^{1}

^{2}

^{3}

^{4}

^{1}

^{5}

^{6}

^{7}

^{1}

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

In this research paper, our work is connected with one of the most popular models in quantum magnetoplasma applications. The computational wave and numerical solutions of the Atangana conformable derivative (

In the existence of a magnetized e-p-i plasma [

Solving this kind of models has attracted many researchers in various areas, chemical physics [

For the fractional models, many analytical and numerical methods with various fractional operators have been derived such as the exponential expansion method, Khater method, Kudryashov method, simplest equation method,

This paper studies the analytical and numerical solutions of the Atangana conformable derivative (

Integrating Equation (

Through the balancing principle, the terms

Balancing between the terms of Equation (

The outline of this research paper is given as follows. Section

In this section, we employ three recent analytical schemes to find the explicit wave solutions of the Atangana conformable derivative (

This section gives a transitory elucidation of the mK method. We now explore a nontrivial solution for Equation (

Family I

Family II

Family III

Family IV

Family VI

Thus, using the above families leads to the new exact solitary wave solutions to the Atangana conformable derivative (

For

For

For

For

For

For

For

For

For

For

where

Here, we use three different analytical solutions Equations (

For

Substituting Equation (

In this section, the stability property has been tested of the obtained results based on the Hamiltonian system characteristics. This system imposes a single condition to ensure the stability of the solution. This condition is given by

Applying the stability check of Equation (

Consequently, this solution is not stable and applying the same steps to other obtained solutions investigates their stability property.

Here, we discuss our obtained solutions of the Atangana conformable derivative (

Computational solutions

Applying the modified Khater method to the Atangana conformable derivative (

The difference between our obtained solutions and that have been obtained in [

Numerical solutions

Applying the septic B-spline scheme to the Atangana conformable derivative (

This section illustrates our explained Figures

Figure

Figure

Figure

Exact, and numerical solutions based on the obtained analytical solution Equation (

Exact and numerical solutions based on the obtained analytical solution Equation (

Exact and numerical solutions based on the obtained analytical solution Equation (

Exact and numerical value of the Atangana conformable derivative (

Value of | Exact | Numerical | Absolute error |
---|---|---|---|

0 | 0. -0.408248 I | 0. -0.0000110311 I | 0.408237 |

0.0001 | 0. -0.408289 I | 0. -8.37587 | 0.408281 |

0.0002 | 0. -0.40833 I | 0. -5.50221 | 0.408324 |

0.0003 | 0. -0.408371 I | 0. -2.53611 | 0.408368 |

0.0004 | 0. -0.408412 I | 0. -1.13958 | 0.408411 |

0.0005 | 0. -0.408453 I | 0. +1.58999 | 0.408453 |

0.0006 | 0. -0.408493 I | 0. -1.01297 | 0.408492 |

0.0007 | 0. -0.408534 I | 0. -2.41857 | 0.408532 |

0.0008 | 0. -0.408575 I | 0. -5.26488 | 0.40857 |

0.0009 | 0. -0.408616 I | 0. -7.96518 | 0.408608 |

0.001 | 0. -0.408657 I | 0. -0.0000103138 I | 0.408647 |

Exact and numerical value of the Atangana conformable derivative (

Value of | Exact | Numerical | Absolute error |
---|---|---|---|

0 | 0. -0.2 I | 0. -0.0000794823 I | 0.199921 |

0.0001 | 0. -0.19995 I | 0. -0.0000595992 I | 0.19989 |

0.0002 | 0. -0.1999 I | 0. -0.0000399605 I | 0.19986 |

0.0003 | 0. -0.19985 I | 0. -0.0000214 I | 0.199829 |

0.0004 | 0. -0.1998 I | 0. -8.76169 | 0.199792 |

0.0005 | 0. -0.19975 I | 0. +1.47924 | 0.199752 |

0.0006 | 0. -0.199701 I | 0. -7.5007 | 0.199693 |

0.0007 | 0. -0.199651 I | 0. -0.0000199929 I | 0.199631 |

0.0008 | 0. -0.199601 I | 0. -0.0000376969 I | 0.199564 |

0.0009 | 0. -0.199552 I | 0. -0.000056177 I | 0.199495 |

0.001 | 0. -0.199502 I | 0. -0.0000736582 I | 0.199428 |

Exact and numerical value of the Atangana conformable derivative (

Value of | Exact | Numerical | Absolute error |
---|---|---|---|

0 | 0 | 3.46945 | 3.46945 |

0.0001 | 0.0027 | 0.000779236 | 0.00192076 |

0.0002 | 0.0054 | -8.67362 | 0.0054 |

0.0003 | 0.0081 | 0.00489544 | 0.00320455 |

0.0004 | 0.0108 | 0 | 0.0108 |

0.0005 | 0.0135 | 6.50521 | 0.0135 |

0.0006 | 0.0162 | 0 | 0.0162 |

0.0007 | 0.0189 | -4.33681 | 0.0189 |

0.0008 | 0.0216 | -8.67362 | 0.0216 |

0.0009 | 0.0242999 | 0.0141565 | 0.0101435 |

0.001 | 0.0269999 | 0.0269999 | 3.46945 |

This paper has succeeded in the implementation of the mK method and septic B-spline scheme to the Atangana conformable derivative (

No data were used to support this study.

The authors declare that there is no conflict of interests regarding the publication of this article.

The work was supported by the Natural Science Foundation of China (Grant Nos. 61673169, 11301127, 11701176, 11626101, and 11601485).