# An Enhanced Adaptive Bernstein Collocation Method for Solving Systems of ODEs

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## Abstract

**:**

## 1. Introduction

## 2. Existence and Uniqueness Theorem

**Theorem**

**1.**

## 3. Bernstein Formulas and Their Operational Matrices

#### 3.1. Bernstein Polynomials

#### 3.2. Generalized Bernstein Functions

#### 3.3. Approximation of Functions

## 4. Applications of Operational Matrices

#### 4.1. The Approximate Solutions Obtained by Tau Method

#### 4.1.1. Residual Correction Procedure for Bernstein Tau Method and GBF Tau Method

#### 4.2. Approximate Solutions Obtained by Collocation Method

#### Residual Correction Procedure for Bernstein Collocation Method and GBF Collocation Method

## 5. Numerical Experiments

#### 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

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The absolute error, estimation of absolute error and the corrected absolute error to Example 1 for $m=6$.

**Figure 2.**The absolute error, the corrected absolute error and corrected approximate solutions to Example 2 for $m=6$.

**Table 1.**The maximum absolute errors on the interval $[0,1]$ for different values of m for different m values and Example 1.

Method | m | 5 | 10 | 15 |
---|---|---|---|---|

Tau method | ${\u2225{u}_{1}-{u}_{1,m}\u2225}_{\infty}$ | $1.2$ × 10${}^{-5}$ | $3.5$ × 10${}^{-13}$ | $6.6$ × 10${}^{-21}$ |

Tau method | ${\u2225{u}_{2}-{u}_{2,m}\u2225}_{\infty}$ | $6.8$ × 10${}^{-6}$ | $1.3$ × 10${}^{-12}$ | $1.2$ × 10${}^{-20}$ |

Coll. method | ${\u2225{u}_{1}-{u}_{1,m}\u2225}_{\infty}$ | $2.0$ × 10${}^{-5}$ | $6.8$ × 10${}^{-13}$ | $1.1$ × 10${}^{-20}$ |

Coll. method | ${\u2225{u}_{2}-{u}_{2,m}\u2225}_{\infty}$ | $1.2$ × 10${}^{-5}$ | $2.2$ × 10${}^{-12}$ | $1.9$ × 10${}^{-20}$ |

**Table 2.**The maximum absolute errors on the interval $[0,1]$ for different values of m and Example 2.

Method | m | 5 | 10 | 15 |
---|---|---|---|---|

Tau method | ${\u2225{u}_{1}-{u}_{1,m}\u2225}_{\infty}$ | $6.9$ × 10${}^{-5}$ | $4.8$ × 10${}^{-11}$ | $7.2$ × 10${}^{-16}$ |

Tau method | ${\u2225{u}_{2}-{u}_{2,m}\u2225}_{\infty}$ | $6.4$ × 10${}^{-7}$ | $4.8$ × 10${}^{-14}$ | $3.3$ × 10${}^{-16}$ |

Coll. method | ${\u2225{u}_{1}-{u}_{1,m}\u2225}_{\infty}$ | $6.1$ × 10${}^{-5}$ | $3.5$ × 10${}^{-11}$ | $8.1$ × 10${}^{-16}$ |

Coll. method | ${\u2225{u}_{2}-{u}_{2,m}\u2225}_{\infty}$ | $1.0$ × 10${}^{-6}$ | $4.3$ × 10${}^{-14}$ | $3.3$ × 10${}^{-16}$ |

**Table 3.**Differences between Bernstein series solution and RK4 solutions in the case $a=1.2$, $b=2.92$, $c=6$, for $i=1,2,3$.

x | $\mathsf{\Delta}=|{\mathit{u}}_{\mathit{i},6}-{\mathbf{RK}4}_{0.001}|$ | $\mathsf{\Delta}=|{\mathit{u}}_{\mathit{i},6}+{\mathit{e}}_{\mathit{i},29}-{\mathbf{RK}4}_{0.001}|$ | ||||
---|---|---|---|---|---|---|

$\Delta {u}_{1}$ | $\Delta {u}_{2}$ | $\Delta {u}_{3}$ | $\Delta {u}_{1}$ | $\Delta {u}_{2}$ | $\Delta {u}_{3}$ | |

0.0 | 0 | 0 | 0 | 0 | 0 | 0 |

0.1 | 0.14 × 10${}^{-6}$ | 0.50 × 10${}^{-6}$ | 0.23 × 10${}^{-6}$ | 0.15 × 10${}^{-18}$ | 0.13 × 10${}^{-18}$ | 0.21 × 10${}^{-18}$ |

0.2 | 0.18 × 10${}^{-6}$ | 0.42 × 10${}^{-6}$ | 0.46 × 10${}^{-6}$ | 0.33 × 10${}^{-18}$ | 0.32 × 10${}^{-18}$ | 0.67 × 10${}^{-18}$ |

