# A Novel Numerical Method for Solving Nonlinear Fractional-Order Differential Equations and Its Applications

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## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

**Definition**

**2.**

- The development of a fast algorithm that can be applied to numerical methods for solving ABC FDEs.
- The development of a fast PCM and its application to ABC fractional-order PDEs (FPDEs) and fractional dynamical systems.
- Error estimates for both conventional and fast PCMs.

- The conventional PCM for solving ABC FDEs is suggested.
- The fast algorithm for the computation of the memory term is proposed, and the fast PCM only requires a computational cost of $\mathcal{O}\left(n\right)$.
- Truncation and global error analyses for both conventional and fast PCMs are provided, achieving a uniform accuracy of order $\mathcal{O}\left({h}^{2}\right)$ regardless of the fractional order $\nu $.
- We apply the proposed fast PCM to sub-diffusion FPDEs to demonstrate the efficiency of the proposed method.
- The proposed fast PCM is implemented to handle the fractional Rössler dynamical system, and the performance of the fast algorithm is verified.

## 2. Fast Predictor-Corrector Scheme

#### 2.1. Description of Predictor-Corrector Scheme

#### 2.2. Sum-of-Exponentials Approximation of the Power Function

**Lemma**

**1**

**.**For any $\beta >0$,

**Lemma**

**2**

**.**For $t\ge \delta >0$, there exists $p>0$ such that

**Theorem**

**1**

**.**thmmsoeapp Let $\u03f5>0$ be the desired error, and choose $0<\delta \le t\le T$. Let $n=\mathcal{O}(log\frac{1}{\u03f5})$; $\widehat{M}=\mathcal{O}(logT+loglog\frac{1}{\u03f5}+log\frac{1}{\delta})+1$; ${\eta}_{i}^{J}$ and ${\zeta}_{i}^{J}$ be n-point Gauss–Jacobi quadratures on the interval $[0,{2}^{-\widehat{M}}]$; and ${\eta}_{i,j}^{L}$ and ${\zeta}_{i,j}^{L}$ be n-point Gauss–Legendre quadrature points and weights on the intervals $[{2}^{j},{2}^{j+1}]$ for $j=-\widehat{M},\cdots ,-1$. Then, for $t\in [\delta ,T]$ and $\beta \in (0,2)$,

**Remark**

**1.**

#### 2.3. Description of Fast Predictor-Corrector Scheme

**Remark**

**2.**

## 3. Error Analysis

#### 3.1. Truncation and Global Error Analyses for the Conventional PCM

**Lemma**

**3**

**.**Let ${\left\{{a}_{n}\right\}}_{n=0}^{N}$, ${\left\{{b}_{n}\right\}}_{n=0}^{N}$ be non-negative sequences with monotonically increasing ${b}_{n}$, satisfying

**Lemma**

**4.**

**Lemma**

**5**

**.**Suppose that $f\in {C}^{2}[0,T]$. Let the truncation error of the predictor at ${t}_{n+1}$ be

**Lemma**

**6**

**.**Suppose that $f\in {C}^{2}[0,T]$ satisfies the Lipschitz continuity condition in the second argument,

**Lemma**

**7**

**.**Under the same assumptions as those in Lemma 5, the truncation error of the corrector at time ${t}_{n+1}$ is given by

**Lemma**

**8**

**.**Under the same assumptions as those in Lemma 6, the global error ${e}_{n+1}=|{u}_{n+1}-{\tilde{u}}_{n+1}|$ is given by

**Theorem**

**2**

**.**Let the truncation error of predictor ${\tilde{r}}_{n+1}^{P}$ be defined by

**Proof.**

**Theorem**

**3**

**.**Under the same assumptions as those in Lemma 6, the global error for the predictor, ${\tilde{e}}_{n+1}^{P}=|{u}_{n+1}-{\tilde{u}}_{n+1}^{P}|$, can be estimated by

**Proof.**

**Theorem**

**4**

**.**Under the same assumptions as those in Theorem 3, let the truncation error at time $t={t}_{n+1}$ be

**Proof.**

**Theorem**

**5**

**.**Under the same assumptions as those in Theorem 3, the global error ${\tilde{e}}_{n+1}=|{u}_{n+1}-{\tilde{u}}_{n+1}|$ is

**Proof.**

#### 3.2. Global Error Analysis for the Fast PCM

**Lemma**

**9.**

**Theorem**

**6**

**.**Under the same assumptions as those in Theorem 3, the global error ${\widehat{e}}_{n+1}^{P}=|{u}_{n+1}-{\widehat{u}}_{n+1}^{P}|$ has the following inequality:

**Proof.**

**Theorem**

**7**

**.**Under the same assumptions as those in Theorem 3, the global error ${\widehat{e}}_{n+1}=|{u}_{n+1}-{\widehat{u}}_{n+1}|$ is

**Proof.**

## 4. Numerical Results

- Maximum norm error:$${E}_{max}=\underset{0\le j\le N}{max}|{u}_{j}-{u}_{j}^{h}|.$$
- Discrete ${L}^{2}$ norm error:$${E}_{{\ell}^{2}}={\left(h\sum _{j=0}^{N}{|{u}_{j}-{u}_{j}^{h}|}^{2}\right)}^{\frac{1}{2}},$$

#### 4.1. Nonlinear ABC Fractional-Order Initial Value Problems

**Example**

**1.**

**Example**

**2.**

- The numerical results obtained with the ABC-FPCM show little difference from those obtained with the ABC-PCM.
- The computational costs, obtained by measuring the CPU time (in seconds) executed by the conventional PCM and the fast PCM versus the total number of steps N on the log-log scale in Example 2, are depicted in Figure 1. The figure shows that the CPU consumption rate of the fast PCM is $\mathcal{O}\left(N\right)$, whereas that of the conventional PCM is $\mathcal{O}\left({N}^{2}\right)$.

