# Enhancing the Accuracy of Solving Riccati Fractional Differential Equations

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries and Notations

**Definition**

**1.**

**Definition**

**2.**

**Proposition**

**1.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Proposition**

**2.**

## 3. Riemann–Liouville Fractional Integral Operator for Hybrid of Block-Pulse Functions and Bernoulli Polynomials

**Theorem**

**1.**

**Proof of Theorem 1.**

## 4. The Numerical Method and Error Analysis

**Theorem**

**2.**

**Remark**

**1.**

## 5. Illustrative Example

#### 5.1. Example 1

#### 5.2. Example 2

#### 5.3. Example 3

#### 5.4. Example 4

#### 5.5. Example 5

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 1.**Absolute errors for $f\left(x\right)$, $\alpha =1$ (purple) and $\alpha =2$ (blue), for Example 1.

**Figure 3.**$f\left(x\right)$ with $q=\alpha $, where $q=0.7$ (dashed), $q=0.8$ (dotted), $q=0.9$ (dashed-dotted), and exact solution $q=1$ (continuous), for Example 2.

**Figure 4.**$f\left(x\right)$, where $q=0.7$ (dashed), $q=0.8$ (dotted), $q=0.9$ (dashed-dotted), and exact solution $q=1$ (continuous), with $\alpha =0.5$, for Example 3.

**Figure 5.**The absolute errors for $f\left(x\right)$ for $\alpha =1$ (purple) and $\alpha =2$ (blue), for Example 4.

**Figure 6.**$f\left(x\right)$, where $q=0.7$ (dashed), $q=0.8$ (dotted), $q=0.9$ (dashed-dotted), and exact solution $q=1$ (continuous), with $\alpha =q$, for Example 4.

**Figure 7.**The absolute errors for $f\left(x\right)$ for $\alpha =0.5$ (purple) and $\alpha =1$ (blue), for Example 5.

**Figure 8.**$f\left(x\right)$, where $q=0.25$ (dashed), $q=0.5$ (dotted), $q=0.75$ (dashed-dotted), and $q=0.95$ (continuous), with $\alpha =q$, for Example 5.

x | Method [49] | Method [50] | Method [33] | This Method |
---|---|---|---|---|

0.2 | $1.1\times {10}^{-8}$ | $3.2\times {10}^{-7}$ | $2.7\times {10}^{-10}$ | $3.4\times {10}^{-11}$ |

0.4 | $5.4\times {10}^{-6}$ | $5.0\times {10}^{-6}$ | $2.5\times {10}^{-10}$ | $2.9\times {10}^{-11}$ |

0.6 | $1.9\times {10}^{-4}$ | $1.9\times {10}^{-4}$ | $2.1\times {10}^{-10}$ | $2.4\times {10}^{-11}$ |

0.8 | $2.3\times {10}^{-3}$ | $2.3\times {10}^{-3}$ | $2.9\times {10}^{-10}$ | $1.8\times {10}^{-11}$ |

1.0 | $1.6\times {10}^{-2}$ | $1.6\times {10}^{-2}$ | $6.8\times {10}^{-8}$ | $1.5\times {10}^{-11}$ |

x | Method [22] | This Method | Method [22] | This Method |
---|---|---|---|---|

$\mathit{q}=\mathbf{0}.\mathbf{5}$ | $\mathit{q}=\mathbf{0}.\mathbf{5}$ | $\mathit{q}=\mathbf{0}.\mathbf{8}$ | $\mathit{q}=\mathbf{0}.\mathbf{8}$ | |

0 | $8.4\times {10}^{-10}$ | 0 | $5.0\times {10}^{-5}$ | 0 |

0.1 | $1.4\times {10}^{-9}$ | $6.9\times {10}^{-18}$ | $6.9\times {10}^{-6}$ | $8.6\times {10}^{-18}$ |

0.2 | $1.7\times {10}^{-9}$ | $1.3\times {10}^{-17}$ | $4.6\times {10}^{-6}$ | $6.9\times {10}^{-18}$ |

0.3 | $2.0\times {10}^{-8}$ | 0 | $4.3\times {10}^{-6}$ | $1.3\times {10}^{-17}$ |

0.4 | $1.5\times {10}^{-8}$ | $2.7\times {10}^{-17}$ | $4.2\times {10}^{-6}$ | 0 |

0.5 | $1.1\times {10}^{-8}$ | $5.5\times {10}^{-17}$ | $3.4\times {10}^{-5}$ | 0 |

0.6 | $9.0\times {10}^{-9}$ | 0 | $3.1\times {10}^{-5}$ | 0 |

0.7 | $7.5\times {10}^{-9}$ | 0 | $3.3\times {10}^{-5}$ | $5.5\times {10}^{-17}$ |

0.8 | $7.0\times {10}^{-9}$ | 0 | $5.3\times {10}^{-5}$ | $5.5\times {10}^{-17}$ |

0.9 | $7.8\times {10}^{-9}$ | $2.2\times {10}^{-16}$ | $1.3\times {10}^{-4}$ | 0 |

1 | $1.0\times {10}^{-8}$ | $1.1\times {10}^{-16}$ | $3.6\times {10}^{-4}$ | 0 |

$\mathit{\alpha}$ | Method [51] | This Method |
---|---|---|

$0.25$ | $1.7\times {10}^{-11}$ | $4.2\times {10}^{-12}$ |

$0.50$ | $9.3\times {10}^{-12}$ | $1.2\times {10}^{-13}$ |

$0.75$ | $7.4\times {10}^{-12}$ | $1.3\times {10}^{-13}$ |

$0.95$ | $3.2\times {10}^{-14}$ | $1.1\times {10}^{-14}$ |

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**MDPI and ACS Style**

Toma, A.; Dragoi, F.; Postavaru, O.
Enhancing the Accuracy of Solving Riccati Fractional Differential Equations. *Fractal Fract.* **2022**, *6*, 275.
https://doi.org/10.3390/fractalfract6050275

**AMA Style**

Toma A, Dragoi F, Postavaru O.
Enhancing the Accuracy of Solving Riccati Fractional Differential Equations. *Fractal and Fractional*. 2022; 6(5):275.
https://doi.org/10.3390/fractalfract6050275

**Chicago/Turabian Style**

Toma, Antonela, Flavius Dragoi, and Octavian Postavaru.
2022. "Enhancing the Accuracy of Solving Riccati Fractional Differential Equations" *Fractal and Fractional* 6, no. 5: 275.
https://doi.org/10.3390/fractalfract6050275