# Application of the Explicit Euler Method for Numerical Analysis of a Nonlinear Fractional Oscillation Equation

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Remark**

**1.**

**Remark**

**2.**

**Definition**

**3.**

**Remark**

**3.**

**Remark**

**4.**

**Definition**

**4.**

## 3. Problem Statement

**Remark**

**6.**

**Remark**

**7.**

**Remark**

**8.**

**Remark**

**9.**

## 4. Numerical Solution

**Theorem**

**1.**

**Proof**

**of Theorem 1.**

**Remark**

**10.**

**Lemma**

**1.**

**Proof**

**of Lemma 1.**

**Remark**

**11.**

## 5. Convergence and Stability Issues

**Definition**

**5.**

**Theorem**

**2**

**Proof**

**of Theorem 2.**

**Theorem**

**3**

**Proof**

**of Theorem 3.**

## 6. The Adams–Bashford–Moulton Method

**Theorem**

**4.**

**Proof.**

## 7. Simulation Results and Some Applications

#### 7.1. Test Examples

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

#### 7.2. Fractional Duffing Oscillator

**Example**

**4.**

**Example**

**5.**

#### 7.3. Amplitude–Frequency Characteristic of a Fractional Duffing oscillator

## 8. The Study of Chaotic and Regular Modes

**Example**

**6.**

**Example**

**7.**

## 9. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Test example. Numerical and exact solutions of problem (23) in case of fulfillment of the condition of Theorems 2 and 3.

**Figure 2.**Test example. Numerical and exact solutions of problem (20) when the conditions of Theorems 2 and 3 are not fulfilled.

**Figure 4.**Phase trajectory (

**a**) and oscillogram (

**b**) for the Cauchy problem (1) under the conditions of Theorems 2 and 3 at $N=1500$.

**Figure 5.**Amplitude–frequency characteristic of the fractional Duffing oscillator for various types of the $q\left(t\right)$ function.

**Figure 7.**The spectrum of Lyapunov exponents from $\lambda $ (

**a**) and phase trajectories at (

**b**) $\lambda =0.18$, (

**c**) $\lambda =0.6$. The dots represent the Poincaré sections.

**Figure 8.**The spectrum of Lyapunov exponents from $q\left(t\right)$ (

**a**) and phase trajectories at (

**b**) $q\left(t\right)=0.15{cos}^{2}\left(0.2\phantom{\rule{3.33333pt}{0ex}}t\right)$, (

**c**) $q\left(t\right)=0.8{cos}^{2}\left(0.5\phantom{\rule{3.33333pt}{0ex}}t\right)$. The dots represent the Poincaré sections.

**Table 1.**Error and computational accuracy of the numerical scheme (10).

N | h | $\mathit{\epsilon}$ (25) | $\mathit{\alpha}$ (26) |
---|---|---|---|

10 | 0.1 | 0.010416664 | - |

20 | 0.05 | 0.005137254 | 1.019824008 |

40 | 0.025 | 0.002556096 | 1.007055385 |

80 | 0.0125 | 0.001275018 | 1.003424407 |

160 | 0.00625 | 0.000636774 | 1.001664279 |

320 | 0.003125 | 0.000318209 | 1.00080679 |

640 | 0.0015625 | 0.000159059 | 1.000412635 |

**Table 2.**Error and computational accuracy of the numerical scheme (10).

N | h | $\mathit{\epsilon}$ (18) | $\mathit{\alpha}$ (19) |
---|---|---|---|

10 | 0.1 | 216.8838 | - |

20 | 0.05 | 11004.94 | −5.665085173 |

40 | 0.025 | 22.6488 | 8.92450095 |

80 | 0.0125 | 0.012252099 | 10.85218997 |

160 | 0.00625 | 0.006278409 | 0.96455801 |

320 | 0.003125 | 0.0031782 | 0.9821891 |

640 | 0.0015625 | 0.0015990701 | 0.990976729 |

**Table 3.**Error and computational accuracy of the numerical scheme (10) and “predictor–corrector” method.

N | h = T/N | $\mathit{\u03f5}$ (Finite-Difference Scheme (10)) | $\mathit{\alpha}$ (Finite-Difference Scheme (10)) | $\mathit{\u03f5}$ (Predictor–Corrector (19)) | $\mathit{\alpha}$ (Predictor–Corrector (19)) |
---|---|---|---|---|---|