#### 4.2. Application to Fractional Order PDEs

**Example**

**3.**

**Example**

**4.**

- Theoretically, the rate of convergence for the second-order central difference quotient is $\mathcal{O}\left({\tau}^{2}\right)$, and the convergence rates for the proposed methods are shown to be 2. Thus, the global order of convergence for both (25) and (26) is expected to be 2 when either $\tau $ or h is fixed. In the tables, one can see that the rates of convergence computed by the ABC-FPCM and ABC-PCM are approximately 2 in both cases where $\tau $ is fixed and h is fixed. This verifies that the global estimates of our proposed methods are valid in solving sub-diffusion FPDEs.
- The gap between the CPU time executed by the ABC-PCM and that executed by the ABC-FPCM is evident in the tables. Particularly, the difference between them drastically increases as $h=1/5000$ is fixed. This verifies that the proposed fast PCM is more efficient compared to the conventional PCM.

**Example**

**5.**

- Table 7 shows the CPU times for Scheme 2 in [37], the ABC-PCM, and the ABC-FPCM for Example 5. To measure the rate of the CPU time, the time T increases twice from T = 125 to 1000, and we can observe that the rate of the CPU time for Scheme 2 and the ABC-FPCM is $\mathcal{O}\left(N\right)$, whereas for the ABC-PCM, it is $\mathcal{O}\left({N}^{2}\right)$.
- Furthermore, the ABC-FPCM is much more efficient in terms of memory management compared to the existing method because it requires storing all previous values (${U}_{n}$, $\mathbf{L}{U}_{n}$, and ${G}_{n}$) to calculate ${U}_{n+1}$ by using the ABC-PCM efficiently. On the other hand, the APC-FPCM requires only local values.
- Figure 2 shows the effect of fractional order and types of fractional derivatives for $T=250,500,1000$, $h=1/3$, $L=400$, and $\tau =1$. The first row depicts the evolution of the system for $\nu =1$. We can see that the spiral pattern is broken when $T=250$, and an irregular pattern appears when $T=500,1000$.
- The second and third rows describe the evolution of the system for the ABC and Liouville–Caputo fractional derivatives of order $\nu =0.98$, respectively. In the fractional-order system, similar to the case of $\nu =1$, the spiral pattern is broken over time.
- In addition, in the fractional systems, a spiral pattern exists at $T=250,1000$, but the spiral pattern disappears at $T=1000$. This phenomenon is similar to the case where $\nu =1$, $T=250$, and has a wider pattern.
- It can be seen that the solution of the problem equipped with the ABC derivative is slower and wider than the solution of the problem equipped with the Liouville–Caputo derivative.

#### 4.3. Application to Fractional Dynamical Systems

**Example**

**6.**

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Additional Information on the Sum-of-Exponentials Approximation

**Definition**

**A1**

**.**When determining the specific situation of the system, we use Gramians. As an example, Gramians are used to determine whether a system is controllable or reachable. The reachability Gramian and the observability Gramian of the system (A1) are defined as follows:

- The reachability Gramian$$P:={\int}_{0}^{{t}_{f}}{e}^{\tau A}B{B}^{T}{e}^{\tau {A}^{T}}d\tau .$$
- The observability Gramian$$Q:={\int}_{0}^{{t}_{f}}{e}^{\tau A}{C}^{T}C{e}^{\tau {A}^{T}}d\tau .$$

**Definition**

**A2**

**.**The square roots of the eigenvalues (singular values) of the product $PQ$ are the so-called Hankel singular values of the system (A1):

- ${\sigma}_{i}$ are basis independent.
- In a lot of instances, not only the eigenvalues of P and Q but also the Hankel singular values decrease very quickly.
- Note that one of properties of the balanced basis is that hard-to-reach states are hard to observe. Therefore, we obtain the reduced model by using the Hankel singular values, except for the small ones.

**Table A1.**The number of exponentials and relative errors for the approximation of ${t}^{-\beta}$ with $T=1$.

$\mathit{\delta}$ | ${10}^{-2}$ | ${10}^{-3}$ | ${10}^{-4}$ | ${10}^{-5}$ | ||||
---|---|---|---|---|---|---|---|---|

$\mathit{\u03f5}$ | ${\mathit{N}}_{exp}$ | ${\mathit{E}}_{\mathrm{rel}}$ | ${\mathit{N}}_{exp}$ | ${\mathit{E}}_{\mathrm{rel}}$ | ${\mathit{N}}_{exp}$ | ${\mathit{E}}_{\mathrm{rel}}$ | ${\mathit{N}}_{exp}$ | ${\mathit{E}}_{\mathrm{rel}}$ |

