# Numerical Solutions of a Differential System Considering a Pure Hybrid Fuzzy Neutral Delay Theory

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## Abstract

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## 1. Introduction

## 2. Hybrid Fuzzy Neutral Delay Differential Equations

## 3. The RK-4 Method and Pure Hybrid Fuzzy Neutral Delay Differential Equations

## 4. Numerical Example

## 5. Results, Discussion and Conclusions

- We developed hybrid fuzzy neutral delay differential equations and pure hybrid fuzzy neutral delay differential equations as governing equations for new systems based on fuzzy differential equations.
- Many authors have extended the fuzzy differential equations so far to include fuzzy hybrid differential equations and fuzzy delay differential equations. However, in this study, the fuzzy differential equations are extended to combinations of hybrid and delay differential equations, particularly neutral delay differential equations.
- The theoretical details were applied to a numerical example, and both analytical and numerical solutions were found. In this way, we generalized the approximate solutions algebraically.
- Though it is dealing with fuzzy solutions, it is enough to provide fuzzy plots. Nevertheless, we provided both non-fuzzy and fuzzy types of solutions. In both the non-fuzzy and fuzzy types of solutions, the coincidence of the exact and approximate solutions was shown graphically. For different values of t, the fuzzy valued plots were provided.
- The most important part of the paper involved using the Runge–Kutta method for solving non-pure (that is, including $y\left(t\right)$) hybrid fuzzy neutral delay differential equations, which we employed to solve a pure form of it (that is, without $y\left(t\right)$).
- We evaluated analytical solutions to the problem that we offered in the study, even though we dealt with numerical answers. The numerical solutions obtained by means of the fourth-order Runge–Kutta method were generalized and mentioned in the problem. Thus, when increasing the order to solve a different problem, the readers themselves can find the numerical solution. For the numerical results, we compared the numerical solutions obtained by means of the fourth-order Runge–Kutta and Euler methods with the exact solutions.
- Note that the fourth-order Runge–Kutta method gives better accuracy than the Euler method. Nonetheless, we compared the result with the Euler method to establish that the system obeys even the lower order methods.
- Though it is a more complicated form of a fuzzy differential equation, we also provided a numerical example to verify the theoretical results.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**Graphical comparison of the approximate and exact solutions for $h=0.1$, $\alpha =1$ and $t\in [0,3]$.

**Figure 5.**Graphical comparison between the approximate and exact solutions for $h=0.1$, $\alpha \in [0,1]$ and $t=3$.

**Figure 6.**Approximate solution obtained with the RK-4 method for $h=0.14$, $\alpha \in [0,1]$ and $t\in [0,3]$.

**Table 1.**Values of exact and approximate solutions by using the RK-4 method for the indicated configuration.

Approximate | Exact | |||
---|---|---|---|---|

$\mathit{\alpha}$ | $\underline{\mathit{y}}\left(\right)open="("\; close=")">\mathit{n};\mathit{\alpha}$ | $\overline{\mathit{y}}\left(\right)open="("\; close=")">\mathit{n};\mathit{\alpha}$ | $\underline{\mathit{Y}}\left(\right)open="("\; close=")">\mathit{t};\mathit{\alpha}$ | $\overline{\mathit{Y}}\left(\right)open="("\; close=")">\mathit{t};\mathit{\alpha}$ |

0.0 | 4.76804606005959 | 7.15206909008938 | 4.76803919566105 | 7.15205879349158 |

0.1 | 4.92698092872824 | 7.07260165575506 | 4.92697383551642 | 7.07259147356389 |

0.2 | 5.08591579739689 | 6.99313422142073 | 5.08590847537179 | 6.99312415363621 |

0.3 | 5.24485066606555 | 6.91366678708640 | 5.24484311522715 | 6.91365683370852 |

0.4 | 5.40378553473420 | 6.83419935275208 | 5.40377775508252 | 6.83418951378084 |

0.5 | 5.56272040340285 | 6.75473191841775 | 5.56271239493789 | 6.75472219385315 |

0.6 | 5.72165527207151 | 6.67526448408342 | 5.72164703479326 | 6.67525487392547 |

0.7 | 5.88059014074016 | 6.59579704974910 | 5.88058167464863 | 6.59578755399779 |

0.8 | 6.03952500940881 | 6.51632961541477 | 6.03951631450400 | 6.51632023407010 |

0.9 | 6.19845987807746 | 6.43686218108044 | 6.19845095435937 | 6.43685291414242 |

1.0 | 6.35739474674612 | 6.35739474674612 | 6.35738559421473 | 6.35738559421473 |