10 | 0.1 | 0.010416664 | - | 0.022014375 | - |

20 | 0.05 | 0.005137254 | 1.019824008 | 0.006517327 | 0.758163925 |

40 | 0.025 | 0.002556096 | 1.007055385 | 0.002125805 | 0.817941863 |

80 | 0.0125 | 0.001275018 | 1.003424407 | 0.004337276 | 1.131071553 |

160 | 0.00625 | 0.000636774 | 1.001664279 | 0.005145645 | 1.03243208 |

320 | 0.003125 | 0.000318209 | 1.00080679 | 0.005452087 | 1.011099493 |

640 | 0.0015625 | 0.000159059 | 1.000412635 | 0.005571391 | 1.004170658 |

**Table 4.**Error and computational accuracy of scheme (10).

N | h | $\mathit{\epsilon}$ (28) | $\mathit{\alpha}$ (16) |
---|---|---|---|

10 | 0.1 | 0.010416664 | - |

20 | 0.05 | 0.005137254 | 1.019824008 |

40 | 0.025 | 0.002556096 | 1.007055385 |

80 | 0.0125 | 0.001275018 | 1.003424407 |

160 | 0.00625 | 0.000636774 | 1.001664279 |

320 | 0.003125 | 0.000318209 | 1.00080679 |

640 | 0.0015625 | 0.000159059 | 1.000412635 |

**Table 5.**Error and computational accuracy of the scheme (10).

N | h | $\mathit{\epsilon}$ (20) | $\mathit{\alpha}$ (19) |
---|---|---|---|

10 | 0.1 | 0.010416664 | - |

20 | 0.05 | 0.055137254 | −2.404134099 |

40 | 0.025 | 0.006556096 | 3.072118534 |

80 | 0.0125 | 0.003275018 | 1.001334144 |

160 | 0.00625 | 0.001536774 | 1.09159782 |

320 | 0.003125 | 0.00078209 | 0.974498474 |

640 | 0.0015625 | 0.000039059 | 1.001679624 |

**Table 6.**Error and computational accuracy of the numerical scheme (10) and “predictor–corrector” method.

N | $\mathit{h}=\mathit{T}/\mathit{N}$ | $\mathit{\u03f5}$ (Finite-Difference Scheme (10)) | $\mathit{\alpha}$ (Finite-Difference Scheme (10)) | $\mathit{\u03f5}$ (Predictor–Corrector (19)) | $\mathit{\alpha}$ (Predictor–Corrector (19)) |
---|---|---|---|---|---|

10 | 0.1 | 0.010416664 | - | 0.012159916 | - |

20 | 0.05 | 0.005137254 | 1.019824008 | 0.005933937 | 1.035071832 |

40 | 0.025 | 0.002556096 | 1.007055385 | 0.002947911 | 1.009296592 |

80 | 0.0125 | 0.001275018 | 1.003424407 | 0.001468966 | 1.004891965 |

160 | 0.00625 | 0.000636774 | 1.001664279 | 0.000733796 | 1.001351042 |

320 | 0.003125 | 0.000318209 | 1.00080679 | 0.000366708 | 1.000745531 |

640 | 0.0015625 | 0.000159059 | 1.000412635 | 0.00018197 | 1.000434281 |

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**MDPI and ACS Style**

Kim, V.A.; Parovik, R.I.
Application of the Explicit Euler Method for Numerical Analysis of a Nonlinear Fractional Oscillation Equation. *Fractal Fract.* **2022**, *6*, 274.
https://doi.org/10.3390/fractalfract6050274

**AMA Style**

Kim VA, Parovik RI.
Application of the Explicit Euler Method for Numerical Analysis of a Nonlinear Fractional Oscillation Equation. *Fractal and Fractional*. 2022; 6(5):274.
https://doi.org/10.3390/fractalfract6050274

**Chicago/Turabian Style**

Kim, Valentine Aleksandrovich, and Roman Ivanovich Parovik.
2022. "Application of the Explicit Euler Method for Numerical Analysis of a Nonlinear Fractional Oscillation Equation" *Fractal and Fractional* 6, no. 5: 274.
https://doi.org/10.3390/fractalfract6050274