Theorem 1 | ||||||||

${10}^{-3}$ | 84 | 2.85 $\times \phantom{\rule{4pt}{0ex}}{10}^{-10}$ | 119 | 2.74 $\times \phantom{\rule{4pt}{0ex}}{10}^{-11}$ | 147 | 2.87 $\times \phantom{\rule{4pt}{0ex}}{10}^{-11}$ | 182 | 2.91 $\times \phantom{\rule{4pt}{0ex}}{10}^{-11}$ |

${10}^{-6}$ | 182 | 5.29 $\times \phantom{\rule{4pt}{0ex}}{10}^{-16}$ | 238 | 6.15 $\times \phantom{\rule{4pt}{0ex}}{10}^{-16}$ | 308 | 7.97 $\times \phantom{\rule{4pt}{0ex}}{10}^{-16}$ | 378 | 9.39 $\times \phantom{\rule{4pt}{0ex}}{10}^{-16}$ |

${10}^{-9}$ | 273 | 8.01 $\times \phantom{\rule{4pt}{0ex}}{10}^{-16}$ | 378 | 1.07 $\times \phantom{\rule{4pt}{0ex}}{10}^{-15}$ | 462 | 1.10 $\times \phantom{\rule{4pt}{0ex}}{10}^{-15}$ | 567 | 1.30 $\times \phantom{\rule{4pt}{0ex}}{10}^{-15}$ |

$\beta =0.2$ | ||||||||

${10}^{-3}$ | 15 | 1.01 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 19 | 9.18 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 23 | 1.11 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 27 | 1.14 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ |

${10}^{-6}$ | 26 | 1.60 $\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 32 | 1.50 $\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 39 | 9.14 $\times \phantom{\rule{4pt}{0ex}}{10}^{-8}$ | 46 | 1.21 $\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ |

${10}^{-9}$ | 37 | 1.10 $\times \phantom{\rule{4pt}{0ex}}{10}^{-10}$ | 47 | 8.56 $\times \phantom{\rule{4pt}{0ex}}{10}^{-11}$ | 54 | 7.89 $\times \phantom{\rule{4pt}{0ex}}{10}^{-11}$ | 64 | 1.16 $\times \phantom{\rule{4pt}{0ex}}{10}^{-10}$ |

$\beta =0.5$ | ||||||||

${10}^{-3}$ | 14 | 1.62 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 19 | 1.52 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 22 | 1.72 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 26 | 1.75 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ |

${10}^{-6}$ | 25 | 3.23 $\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 31 | 2.99 $\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 38 | 1.57 $\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 45 | 2.20 $\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ |

${10}^{-9}$ | 36 | 2.12 $\times \phantom{\rule{4pt}{0ex}}{10}^{-10}$ | 46 | 1.64 $\times \phantom{\rule{4pt}{0ex}}{10}^{-10}$ | 53 | 1.44 $\times \phantom{\rule{4pt}{0ex}}{10}^{-10}$ | 62 | 2.10 $\times \phantom{\rule{4pt}{0ex}}{10}^{-10}$ |

$\beta =0.8$ | ||||||||

${10}^{-3}$ | 14 | 3.19 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 18 | 2.96 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 21 | 3.35 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 25 | 3.41 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ |

${10}^{-6}$ | 25 | 6.46 $\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 30 | 5.62 $\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 37 | 3.56 $\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 43 | 4.48 $\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ |

${10}^{-9}$ | 35 | 5.44 $\times \phantom{\rule{4pt}{0ex}}{10}^{-10}$ | 44 | 4.22 $\times \phantom{\rule{4pt}{0ex}}{10}^{-10}$ | 52 | 4.47 $\times \phantom{\rule{4pt}{0ex}}{10}^{-10}$ | 61 | 4.31 $\times \phantom{\rule{4pt}{0ex}}{10}^{-10}$ |

**Table A2.**The number of exponentials and relative errors for the approximation of ${t}^{-\beta}$ with $\delta ={10}^{-2}$.

T | 10 | ${10}^{2}$ | ${10}^{3}$ | ${10}^{4}$ | ||||
---|---|---|---|---|---|---|---|---|

$\mathit{\u03f5}$ | ${\mathit{N}}_{exp}$ | ${\mathit{E}}_{\mathrm{rel}}$ | ${\mathit{N}}_{exp}$ | ${\mathit{E}}_{\mathrm{rel}}$ | ${\mathit{N}}_{exp}$ | ${\mathit{E}}_{\mathrm{rel}}$ | ${\mathit{N}}_{exp}$ | ${\mathit{E}}_{\mathrm{rel}}$ |

Theorem 1 | ||||||||

${10}^{-3}$ | 98 | 2.85 $\times \phantom{\rule{4pt}{0ex}}{10}^{-10}$ | 119 | 2.74 $\times \phantom{\rule{4pt}{0ex}}{10}^{-11}$ | 133 | 2.74 $\times \phantom{\rule{4pt}{0ex}}{10}^{-11}$ | 147 | 2.74 $\times \phantom{\rule{4pt}{0ex}}{10}^{-11}$ |

${10}^{-6}$ | 210 | 6.09 $\times \phantom{\rule{4pt}{0ex}}{10}^{-16}$ | 238 | 6.15 $\times \phantom{\rule{4pt}{0ex}}{10}^{-16}$ | 280 | 6.15 $\times \phantom{\rule{4pt}{0ex}}{10}^{-16}$ | 308 | 6.15 $\times \phantom{\rule{4pt}{0ex}}{10}^{-16}$ |