**Table 2.**Values of exact and approximate solutions by using the Euler method for the indicated configuration.

Approximate | Exact | |||
---|---|---|---|---|

$\mathit{\alpha}$ | $\underline{\mathit{y}}\left(\right)open="("\; close=")">\mathit{n};\mathit{\alpha}$ | $\overline{\mathit{y}}\left(\right)open="("\; close=")">\mathit{n};\mathit{\alpha}$ | $\underline{\mathit{Y}}\left(\right)open="("\; close=")">\mathit{t};\mathit{\alpha}$ | $\overline{\mathit{Y}}\left(\right)open="("\; close=")">\mathit{t};\mathit{\alpha}$ |

0.0 | 4.676726124815156 | 7.015089187222735 | 4.76803919566105 | 7.15205879349158 |

0.1 | 4.832616995642328 | 6.937143751809149 | 4.92697383551642 | 7.07259147356389 |

0.2 | 4.988507866469501 | 6.859198316395563 | 5.08590847537179 | 6.99312415363621 |

0.3 | 5.144398737296671 | 6.781252880981976 | 5.24484311522715 | 6.91365683370852 |

0.4 | 5.300289608123844 | 6.703307445568391 | 5.40377775508252 | 6.83418951378084 |

0.5 | 5.456180478951015 | 6.625362010154805 | 5.56271239493789 | 6.75472219385315 |

0.6 | 5.612071349778188 | 6.547416574741219 | 5.72164703479326 | 6.67525487392547 |

0.7 | 5.767962220605360 | 6.469471139327633 | 5.88058167464863 | 6.59578755399779 |

0.8 | 5.923853091432531 | 6.391525703914047 | 6.03951631450400 | 6.51632023407010 |

0.9 | 6.079743962259703 | 6.313580268500461 | 6.19845095435937 | 6.43685291414242 |

1.0 | 6.235634833086875 | 6.235634833086875 | 6.35738559421473 | 6.35738559421473 |

**Table 3.**Error analysis of approximate solutions by using the RK-4 and Euler methods for the indicated configuration.

RK-4 Method | Euler Method | |||
---|---|---|---|---|

$\mathit{\alpha}$ | $\underline{\mathit{y}}\left(\right)open="("\; close=")">\mathit{n};\mathit{\alpha}$ | $\overline{\mathit{y}}\left(\right)open="("\; close=")">\mathit{n};\mathit{\alpha}$ | $\underline{\mathit{y}}\left(\right)open="("\; close=")">\mathit{t};\mathit{\alpha}$ | $\overline{\mathit{y}}\left(\right)open="("\; close=")">\mathit{t};\mathit{\alpha}$ |

0.0 | 0.000006864 | 0.000010297 | 0.091313071 | 0.136969606 |

0.1 | 0.000007093 | 0.000010182 | 0.094356840 | 0.135447722 |

0.2 | 0.000007322 | 0.000010068 | 0.097400609 | 0.133925837 |

0.3 | 0.000007551 | 0.000009953 | 0.100444378 | 0.132403953 |

0.4 | 0.000007780 | 0.000009839 | 0.103488147 | 0.130882068 |

0.5 | 0.000008008 | 0.000009725 | 0.106531916 | 0.129360184 |

0.6 | 0.000008237 | 0.000009610 | 0.109575685 | 0.127838299 |

0.7 | 0.000008466 | 0.000009496 | 0.112619454 | 0.126316415 |

0.8 | 0.000008695 | 0.000009381 | 0.115663223 | 0.124794530 |

0.9 | 0.000008924 | 0.000009267 | 0.118706992 | 0.123272646 |

1.0 | 0.000009153 | 0.000009153 | 0.121750761 | 0.121750761 |

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**MDPI and ACS Style**

Dhandapani, P.B.; Thippan, J.; Martin-Barreiro, C.; Leiva, V.; Chesneau, C.
Numerical Solutions of a Differential System Considering a Pure Hybrid Fuzzy Neutral Delay Theory. *Electronics* **2022**, *11*, 1478.
https://doi.org/10.3390/electronics11091478

**AMA Style**

Dhandapani PB, Thippan J, Martin-Barreiro C, Leiva V, Chesneau C.
Numerical Solutions of a Differential System Considering a Pure Hybrid Fuzzy Neutral Delay Theory. *Electronics*. 2022; 11(9):1478.
https://doi.org/10.3390/electronics11091478

**Chicago/Turabian Style**

Dhandapani, Prasantha Bharathi, Jayakumar Thippan, Carlos Martin-Barreiro, Víctor Leiva, and Christophe Chesneau.
2022. "Numerical Solutions of a Differential System Considering a Pure Hybrid Fuzzy Neutral Delay Theory" *Electronics* 11, no. 9: 1478.
https://doi.org/10.3390/electronics11091478