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In this paper, the Mohand transformbased homotopy perturbation method is proposed to solve twodimensional linear and nonlinear shallow water wave equations. This approach has been proved suitable for a broad variety of nonlinear differential equations in science and engineering. The variation trend of the water surface elevation at different time levels and depths are given by some graphs. Moreover, the obtained solutions are compared with the existing results, which show higher efficiency and fewer computations than other approaches studied in the literature.
Water waves that have a horizontal scale much larger than the depth of the fluid are considered shallow water waves (SWWs). SWWs describe the evolution of incompressible flow, neglecting density change along the depth, which are widely used to simulate the propagation of tsunami waves, tidal currents, storm floods, and shock waves [
Mohand transform (MT) [
The homotopy perturbation method (HPM) [
The main purpose of the present work is to solve twodimensional linear and nonlinear shallow water wave equations by coupling Mohand transform with the homotopy perturbation method. The remainder of the paper is organized as follows: in
The linear SWWEs in two dimensions [
The Mohand transform of the function
Mohand transform of some useful fundamental functions is given in
MT of frequently encountered functions.




1  1 

2 

1 
3 


4 


5 


The linearity property of MT is given as follows [
Let
The function
Inverse MT of frequently encountered functions.
S. No. 



1 

1 
2  1 

3 


4 


5 


To illustrate the basic ideas of the HPM [
The operator
Using the homotopy technique, one may construct a homotopy
We consider the solution to Eq.
The approximate solution to Eq.
In this section, we obtain the series solution to linear twodimensional shallow water Eq.
Let us consider the Gaussian initial conditions for
Applying Mohand transform in Eq.
Using the derivative characteristics of Mohand transform, Eq.
Taking the inverse of Mohand transform on both sides of Eq.
Similarly, on solving Eqs
Now using the HPM technique, we may construct a homotopy for Eqs
By simplifying the aforementioned formulas, we may obtain
Let us consider the solution to Eq.
Comparing the coefficients of
In the same way, by correlating the coefficients of
We can also get
In this section, we obtain the series solution to the nonlinear shallow water wave equation in twodimensional Eq.
Applying the MT and HPM in Eq.
Here, we again consider the power series solution to Eq.
By continuing to compare the coefficients of
In this section, the results obtained for both linear and nonlinear SWWEs have been presented, and the graphs of solutions obtained by the MHPM for various time levels (
Termwise solutions to linear SWWEs obtained by the MHPM.
The termwise solutions in
MHPM at
Number of terms considered  Water surface elevation ( 

3term solution  23.83 
4term solution  23.82 
5term solution  24.97 
6term solution  24.97 
7term solution  24.98 
8term solution  24.98 
WSE of linear SWWEs at
The water surface elevation (WSE) (
Diagrams in
WSE of linear SWWEs at
WSE of linear SWWEs at










23.0  26.8973  186.4153  3.2798 × 10^{3}  2.4953 × 10^{4} 
23.2  32.2187  242.9376  4.5436 × 10^{3}  3.7379 × 10^{4} 
23.4  35.5048  276.6909  5.3663 × 10^{3}  4.5951 × 10^{4} 
23.6  36.8368  288.2283  5.7489 × 10^{3}  5.0608 × 10^{4} 
23.8  36.4372  279.7550  5.7251 × 10^{3}  5.1581 × 10^{4} 
24.0  34.6307  254.7271  5.3546 × 10^{3}  4.9340 × 10^{4} 
24.2  31.8003  217.3891  4.7140 × 10^{3}  4.4520 × 10^{4} 
24.4  28.3449  172.3047  3.8886 × 10^{3}  3.7848 × 10^{4} 
24.6  24.6413  123.9262  2.9639 × 10^{3}  3.0068 × 10^{4} 
24.8  21.0149  76.2404  2.0189 × 10^{3}  2.1883 × 10^{4} 
25.0  17.7191  32.5119  1.1204 × 10^{3}  1.3901 × 10^{4} 
WSE of nonlinear SWWEs at
WSE of nonlinear SWWEs at
WSE of nonlinear SWWEs in










17.0  0.3977 × 10^{4}  0.2515 × 10^{5}  0.2576 × 10^{6}  1.6124 × 10^{6} 
17.2  1.1673 × 10^{4}  0.3689 × 10^{5}  0.7504 × 10^{6}  2.3641 
17.4  2.0858 × 10^{4}  0.4985 × 10^{5}  1.3385 × 10^{6}  3.1944 × 10^{6} 
17.6  3.1173 × 10^{4}  0.6351 × 10^{5}  1.9989 × 10^{6}  4.0684 × 10^{6} 
17.8  4.2052 × 10^{4}  0.7709 × 10^{5}  2.6953 × 10^{6}  4.9382 × 10^{6} 
18.0  5.2745 × 10^{4}  0.8969 × 10^{5}  3.3799 × 10^{6}  5.7447 × 10^{6} 
18.2  6.2365 × 10^{4}  1.0025 × 10^{5}  3.9958 × 10^{6}  6.4214 × 10^{6} 
18.4  6.9961 × 10^{4}  1.0772 × 10^{5}  4.4821 × 10^{6}  6.8999 × 10^{6} 
18.6  7.4621 × 10^{4}  1.1111 × 10^{5}  4.7804 × 10^{6}  7.1171 × 10^{6} 
18.8  7.5578 × 10^{4}  1.0962 × 10^{5}  4.8417 × 10^{6}  7.0220 × 10^{6} 
19.0  7.2314 × 10^{4}  1.0277 × 10^{5}  4.6328 × 10^{6}  6.5834 × 10^{6} 
WSE of linear SWWEs by the MHPM and HPM at
WSE of nonlinear SWWEs by the MHPM and HPM at
From
The main goal of this work is to solve shallow water wave equations using the Mohand transformbased homotopy perturbation method. The variation in the water surface height at different time levels and depths are given in this paper; the results are consistent with the characteristics of shallow water waves. It is proved that this method is a very good tool for solving SWWEs, which can easily be applied in finding out the approximate analytic solutions. The main advantage of this method over the HPM is that it is a powerful and efficient method to determine the analytical solution of the wave equation. In future research, we can also explore the effectiveness of this method in solving other problems, so as to improve the problemsolving efficiency.
The original contributions presented in the study are included in the article/supplementary material; further inquiries can be directed to the corresponding authors.
Conceptualization, YL, YZ, and JP; methodology, YL, YZ, and JP; software, YL and YZ; validation, YL and JP; writing—original draft preparation, YL, YZ, and JP; writing—review and editing, YL, YZ, and JP; supervision, JP; funding acquisition, JP. All authors have read and agreed to the published version of the manuscript.
This research was supported by the National Natural Science Foundation of China (Grant No. 10561151), the Basic Science Research Fund in the universities directly under the Inner Mongolia Autonomous Region (Grant No. JY20220003), and the Scientific Research Project of Hetao College (Grant No. HYZQ202122).
The authors would like to express their gratitude to the reviewers and editors for their insightful remarks and ideas, which helped them improve the paper’s quality.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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