${10}^{-9}$ | 336 | 8.84 $\times \phantom{\rule{4pt}{0ex}}{10}^{-16}$ | 378 | 1.07 $\times \phantom{\rule{4pt}{0ex}}{10}^{-15}$ | 420 | 1.07 $\times \phantom{\rule{4pt}{0ex}}{10}^{-15}$ | 462 | 1.07 $\times \phantom{\rule{4pt}{0ex}}{10}^{-15}$ |

$\beta =0.2$ | ||||||||

${10}^{-3}$ | 16 | 2.24 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 19 | 1.48 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 21 | 2.33 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 23 | 1.76 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ |

${10}^{-6}$ | 29 | 1.15 $\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 32 | 2.78 $\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 36 | 2.51 $\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 39 | 1.89 $\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ |

${10}^{-9}$ | 42 | 1.39 $\times \phantom{\rule{4pt}{0ex}}{10}^{-10}$ | 47 | 1.14 $\times \phantom{\rule{4pt}{0ex}}{10}^{-10}$ | 50 | 2.16 $\times \phantom{\rule{4pt}{0ex}}{10}^{-10}$ | 54 | 2.20 $\times \phantom{\rule{4pt}{0ex}}{10}^{-10}$ |

$\beta =0.5$ | ||||||||

${10}^{-3}$ | 16 | 3.89 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 19 | 3.37 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 20 | 6.79 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 22 | 5.16 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ |

${10}^{-6}$ | 29 | 2.54 $\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 31 | 6.58 $\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 35 | 8.26 $\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 38 | 6.23 $\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ |

${10}^{-9}$ | 41 | 2.77 $\times \phantom{\rule{4pt}{0ex}}{10}^{-10}$ | 46 | 2.56 $\times \phantom{\rule{4pt}{0ex}}{10}^{-10}$ | 49 | 5.90 $\times \phantom{\rule{4pt}{0ex}}{10}^{-10}$ | 53 | 7.82 $\times \phantom{\rule{4pt}{0ex}}{10}^{-10}$ |

$\beta =0.8$ | ||||||||

${10}^{-3}$ | 15 | 7.04 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 18 | 4.08 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 19 | 1.23 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 21 | 9.38 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ |

${10}^{-6}$ | 28 | 5.32 $\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 30 | 1.05 $\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 34 | 1.66 $\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 37 | 1.27 $\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ |

${10}^{-9}$ | 40 | 6.80 $\times \phantom{\rule{4pt}{0ex}}{10}^{-10}$ | 44 | 6.08 $\times \phantom{\rule{4pt}{0ex}}{10}^{-10}$ | 48 | 1.27 $\times \phantom{\rule{4pt}{0ex}}{10}^{-9}$ | 52 | 1.70 $\times \phantom{\rule{4pt}{0ex}}{10}^{-9}$ |

**Theorem**

**A1**

**.**Assume that the asymptotically stable, minimal realization $(A,B,C)$ of the full-order model transfer function, $G\left(s\right)=C{(Is-A)}^{-1}B$, is internally balanced, i.e.,

Algorithm A1: Balanced truncation method [42] |

Data: The desired error $\u03f5$, quadrature points $\mathit{\eta}=({\eta}_{1},{\eta}_{2},\cdots ,{\eta}_{N})$, and weights $\mathit{\zeta}=({\zeta}_{1},\cdots ,{\zeta}_{N})$. Result: Reduced quadrature points $\widehat{\mathit{\eta}}$ and weights $\widehat{\mathit{\zeta}}$.Set $A=-\mathrm{diag}\left(\mathit{\eta}\right)$, $C=\sqrt{\mathit{\zeta}}$, and $B={C}^{T}$; Solve two Lyapunov equations $AP+P{A}^{T}+B{B}^{T}=0$ and $AQ+Q{A}^{T}+{C}^{T}C=0$; Compute two singular-value decompositions of $P={U}_{P}{\Sigma}_{P}{V}_{P}$ and $Q={U}_{Q}{\Sigma}_{Q}{V}_{Q}$; Set $S={U}_{P}{\Sigma}_{P}^{1/2}$ and $L={U}_{Q}{\Sigma}_{Q}^{1/2}$; Compute a singular-value decomposition of $L{S}^{T}=U\Sigma V$, where $\Sigma =\mathrm{diag}({\sigma}_{1},\cdots ,{\sigma}_{N})$; Find k such that $2{\sum}_{j=k+1}^{N}{\sigma}_{j}\le \u03f5$; Form a $N\times k$ matrix J, where ${J}_{ii}={\sigma}_{i}^{-1/2}$, $i=1,\cdots ,k$ and ${J}_{ij}=0$ otherwise; Set ${T}_{l}={L}^{T}UJ$ and ${T}_{r}={S}^{T}VJ$; Set $\widehat{A}={T}_{l}^{T}A{T}_{r}$, $\widehat{B}={T}_{l}^{T}B$, and $\widehat{C}=C{T}_{r}$; Compute the eigenvalue decomposition of $\widehat{A}=X\Lambda {X}^{-1}$; Set $\widehat{\mathit{\eta}}=({\Lambda}_{11},\cdots ,{\Lambda}_{kk})$; Form $\tilde{B}={X}^{-1}{\widehat{B}}^{T}$ and $\tilde{C}=\widehat{C}X$; Set $\widehat{\mathit{\zeta}}={\left({\tilde{B}}_{i}{\tilde{C}}_{i}\right)}_{i=1}^{k}$; |

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**Figure 1.**Log-log plot of CPU time (CT(s)) versus the number of steps N with $\nu =0.5$ for Example 2.

