We employed different iteration methods like Homotopy Analysis Method (HAM), Adomian Decomposition Method (ADM), and Variational Iteration Method (VIM) to find the approximate solution to the ZabolotskayaKhokhlov (ZK) equation. Iteration methods are used to solve linear and nonlinear PDEs whose classical methods are either very complex or too limited to apply. A comparison study has been made to see which of these methods converges to the approximate solution rapidly. The result revealed that, amongst these methods, ADM is more effective and simpler tool in its nature which does not require any transformation or linearization.
In science, physics, and engineering, many problems are modeled by the means of nonlinear evolutions equations especially in plasma physics, fluid dynamics, biology, and nonlinear acoustics. In this paper, we consider one such nonlinear partial differential equation which is (1 + 1) dimensional ZabolotskayaKhokhlov (ZK) equation (ZK for short)
The solution of the ZK equation will shed light on some new features in the behavior of nonlinear beam. Many authors [
As we know that many physical problems are modeled by means of partial differential equations (PDEs) like Helmholtz equation, heat equation, wave equation, gas equation, and so forth [
Consider the initial and boundary conditions for
Integrating
In this section, we will analyze uniqueness and existence of the solution to the ZK equation.
Equation
We suppose that
In this section, we will introduce the iteration methods which we will apply on the ZK equation. We will apply Homotopy Analysis Method (HAM), Adomian Decomposition Method (ADM), and Variational Iteration Method (VIM) to solve the said equation.
To illustrate the basic idea of the Homotopy Analysis Method, we consider the following nonlinear differential equation:
Using Taylor theorem, we can express
As we see, the initial and boundary conditions play vital role in determining the series solution and different set of conditions will result in different series solutions. Though we do not have general criteria of imposing restrictions at these conditions at the start, one should be careful in choosing initial conditions.
Adomian Decomposition Method was first introduced by Adomian in 1984 [
Consider the following equation:
By the Adomian Decomposition Method [
Under Variational Iteration Method, we break the given (
Next we find the value of Lagrange multiplier by taking variation on (
The series solution
Consider
In this section, we solve the ZK equation by the abovementioned methods. We also compare the results obtained to see which method is more sufficient and converges rapidly to the exact solution.
Consider ZK equation
The values of



Exact  


0.01040032  0.01042119  0.01011233  0.01042119 

0.04321024  0.04356014  0.04090019  0.04356076 

0.10087776  0.10274115  0.09304346  0.10275264 

0.18592772  0.19214213  0.16722674  0.19223594 

0.30100029  0.31705000  0.26413946  0.31754163 

0.44888979  0.4841644  0.3844759  0.48612181 

0.63258414  0.70194985  0.5289352  0.70843802 

0.85530539  0.98103624  0.69822148  1.00000000 

1.12055235  1.33466772  0.89304379  1.38562172 
Table
The absolute error of


 


0.00002087171  0.00000000451  0.00030886334 

0.00035052253  0.00000062180  0.00266056974 

0.00187487820  0.00001149230  0.00029081445 

0.00187480000  0.00001150000  0.00970910000 

0.00630820000  0.00009380000  0.02500920000 

0.01654130000  0.00049160000  0.05340210000 

0.03723200000  0.00195740000  0.10164590000 

0.07585390000  0.00648810000  0.17950280000 

0.14469460000  0.01896380000  0.30177850000 
In Figure
The exact solution is in purple line, HAM is in blue, ADM is in red, and VIM is in green.
The error graph of HAM is in blue, ADM is in red, and VIM is in green.
A comparative study has been made to seek the semianalytical solution of the ZK equation with initial and boundary conditions. The solution of the ZK equation is obtained in the form of an infinite series that converges rapidly in its domain. This form of solution gives the acoustic pressure of the beam in terms of propagation coordinate and time more explicitly. Since such form of the solution has not been obtained before, we believe that this will open new ways to understand the propagation of the confined beam in the nonlinear medium. Different iteration methods like HAM, ADM, and VIM have been applied and compared. The result revealed that ADM generated surprisingly effective results.
The authors declare that they there is no conflict of interests regarding the publication of this paper.