In the last few decades, it has become increasingly clear that fractional calculus always plays a very significant role in various branches of applied sciences. For this reason, fractional partial differential equations (FPDEs) are of more importance to model the different physical processes in nature more accurately. Therefore, the analytical or numerical solutions to these problems are taken into serious consideration and several techniques or algorithms have been developed for their solution. In the current work, the idea of fractional calculus has been used, and fractional Fornberg Whitham equation (FFWE) is represented in its fractional view analysis. A well-known method which is residual power series method (RPSM), is then implemented to solve FFWE. The RPSM results are discussed through graphs and tables which conform to the higher accuracy of the proposed technique. The solutions at different fractional orders are obtained and shown to be convergent toward an integer-order solution. Because the RPSM procedure is simple and straightforward, it can be extended to solve other FPDEs and their systems.

In mathematical physics, the Fornberg-Whitham equation is a fundamental mathematical model. The Fornberg-Whitham equation [

This equation was introduced to investigate how non-linear dispersive water waves break. The Fornberg-Whitham equation is shown to allow peakon solutions, as well as the occurrence of wave breaking, as a mathematical model for waves of limiting height. Fractional calculus (FC) is now widely used and accepted, owing to its well-established applications in a wide range of seemingly disparate domains of science and engineering [

FC has the potential to explain various difficult phenomena like memory and heredity. In recent years, researchers have taken a keen interest in the subject of fractional differential equations (FDEs), such as viscoelasticity, fluid mechanics, nanotechnology, electrochemistry, modelling for shape memory polymers, biological population models, optics and signal processing, modelling control theory, the damping behaviour of materials, economics and chemistry, signal processing, creeping and relaxation for viscoelastic materials and diffusion and reaction processes [

Analytical and numerical techniques are frequently used for the solution of FPDEs and their systems. The fractional problems that have been modelled by using FPDEs are found in various disciplines, because the mathematical modelling of real-life phenomena is usually modelled accurately by using FPDEs.

In this connection, the important fractional mathematical models are solved by using various techniques such as Chun-hui He’s algorithm [

Many researchers have worked hard to find the solutions to FPDEs by using RPSM and other novel techniques that have been used for the solutions of FPDEs like: Senol et al. [

In the current work, the solution of FFWE is investigated by RPSM. The RPSM was found to be a very effective technique for finding the analytical solution of FFWE [

In this section, we discussed some preliminaries and definitions.

The integral operator of Reimann-Liouville having order

Its fractional derivative for

The Caputo FD operator of the fractional order

The operators

A Power Series (PS) expansion is defined as [

Assume that

If

To the understand the procedure of RPSM [_{o}

Let _{k}

Also obtain

Let us consider the time fractional F-W equation,

Let

We define the

As

For the first step,

Now we can write

Differentiating of

Step 2. for

Now we can written

Similarly,

Now using

Putting

For the third step, putting

From

Applying

Using

Equivalence of the 3rd-order approximate solution of

In

Exact | ||||||||
---|---|---|---|---|---|---|---|---|

−4 | 0.01 | 0.1792480649 | 0.1795470609 | 0.1789669892 | 0.1780335597 | 0.1765509395 | 0.1742355152 | 0.1707005599 |

−3 | 0.01 | 0.2955300975 | 0.2960230583 | 0.2950666817 | 0.2935277167 | 0.2910832893 | 0.2872658000 | 0.2814376440 |

−2 | 0.01 | 0.4872467579 | 0.4880595131 | 0.4864827145 | 0.4839453902 | 0.4799152106 | 0.4736212348 | 0.4640122301 |

−1 | 0.01 | 0.8033340940 | 0.8046741005 | 0.8020743991 | 0.7978910586 | 0.7912464160 | 0.7808694041 | 0.7650268336 |

0 | 0.01 | 1.3244740080 | 1.3266833050 | 1.3223971220 | 1.3154999600 | 1.3045447970 | 1.2874359960 | 1.2613160130 |

1 | 0.01 | 2.1836884690 | 2.1873309860 | 2.1802642650 | 2.1688927670 | 2.1508307550 | 2.1226231120 | 2.0795585410 |

2 | 0.01 | 3.6002936290 | 3.6062991210 | 3.5946480680 | 3.5758996370 | 3.5461204150 | 3.4996138740 | 3.4286123980 |

3 | 0.01 | 5.9358806850 | 5.9457820710 | 5.9265727310 | 5.8956617930 | 5.8465641560 | 5.7698878340 | 5.6528261920 |

4 | 0.01 | 9.7866127450 | 9.8029373700 | 9.7712665230 | 9.7203030030 | 9.6393546840 | 9.5129368000 | 9.3199347810 |

In

He’s |
|||
---|---|---|---|

0.3 | 0.2 | 0.002395942 | 0.000190752 |

0.4 | 0.1 | 0.000653162 | 0.000195353 |

0.8 | 0.5 | 0.017409892 | 0.000332255 |

0.6 | 0.9 | 0.0459783993 | 0.0006452894 |

FPDEs | Fractional partial differential equations |

FDEs | Fractional differential equations |

DEs | Differential equations |

RPSM | Residual power series method |

FFWE | Fractional fornberg whitham equations |

In this work, we have implemented an efficient analytical technique, which is known as RPSM, to get an approximate series solution of FFWE with initial conditions. The suggested problems are first converted into their fractional form of the derivative and then incorporated the Caputo definition into the given problem to define FD. The general formulation for the proposed problem is discussed and then implemented for the solutions of FFWE. The proposed technique is applied to both fractional and integer orders of the suggested problem. It is observed that the procedure of the present technique is very effective and straight-forward. For verification and a better understanding of the obtained solutions, the graphical and tabular scenarios are presented. In