**Figure 2.**Comparison of numerical solution of Example 5 at $T=250,500,1000$ with the parameters $a=0.4$, $b=2.0$, $c=0.6$, $\delta =1$, $L=400$, $\tau =1$, and $h=1/3$. Each row represents $\nu =1$, $\nu =0.98$ with the Liouville–Caputo FD, and $\nu =0.98$ with the ABC FD.

**Table 1.**Comparison of the number of exponentials ${N}_{exp}$ for approximating ${t}^{-1-\nu}$ with $\delta ={10}^{-3}$.

$\mathit{T}/\mathit{\delta}$ | ${10}^{3}$ | ${10}^{4}$ | ${10}^{5}$ | ${10}^{6}$ | ${10}^{3}$ | ${10}^{4}$ | ${10}^{5}$ | ${10}^{6}$ |
---|---|---|---|---|---|---|---|---|

$\u03f5$ | $\nu =0.2$ | $\nu =0.5$ | ||||||

The number of exponentials ${N}_{\mathrm{exp}}$ in [35] | ||||||||

${10}^{-3}$ | 25 | 29 | 33 | 37 | 30 | 35 | 40 | 44 |

${10}^{-6}$ | 34 | 40 | 45 | 51 | 39 | 46 | 52 | 59 |

${10}^{-9}$ | 43 | 51 | 58 | 66 | 49 | 57 | 65 | 74 |

The number of exponentials ${N}_{\mathrm{exp}}$ in our method | ||||||||

${10}^{-3}$ | 18 | 19 | 21 | 23 | 19 | 21 | 23 | 25 |

${10}^{-6}$ | 30 | 34 | 37 | 43 | 31 | 34 | 39 | 43 |

${10}^{-9}$ | 45 | 50 | 53 | 60 | 45 | 50 | 54 | 60 |

**Table 2.**Comparison of the number of exponentials ${N}_{exp}$ for approximating ${t}^{-1-\nu}$ with $T=1$.

$\mathit{\delta}$ | ${10}^{-3}$ | ${10}^{-4}$ | ${10}^{-5}$ | ${10}^{-6}$ | ${10}^{-3}$ | ${10}^{-4}$ | ${10}^{-5}$ | ${10}^{-6}$ |
---|---|---|---|---|---|---|---|---|

$\u03f5$ | $\nu =0.2$ | $\nu =0.5$ | ||||||

The number of exponentials ${N}_{\mathrm{exp}}$ in [35] | ||||||||

${10}^{-3}$ | 25 | 34 | 43 | 55 | 30 | 41 | 52 | 66 |

${10}^{-6}$ | 34 | 45 | 58 | 69 | 39 | 51 | 65 | 81 |

${10}^{-8}$ | 40 | 51 | 64 | 79 | 43 | 57 | 72 | 84 |

The number of exponentials ${N}_{\mathrm{exp}}$ in our method | ||||||||

${10}^{-3}$ | 18 | 21 | 25 | 30 | 19 | 21 | 27 | 30 |

${10}^{-6}$ | 30 | 37 | 43 | 49 | 31 | 37 | 44 | 50 |

${10}^{-8}$ | 40 | 48 | 56 | 82 | 41 | 48 | 58 | 82 |

**Table 3.**Errors and rates of convergence versus h for Example 1 with $\u03f5={10}^{-9}$, $\nu =0.2,0.5,0.8$.

ABC-FPCM | ABC-PCM | |||||||
---|---|---|---|---|---|---|---|---|

${\mathit{E}}_{max}$ | roc | ${\mathit{E}}_{{\ell}_{\mathbf{2}}}$ | roc | ${\mathit{E}}_{max}$ | roc | ${\mathit{E}}_{{\ell}_{\mathbf{2}}}$ | roc | |

h | $\nu =0.2$ | |||||||

1/10 | 3.37 $\times \phantom{\rule{4pt}{0ex}}{10}^{-2}$ | - | 1.85 $\times \phantom{\rule{4pt}{0ex}}{10}^{-2}$ | - | 3.37 $\times \phantom{\rule{4pt}{0ex}}{10}^{-2}$ | - | 1.85 $\times \phantom{\rule{4pt}{0ex}}{10}^{-2}$ | - |

1/20 | 9.03 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 1.90 | 4.92 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 1.91 | 9.03 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 1.90 | 4.92 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 1.91 |

1/40 | 1.71 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 2.40 | 1.14 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 2.11 | 1.71 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 2.40 | 1.14 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 2.11 |

1/80 | 3.98 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.10 | 2.67 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.09 | 3.98 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.10 | 2.67 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.09 |

1/160 | 9.35 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.09 | 6.33 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.08 | 9.35 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.09 | 6.33 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.08 |

