# An Operational Matrix Method Based on Poly-Bernoulli Polynomials for Solving Fractional Delay Differential Equations

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. The Atangana–Baleanu Derivative

**Definition**

**1.**

**Definition**

**2.**

**Lemma**

**1.**

**Proof.**

#### 2.2. The Properties of Poly-Bernoulli Polynomials

#### 2.3. The Poly-Bernoulli Delay Operational Matrix

## 3. An Operational Matrix Based on Poly-Bernoulli Polynomials for an ABC-Fractional Derivative

**Theorem**

**1.**

**Proof.**

#### Error Bound

**Theorem**

**2.**

**Proof.**

## 4. An Application to Solving Variable Coefficients of Fractional Delay Differential Equations in the ABC-Derivative

#### 4.1. Collocation Scheme

**Step 1**: Approximate each terms in Equation (29) by using poly-Bernoulli polynomials. For the derivative in an ABC-sense, i.e., ${}^{ABC}{D}_{x}^{\alpha}y\left(x\right)$, we use Equations (14) and (15). The delay term $y(x-a)$ is approximate via Equations (11) and (12).

**Step 2**: To find the solution ${y}_{N}\left(x\right)$; we collocate Equation (31) at the collocation points ${x}_{j}=\frac{j}{N+1}$, $j=1,2,\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}N-1$ to obtain

**Step 3**: The solution for Equation (29) is obtained by substituting the poly-Bernoulli coefficient vector $\mathbf{C}$ into Equation (30), i.e., ${y}_{N}\left(x\right)\approx {\displaystyle \sum _{n=0}^{N}}{c}_{n}{B}_{n}^{\left(k\right)}\left(x\right)=\mathbf{C}{\mathbf{B}}^{\left(k\right)}\left(x\right)$.

#### 4.2. Numerical Examples

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

## 5. Conclusions

- A new operational matrix based on poly-Bernoulli polynomials for ABC-derivative.
- A new delay operational matrix based on poly-Bernoulli polynomials.
- A collocation scheme via operational matrix and delay operational matrix based on poly-Bernoulli polynomials for fractional differential equations in an ABC-sense.

- The proposed scheme can be modified to solve various other problems that use an ABC-derivative.
- The operational matrix can be extend to other poly-type polynomials.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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x | Exact Solution | Propose Method, $\mathit{k}=1$ | Propose Method, $\mathit{k}=2$ |
---|---|---|---|

0 | 0 | 8.80000 × 10${}^{-11}$ | 2.01000 × 10${}^{-11}$ |

0.1 | 0.0076097334 | 1.72599 × 10${}^{-4}$ | 1.90269 × 10${}^{-4}$ |

0.2 | 0.0324547395 | 2.67576 × 10${}^{-5}$ | 4.08169 × 10${}^{-5}$ |

0.3 | 0.0765034239 | 4.61467 × 10${}^{-5}$ | 4.46544 × 10${}^{-5}$ |

0.4 | 0.1412220778 | 1.64420 × 10${}^{-5}$ | 2.74740 × 10${}^{-5}$ |

0.5 | 0.2278397784 | 4.87371 × 10${}^{-5}$ | 3.09000 × 10${}^{-5}$ |

0.6 | 0.3374369990 | 7.40603 × 10${}^{-5}$ | 5.74909 × 10${}^{-5}$ |

0.7 | 0.4709895148 | 1.99067 × 10${}^{-5}$ | 1.17086 × 10${}^{-5}$ |

0.8 | 0.6293940988 | 9.19385 × 10${}^{-5}$ | 8.89538 × 10${}^{-5}$ |

0.9 | 0.8134851328 | 1.61673 × 10${}^{-4}$ | 1.52310 × 10${}^{-4}$ |

1.0 | 1.0240460870 | 0.00000 × 10${}^{0}$ | 0.00000 × 10${}^{0}$ |

x | Exact Solution | Propose Method, $\mathit{k}=2$ | Propose Method, $\mathit{k}=5$ |
---|---|---|---|

0 | 0 | 2.00000 × 10${}^{-11}$ | 3.00000 × 10${}^{-11}$ |

0.1 | 0.0028633363 | 4.328440 × 10${}^{-4}$ | 3.34223 × 10${}^{-4}$ |

0.2 | 0.0132584457 | 3.51223 × 10${}^{-4}$ | 4.63151 × 10${}^{-4}$ |

0.3 | 0.0334875577 | 4.19672 × 10${}^{-5}$ | 4.52455 × 10${}^{-4}$ |

0.4 | 0.0656052327 | 1.97477 × 10${}^{-4}$ | 4.43244 × 10${}^{-4}$ |

0.5 | 0.1115273498 | 1.68033 × 10${}^{-4}$ | 5.43080 × 10${}^{-4}$ |

0.6 | 0.1730740180 | 1.88093 × 10${}^{-4}$ | 7.83066 × 10${}^{-4}$ |

0.7 | 0.2519928613 | 7.64132 × 10${}^{-4}$ | 1.09456 × 10${}^{-3}$ |

0.8 | 0.3499735921 | 1.27417 × 10${}^{-3}$ | 1.29460 × 10${}^{-3}$ |

0.9 | 0.4686579538 | 1.24322 × 10${}^{-3}$ | 1.07597 × 10${}^{-3}$ |

1.0 | 0.6096469123 | 4.00000 × 10${}^{-10}$ | 8.00000 × 10${}^{-10}$ |

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**MDPI and ACS Style**

Phang, C.; Toh, Y.T.; Md Nasrudin, F.S.
An Operational Matrix Method Based on Poly-Bernoulli Polynomials for Solving Fractional Delay Differential Equations. *Computation* **2020**, *8*, 82.
https://doi.org/10.3390/computation8030082

**AMA Style**

Phang C, Toh YT, Md Nasrudin FS.
An Operational Matrix Method Based on Poly-Bernoulli Polynomials for Solving Fractional Delay Differential Equations. *Computation*. 2020; 8(3):82.
https://doi.org/10.3390/computation8030082

**Chicago/Turabian Style**

Phang, Chang, Yoke Teng Toh, and Farah Suraya Md Nasrudin.
2020. "An Operational Matrix Method Based on Poly-Bernoulli Polynomials for Solving Fractional Delay Differential Equations" *Computation* 8, no. 3: 82.
https://doi.org/10.3390/computation8030082