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# Modifications of the Optimal Auxiliary Function Method to Fractional Order Fornberg-Whitham Equations

Hakeem Ullah1, Mehreen Fiza1,*, Ilyas Khan2,*, Abd Allah A. Mosa3, Saeed Islam1, Abdullah Mohammed4

1 Department of Mathematics, Abdul Wali Khan University, Mardan, 23200, Pakistan
2 Department of Mathematics, College of Science Al-Zulfi Majmmah University, Al-Majmmah, 11952, Saudi Arabia
3 Department of Mathematics and Statistics, College of Science, Taif University, Taif, 21944, Saudi Arabia
4 University Research Centre, Future University in Egypt, New Cairo, 11745, Egypt

* Corresponding Authors: Mehreen Fiza. Email: ; Ilyas Khan. Email:

(This article belongs to this Special Issue: Fractal-Fractional Models for Engineering & Sciences)

Computer Modeling in Engineering & Sciences 2023, 136(1), 277-291. https://doi.org/10.32604/cmes.2023.022289

## Abstract

In this paper, we present a new modification of the newly developed semi-analytical method named the Optimal Auxilary Function Method (OAFM) for fractional-order equations using the Caputo operator, which is named FOAFM. The mathematical theory of FOAFM is presented and the effectiveness of this method is proven by using it with well-known Fornberg-Whitham Equations (FWE). The FOAFM results are compared with other method results along with their exact solutions with the help of tables and plots to prove the validity of FOAFM. A rapidly convergent series solution is obtained from FOAFM and is validated by comparison with other results. The analysis proves that our method is simply applicable, contains less computational work, and is rapidly convergent to the exact solution at the first iteration. A series solution to the problem is obtained with the help of FOAFM. The validity of FOAFM results is validated by comparing its results with the results available in the literature. It is observed that FOAFM is simply applicable, contains less computational work, and is fastly convergent. The convergence and stability are obtained with the help of optimal constants. FOAFM is very easy in applicability and provides excellent results at the first iteration for complex nonlinear initial/boundary value problems. FOAFM contains the optimal auxiliary constants through which we can control the convergence as FOAFM contains the auxiliary functions in which the optimal constants and the control convergence parameters exist to play an important role in getting the convergent solution which is obtained rigorously. The computational work in FOAFM is less when compared to other methods and even a low-specification computer can do the computational work easily.

## Keywords

Ullah, H., Fiza, M., Khan, I., Allah, A., Islam, S. et al. (2023). Modifications of the Optimal Auxiliary Function Method to Fractional Order Fornberg-Whitham Equations. CMES-Computer Modeling in Engineering & Sciences, 136(1), 277–291.

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