1/320 | 2.21 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.08 | 1.51 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.07 | 2.21 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.08 | 1.51 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.07 |

1/640 | 5.28 $\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 2.07 | 3.61 $\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 2.06 | 5.28 $\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 2.07 | 3.61 $\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 2.06 |

h | $\nu =0.5$ | |||||||

1/10 | 7.35 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | - | 4.93 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | - | 7.35 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | - | 4.93 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | - |

1/20 | 1.55 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 2.24 | 1.11 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 2.15 | 1.55 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 2.24 | 1.11 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 2.15 |

1/40 | 3.53 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.14 | 2.56 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.11 | 3.53 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.14 | 2.56 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.11 |

1/80 | 8.24 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.10 | 6.02 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.09 | 8.24 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.10 | 6.02 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.09 |

1/160 | 1.95 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.08 | 1.43 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.07 | 1.95 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.08 | 1.43 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.07 |

1/320 | 4.70 $\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 2.06 | 3.46 $\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 2.05 | 4.70 $\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 2.06 | 3.46 $\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 2.05 |

1/640 | 1.14 $\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 2.04 | 8.41 $\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 2.04 | 1.14 $\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 2.04 | 8.41 $\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 2.04 |

h | $\nu =0.8$ | |||||||

1/10 | 1.82 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | - | 1.38 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | - | 1.82 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | - | 1.38 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | - |

1/20 | 3.95 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.21 | 2.99 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.21 | 3.95 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.21 | 2.99 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.21 |

1/40 | 9.17 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.11 | 6.84 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.13 | 9.17 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.11 | 6.84 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.13 |

1/80 | 2.21 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.05 | 1.62 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.08 | 2.21 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.05 | 1.62 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.08 |

1/160 | 5.41 $\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 2.03 | 3.93 $\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 2.04 | 5.42 $\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 2.03 | 3.94 $\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 2.04 |

1/320 | 1.35 $\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 2.01 | 9.68 $\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 2.02 | 1.34 $\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 2.01 | 9.67 $\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 2.03 |

1/640 | 3.39 $\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 1.99 | 2.41 $\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 2.01 | 3.33 $\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 2.01 | 2.39 $\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 2.01 |

**Table 4.**Errors and rates of convergence versus h for Example 2 with $\u03f5={10}^{-9}$, $\nu =0.2,0.5,0.8$.

ABC-FPCM | ABC-PCM | |||||||
---|---|---|---|---|---|---|---|---|

${\mathit{E}}_{max}$ | roc | ${\mathit{E}}_{{\ell}_{\mathbf{2}}}$ | roc | ${\mathit{E}}_{max}$ | roc | ${\mathit{E}}_{{\ell}_{\mathbf{2}}}$ | roc | |

h | $\nu =0.2$ | |||||||

1/10 | 5.14 $\times \phantom{\rule{4pt}{0ex}}{10}^{-1}$ | - | 2.47 $\times \phantom{\rule{4pt}{0ex}}{10}^{-1}$ | - | 5.14 $\times \phantom{\rule{4pt}{0ex}}{10}^{-1}$ | - | 2.47 $\times \phantom{\rule{4pt}{0ex}}{10}^{-1}$ | |

1/20 | 2.00 $\times \phantom{\rule{4pt}{0ex}}{10}^{-1}$ | 1.36 | 8.53 $\times \phantom{\rule{4pt}{0ex}}{10}^{-2}$ | 1.54 | 2.00 $\times \phantom{\rule{4pt}{0ex}}{10}^{-1}$ | 1.36 | 8.53 $\times \phantom{\rule{4pt}{0ex}}{10}^{-2}$ | 1.54 |

1/40 | 5.59 $\times \phantom{\rule{4pt}{0ex}}{10}^{-2}$ | 1.84 | 2.29 $\times \phantom{\rule{4pt}{0ex}}{10}^{-2}$ | 1.90 | 5.59 $\times \phantom{\rule{4pt}{0ex}}{10}^{-2}$ | 1.84 | 2.29 $\times \phantom{\rule{4pt}{0ex}}{10}^{-2}$ | 1.90 |

1/80 | 1.34 $\times \phantom{\rule{4pt}{0ex}}{10}^{-2}$ | 2.06 | 5.56 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 2.04 | 1.34 $\times \phantom{\rule{4pt}{0ex}}{10}^{-2}$ | 2.06 | 5.56 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 2.04 |

1/160 | 3.20 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 2.07 | 1.36 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 2.03 | 3.20 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 2.07 | 1.36 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 2.03 |

1/320 | 7.81 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.03 | 3.36 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.02 | 7.81 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.03 | 3.36 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.02 |

1/640 | 1.93 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.02 | 8.31 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.02 | 1.93 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.02 | 8.31 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.02 |

h | $\nu =0.5$ | |||||||

1/10 | 1.09 $\times \phantom{\rule{4pt}{0ex}}{10}^{-1}$ | - | 5.72 $\times \phantom{\rule{4pt}{0ex}}{10}^{-2}$ | - | 1.09 $\times \phantom{\rule{4pt}{0ex}}{10}^{-1}$ | - | 5.72 $\times \phantom{\rule{4pt}{0ex}}{10}^{-2}$ | - |

