7}$ | 2.008 | 3.131 × 10${}^{-7}$ | 2.001 |

**Table 6.**Table showing the maximum absolute error and EOC for solving Example 3 using the predictor–corrector method for $\alpha <1$.

N | $\mathit{\alpha}=0.3$ | EOC | $\mathit{\alpha}=0.6$ | EOC | $\mathit{\alpha}=0.9$ | EOC |
---|---|---|---|---|---|---|

10 | 1.353 × 10${}^{-3}$ | 6.520 × 10${}^{-4}$ | 3.414 × 10${}^{-4}$ | |||

20 | 4.324 × 10${}^{-4}$ | 1.646 | 1.937 × 10${}^{-4}$ | 1.751 | 8.704 × 10${}^{-5}$ | 1.972 |

40 | 1.466 × 10${}^{-4}$ | 1.560 | 6.019 × 10${}^{-5}$ | 1.686 | 2.270 × 10${}^{-5}$ | 1.939 |

80 | 5.159 × 10${}^{-5}$ | 1.507 | 1.919 × 10${}^{-5}$ | 1.649 | 5.990 × 10${}^{-6}$ | 1.922 |

160 | 1.863 × 10${}^{-5}$ | 1.470 | 6.211 × 10${}^{-6}$ | 1.628 | 1.591 × 10${}^{-6}$ | 1.913 |

320 | 6.863 × 10${}^{-6}$ | 1.441 | 2.027 × 10${}^{-6}$ | 1.615 | 4.239 × 10${}^{-7}$ | 1.908 |

640 | 2.570 × 10${}^{-6}$ | 1.417 | 6.652 × 10${}^{-7}$ | 1.608 | 1.132 × 10${}^{-7}$ | 1.905 |

1280 | 9.752 × 10${}^{-7}$ | 1.398 | 2.189 × 10${}^{-7}$ | 1.604 | 3.026 × 10${}^{-8}$ | 1.904 |

**Table 7.**Table showing the maximum absolute error and EOC for solving Example 3 using the predictor–corrector method for $\alpha >1$.

N | $\mathit{\alpha}=1.25$ | EOC | $\mathit{\alpha}=1.50$ | EOC | $\mathit{\alpha}=1.75$ | EOC |
---|---|---|---|---|---|---|

10 | 2.457 × 10${}^{-4}$ | 2.108 × 10${}^{-4}$ | 1.507 × 10${}^{-4}$ | |||

20 | 5.847 × 10${}^{-5}$ | 2.071 | 5.141 × 10${}^{-5}$ | 2.036 | 3.723 × 10${}^{-5}$ | 2.017 |

40 | 1.400 × 10${}^{-5}$ | 2.062 | 1.261 × 10${}^{-5}$ | 2.027 | 9.236 × 10${}^{-6}$ | 2.011 |

80 | 3.374 × 10${}^{-6}$ | 2.053 | 3.109 × 10${}^{-6}$ | 2.020 | 2.298 × 10${}^{-6}$ | 2.007 |

160 | 8.213 × 10${}^{-7}$ | 2.039 | 7.693 × 10${}^{-7}$ | 2.015 | 5.728 × 10${}^{-7}$ | 2.004 |

320 | 2.016 × 10${}^{-7}$ | 2.027 | 1.909 × 10${}^{-7}$ | 2.011 | 1.430 × 10${}^{-7}$ | 2.003 |

640 | 4.975 × 10${}^{-8}$ | 2.019 | 4.748 × 10${}^{-8}$ | 2.008 | 3.570 × 10${}^{-8}$ | 2.002 |

1280 | 1.233 × 10${}^{-8}$ | 2.013 | 1.183 × 10${}^{-8}$ | 2.005 | 8.919 × 10${}^{-9}$ | 2.001 |

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**MDPI and ACS Style**

Green, C.W.H.; Yan, Y.
Detailed Error Analysis for a Fractional Adams Method on Caputo–Hadamard Fractional Differential Equations. *Foundations* **2022**, *2*, 839-861.
https://doi.org/10.3390/foundations2040057

**AMA Style**

Green CWH, Yan Y.
Detailed Error Analysis for a Fractional Adams Method on Caputo–Hadamard Fractional Differential Equations. *Foundations*. 2022; 2(4):839-861.
https://doi.org/10.3390/foundations2040057

**Chicago/Turabian Style**

Green, Charles Wing Ho, and Yubin Yan.
2022. "Detailed Error Analysis for a Fractional Adams Method on Caputo–Hadamard Fractional Differential Equations" *Foundations* 2, no. 4: 839-861.
https://doi.org/10.3390/foundations2040057