0.3 | 0.12 × 10${}^{-7}$ | 0.42 × 10${}^{-6}$ | 0.19 × 10${}^{-6}$ | 0.56 × 10${}^{-18}$ | 0.50 × 10${}^{-18}$ | 0.30 × 10${}^{-18}$ |

0.4 | 0.13 × 10${}^{-6}$ | 0.64 × 10${}^{-6}$ | 0.11 × 10${}^{-6}$ | 0.81 × 10${}^{-18}$ | 0.65 × 10${}^{-18}$ | 0.20 × 10${}^{-18}$ |

0.5 | 0.61 × 10${}^{-8}$ | 0.50 × 10${}^{-7}$ | 0.30 × 10${}^{-7}$ | 0.11 × 10${}^{-17}$ | 0.73 × 10${}^{-18}$ | 0.11 × 10${}^{-17}$ |

0.6 | 0.22 × 10${}^{-6}$ | 0.73 × 10${}^{-6}$ | 0.30 × 10${}^{-6}$ | 0.12 × 10${}^{-17}$ | 0.67 × 10${}^{-18}$ | 0.19 × 10${}^{-17}$ |

0.7 | 0.23 × 10${}^{-6}$ | 0.46 × 10${}^{-6}$ | 0.60 × 10${}^{-6}$ | 0.17 × 10${}^{-17}$ | 0.48 × 10${}^{-18}$ | 0.32 × 10${}^{-17}$ |

0.8 | 0.39 × 10${}^{-7}$ | 0.40 × 10${}^{-6}$ | 0.40 × 10${}^{-6}$ | 0.19 × 10${}^{-17}$ | 0.10 × 10${}^{-18}$ | 0.45 × 10${}^{-17}$ |

0.9 | 0.24 × 10${}^{-7}$ | 0.47 × 10${}^{-6}$ | 0.20 × 10${}^{-6}$ | 0.22 × 10${}^{-17}$ | 0.47 × 10${}^{-18}$ | 0.58 × 10${}^{-17}$ |

1.0 | 0.87 × 10${}^{-7}$ | 0.10 × 10${}^{-7}$ | 0.30 × 10${}^{-6}$ | 0.23 × 10${}^{-17}$ | 0.12 × 10${}^{-17}$ | 0.70 × 10${}^{-17}$ |

**Table 4.**The maximum values of the absolute errors by using the estimations of the absolute errors on the interval $[0,1]$ for Example 3.

Method | m | 5 | 10 | 15 |
---|---|---|---|---|

Tau method | ${\u2225{e}_{1,m}\u2225}_{\infty}$ | $1.8$ × 10${}^{-6}$ | $1.2$ × 10${}^{-12}$ | $4.1$ × 10${}^{-18}$ |

Tau method | ${\u2225{e}_{2,m}\u2225}_{\infty}$ | $3.6$ × 10${}^{-6}$ | $7.8$ × 10${}^{-12}$ | $1.6$ × 10${}^{-18}$ |

Tau method | ${\u2225{e}_{3,m}\u2225}_{\infty}$ | $1.4$ × 10${}^{-5}$ | $3.8$ × 10${}^{-11}$ | $4.5$ × 10${}^{-17}$ |

Coll. method | ${\u2225{e}_{1,m}\u2225}_{\infty}$ | $3.4$ × 10${}^{-6}$ | $1.9$ × 10${}^{-12}$ | $6.2$ × 10${}^{-18}$ |

Coll. method | ${\u2225{e}_{2,m}\u2225}_{\infty}$ | $7.2$ × 10${}^{-6}$ | $1.2$ × 10${}^{-11}$ | $4.1$ × 10${}^{-18}$ |

Coll. method | ${\u2225{e}_{3,m}\u2225}_{\infty}$ | $2.3$ × 10${}^{-5}$ | $6.1$ × 10${}^{-11}$ | $6.8$ × 10${}^{-17}$ |

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**MDPI and ACS Style**

Bataineh, A.S.; Isik, O.R.; Oqielat, M.; Hashim, I.
An Enhanced Adaptive Bernstein Collocation Method for Solving Systems of ODEs. *Mathematics* **2021**, *9*, 425.
https://doi.org/10.3390/math9040425

**AMA Style**

Bataineh AS, Isik OR, Oqielat M, Hashim I.
An Enhanced Adaptive Bernstein Collocation Method for Solving Systems of ODEs. *Mathematics*. 2021; 9(4):425.
https://doi.org/10.3390/math9040425

**Chicago/Turabian Style**

Bataineh, Ahmad Sami, Osman Rasit Isik, Moa’ath Oqielat, and Ishak Hashim.
2021. "An Enhanced Adaptive Bernstein Collocation Method for Solving Systems of ODEs" *Mathematics* 9, no. 4: 425.
https://doi.org/10.3390/math9040425