1/20 | 2.88 $\times \phantom{\rule{4pt}{0ex}}{10}^{-2}$ | 1.93 | 1.45 $\times \phantom{\rule{4pt}{0ex}}{10}^{-2}$ | 1.98 | 2.88 $\times \phantom{\rule{4pt}{0ex}}{10}^{-2}$ | 1.93 | 1.45 $\times \phantom{\rule{4pt}{0ex}}{10}^{-2}$ | 1.98 |

1/40 | 6.89 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 2.06 | 3.51 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 2.04 | 6.89 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 2.06 | 3.51 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 2.04 |

1/80 | 1.66 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 2.05 | 8.49 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.05 | 1.66 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 2.05 | 8.49 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.05 |

1/160 | 4.03 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.04 | 2.06 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.04 | 4.03 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.04 | 2.06 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.04 |

1/320 | 9.84 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.04 | 5.03 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.04 | 9.84 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.04 | 5.03 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.04 |

1/640 | 2.41 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.03 | 1.23 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.03 | 2.41 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.03 | 1.23 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.03 |

h | $\nu =0.8$ | |||||||

1/10 | 4.37 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | - | 2.03 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | - | 4.37 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | - | 2.03 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | - |

1/20 | 8.14 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.42 | 3.70 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.46 | 8.14 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.42 | 3.70 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.46 |

1/40 | 1.51 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.43 | 6.97 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.41 | 1.51 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.43 | 6.97 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.41 |

1/80 | 2.94 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.36 | 1.39 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.32 | 2.94 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.37 | 1.39 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.32 |

1/160 | 6.05 $\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 2.28 | 2.97 $\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 2.23 | 6.04 $\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 2.28 | 2.96 $\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 2.23 |

1/320 | 1.29 $\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 2.23 | 6.54 $\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 2.18 | 1.31 $\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 2.20 | 6.63 $\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 2.16 |

1/640 | 2.74 $\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 2.23 | 1.46 $\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 2.17 | 2.99 $\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 2.13 | 1.54 $\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 2.10 |

**Table 5.**Errors and rates of convergence versus h when $\tau =1/5000$ and versus $\tau $ when $h=1/5000$ for Example 3.

$\mathit{\nu}=0.2$ | $\mathit{\nu}=0.8$ | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

ABC-FPCM | ABC-PCM | ABC-FPCM | ABC-PCM | ||||||||||

h | ${E}_{max}$ | roc | CT(s) | ${E}_{max}$ | roc | CT(s) | ${E}_{max}$ | roc | CT(s) | ${E}_{max}$ | roc | CT(s) | |

$\tau =1/5000$ | 1/10 | 1.81 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | - | 1.017 | 1.81 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | - | 1.101 | 1.52 $\times \phantom{\rule{4pt}{0ex}}{10}^{-2}$ | - | 1.005 | 1.52 $\times \phantom{\rule{4pt}{0ex}}{10}^{-2}$ | - | 1.149 |

1/20 | 4.96 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 1.87 | 1.975 | 4.96 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 1.87 | 2.639 | 3.82 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 1.99 | 2.115 | 3.82 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 1.99 | 2.508 | |

1/40 | 1.33 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 1.89 | 3.866 | 1.33 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 1.89 | 4.971 | 9.57 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.00 | 3.935 | 9.57 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.00 | 4.648 | |

1/80 | 3.56 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 1.91 | 7.997 | 3.56 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 1.91 | 10.120 | 2.40 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.00 | 8.206 | 2.40 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.00 | 10.139 | |

1/160 | 9.59 $\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 1.89 | 15.985 | 9.59 $\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 1.89 | 25.011 | 6.02 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 1.99 | 15.744 | 6.02 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 1.99 | 20.843 | |

$\tau $ | ${E}_{max}$ | roc | CT(s) | ${E}_{max}$ | roc | CT(s) | ${E}_{max}$ | roc | CT(s) | ${E}_{max}$ | roc | CT(s) | |

$h=1/5000$ | 1/10 | 9.67 $\times \phantom{\rule{4pt}{0ex}}{10}^{-2}$ | - | 3.761 | 9.67 $\times \phantom{\rule{4pt}{0ex}}{10}^{-2}$ | - | 62.772 | 1.00 $\times \phantom{\rule{4pt}{0ex}}{10}^{-1}$ | - | 3.875 | 1.00 $\times \phantom{\rule{4pt}{0ex}}{10}^{-1}$ | - | 61.249 |

1/20 | 2.42 $\times \phantom{\rule{4pt}{0ex}}{10}^{-2}$ | 2.00 | 7.065 | 2.42 $\times \phantom{\rule{4pt}{0ex}}{10}^{-2}$ | 2.00 | 70.675 | 2.51 $\times \phantom{\rule{4pt}{0ex}}{10}^{-2}$ | 2.00 | 7.113 | 2.51 $\times \phantom{\rule{4pt}{0ex}}{10}^{-2}$ | 2.00 | 69.826 | |

1/40 | 6.05 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 2.00 | 13.893 | 6.05 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 2.00 | 90.210 | 6.27 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 2.00 | 13.590 | 6.27 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 2.00 | 86.519 | |

1/80 | 1.51 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 2.00 | 26.930 | 1.51 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 2.00 | 121.949 | 1.57 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 2.00 | 26.398 | 1.57 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 2.00 | 119.277 | |

1/160 | 3.78 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.00 | 29.859 | 3.78 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.00 | 153.570 | 3.92 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.00 | 21.083 | 3.92 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.00 | 155.221 |

**Table 6.**Errors and rates of convergence versus h when $\tau =1/5000$ and versus $\tau $ when $h=1/5000$ for Example 4.

$\mathit{\nu}=0.2$ | $\mathit{\nu}=0.8$ | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

ABC-FPCM | ABC-PCM | ABC-FPCM | ABC-PCM | ||||||||||

h | ${E}_{max}$ | roc | CT(s) | ${E}_{max}$ | roc | CT(s) | ${E}_{max}$ | roc | CT(s) | ${E}_{max}$ | roc | CT(s) | |

$\tau =1/5000$ | 1/10 | 1.19 $\times \phantom{\rule{4pt}{0ex}}{10}^{-2}$ | - | 0.101 | 1.19 $\times \phantom{\rule{4pt}{0ex}}{10}^{-2}$ | - | 0.079 | 7.64 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | - | 0.101 | 7.64 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | - | 0.069 |

1/20 | 3.44 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 1.79 | 0.165 | 3.44 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 1.79 | 0.123 | 1.96 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 1.97 | 0.162 | 1.96 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 1.97 | 0.131 | |

1/40 | 9.07 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 1.92 | 0.368 | 9.07 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 1.92 | 0.324 | 4.79 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.03 | 0.290 | 4.79 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.03 | 0.312 | |

1/80 | 2.31 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 1.98 | 0.628 | 2.31 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 1.98 | 0.690 | 1.17 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.03 | 0.608 | 1.17 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.03 | 0.720 | |

1/160 | 5.79 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 1.99 | 1.207 | 5.79 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 1.99 | 1.807 | 2.89 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.02 | 1.183 | 2.89 $\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.02 | 1.797 | |

$\tau $ | ${E}_{max}$ | roc | CT(s) | ${E}_{max}$ | roc | CT(s) | ${E}_{max}$ | roc | CT(s) | ${E}_{max}$ | roc | CT(s) | |

$h=1/5000$ | 1/10 | 1.12 $\times \phantom{\rule{4pt}{0ex}}{10}^{-1}$ | - | 0.834 | 1.12 $\times \phantom{\rule{4pt}{0ex}}{10}^{-1}$ | - | 48.032 | 1.08 $\times \phantom{\rule{4pt}{0ex}}{10}^{-1}$ | - | 0.720 | 1.08 $\times \phantom{\rule{4pt}{0ex}}{10}^{-1}$ | - | 53.056 |

1/20 | 2.70 $\times \phantom{\rule{4pt}{0ex}}{10}^{-2}$ | 2.06 | 0.642 | 2.70 $\times \phantom{\rule{4pt}{0ex}}{10}^{-2}$ | 2.06 | 53.879 | 2.60 $\times \phantom{\rule{4pt}{0ex}}{10}^{-2}$ | 2.05 | 0.779 | 2.60 $\times \phantom{\rule{4pt}{0ex}}{10}^{-2}$ | 2.05 | 52.112 | |

1/40 | 6.71 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 2.01 | 0.959 | 6.71 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 2.01 | 60.800 | 6.47 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 2.01 | 0.971 | 6.47 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 2.01 | 60.103 | |

1/80 | 1.68 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 2.00 | 1.391 | 1.68 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 2.00 | 69.584 | 1.62 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 2.00 | 1.479 | 1.62 $\times \phantom{\rule{4pt}{0ex}}{10}^{-3}$ | 2.00 | 64.850 | |

1/160 | 4.20 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.00 | 2.493 | 4.20 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.00 | 81.508 | 4.04 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.00 | 2.515 | 4.04 $\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.00 | 75.244 |

**Table 7.**CPU times of the fractional-order predator–prey interaction (5) with the parameters $a=0.4$, $b=2.0$, $c=0.6$, $\delta =1$, $L=400$, $\tau =1$, and $h=1/3$.

T | 125 | 250 | 500 | 1000 |
---|---|---|---|---|

Scheme 2 [37] | 17.03 | 35.22 | 74.13 | 155.21 |

ABC-PCM | 52.41 | 149.55 | 495.41 | 1668.56 |

ABC-FPCM | 55.23 | 109.69 | 223.01 | 547.09 |

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**MDPI and ACS Style**

Lee, S.; Kim, H.; Jang, B.
A Novel Numerical Method for Solving Nonlinear Fractional-Order Differential Equations and Its Applications. *Fractal Fract.* **2024**, *8*, 65.
https://doi.org/10.3390/fractalfract8010065

**AMA Style**

Lee S, Kim H, Jang B.
A Novel Numerical Method for Solving Nonlinear Fractional-Order Differential Equations and Its Applications. *Fractal and Fractional*. 2024; 8(1):65.
https://doi.org/10.3390/fractalfract8010065

**Chicago/Turabian Style**

Lee, Seyeon, Hyunju Kim, and Bongsoo Jang.
2024. "A Novel Numerical Method for Solving Nonlinear Fractional-Order Differential Equations and Its Applications" *Fractal and Fractional* 8, no. 1: 65.
https://doi.org/10.3390/fractalfract8010065