Optimal Homotopy Asymptotic Method to Nonlinear Damped Generalized Regularized Long-Wave Equation

A new semianalytical technique optimal homology asymptoticmethod (OHAM) is introduced for deriving approximate solution of the homogeneous and nonhomogeneous nonlinear Damped Generalized Regularized Long-Wave (DGRLW) equation. We tested numerical examples designed to confine the features of the proposed scheme.We drew 3D and 2D images of the DGRLW equations and the results are compared with that of variational iteration method (VIM). Results reveal that OHAM is operative and very easy to use.


Introduction
Partial differential equations used in modeling different problems in physics, biology, chemical reactions, and engineering sciences problems are frequently too difficult to be solved exactly, and even if an exact solution is possible, the required calculations may be too difficult.
The DGRLW equation is a partial differential equation that describes the amplitude of long-wave, which takes the following form: Here  > 0,  ≥ 1 is an integer, (, ) is known function, and (, ) is the amplitude of the long-wave at the position  and time .For  ̸ = 0, (1) features a balance between nonlinear and dispersive effects but also takes into account mechanisms of dissipation.In the physical sense, (1) with the dissipative term   is suggested if the good predictive power is preferred; such type of problem arises in the bore propagation as well as for water waves [1].
Marinca and Heris ¸anu proposed semianalytical technique OHAM for deriving approximate solution of nonlinear problems of thin film flow of a fourth grade fluid down a vertical cylinder [17][18][19].The method has been used by many researchers for obtaining numerical approximations of linear and nonlinear differential equations [20][21][22][23][24].The author has successfully applied the OHAM for deriving approximate solution of Equal Width Wave equation, Burger equations, and tenth order boundary value problems [25,26].The convergence criterion of proposed method is similar to that of homotopy analysis method (HAM) and homotopy perturbation method (HPM), but this method is more efficient and flexible.To improve the efficiency and accuracy of OHAM, Mathematical Problems in Engineering Heris ¸anu and Marinca introduced more generalized and new advances in OHAM which shows that the auxiliary function includes the functions of physical parameters in addition to the convergence control parameter [27,28].
Here, we investigate the approximate solution of the DGRLW equation with a variable coefficient using OHAM.The whole paper is divided into 3 sections.Section 2 is devoted to the analysis of the proposed method.In Section 3, solution of homogeneous and non-homogenous (DGRLW) equations is presented by OHAM, and absolute errors are also compared with VIM.The 3D and 2D images of the approximate solution and exact solution are also drawn.In all cases, the proposed method yields very encouraging results.

Fundamental Theory of OHAM
Consider the partial differential equation of the following form: where  is a linear operator and  is nonlinear operator. is boundary operator, (, ) is an unknown function,  and  denote spatial and time variables, respectively, Ω is the problem domain, and (, ) is a known function.
Example 1.Consider (1) with  = 1,  = 2, and (, ) = −(1/6) −(2+4) (, ) = 0 which in the simplest form is given as The initial condition is (, 0) =  − and exact solution given by Zeroth Order Problem.Consider the following: Its solution is given as  under First Order Problem.Consider the following: Its solution is as follows: Second Order Problem.Consider the following: Its solution is under Third Order Problem.Consider the following: Its solution is given as follows: The third order approximate solution is given by the following equation: The constants 1, 2, and 3 are calculated using the Least Squares, we have their optimal values as follows: The 3rd order OHAM solution yields very encouraging results after comparing with 3rd order VIM solution [5].Tables 1(a-c), and Figures 1, 2, 3, and 4 show the effectiveness of OHAM for  = 1.2,  = 1.4 and  = 1.6.

Conclusion
In this paper, the OHAM has been successfully implemented for the approximate solution of solutions of the Nonlinear Damped Generalized Regularized Long-Wave (DGRLW) equations.The results obtained by OHAM are very consistent in comparison with VIM.

Figure 1 :
Figure 1: Plot of of 3rd order approximate solution for the homogeneous DGRLW equation (2).

Table 1 :
(a) Comparison of absolute errors of 3rd order OHAM solution and 3rd order VIM solution for Example 1 at  = 1.2 and various values of t.(b) Comparison of absolute errors of 3rd order OHAM solution and 3rd order VIM solution for Example 1 at  = 1.4 and various values of t.(c) Comparison of absolute errors of 3rd order OHAM solution and 3rd order VIM solution for Example 1 at  = 1.6 and various values of t.

Table 2 :
(a) Absolute error of the solution of Example 2 by optimal homotopy asymptotic method (OHAM) at  = 15,  = 20, and  = 25 and various values of .(b) Absolute error of the solution of Example 2 by optimal homotopy asymptotic method (OHAM) at various values of  and .

Table 3 :
(a) Comparison of absolute errors of 1st order OHAM solution and 1st order VIM solution for Example 3 at  = −2.5 and various values of t.(b) Comparison of absolute errors of 1st order OHAM solution and 1st order VIM solution for Example 3 at  = 0 and various values of t.(c) Comparison of absolute errors of 1st order OHAM solution and 1st order VIM solution for Example 3 at  = −2.5 and various values of t.(d) Comparison of absolute errors of 1st order OHAM solution and 1st order VIM solution for Example 3 at  = −2.5 and various values of t.

Table 4 :
(a) Comparison of absolute errors of 1st order OHAM solution and 1st order VIM solution for Example 4 at  = 0.2 and various values of t.(b) Comparison of absolute errors of 1st order OHAM solution and 1st order VIM solution for Example 4 at  = 0.4 and various values of t.(c) Comparison of absolute errors of 1st order OHAM solution and 1st order VIM solution for Example 4 at  = 0.6 and various values of t.(d) Comparison of absolute errors of 1st order OHAM solution and 1st order VIM solution for Example 4 at  = 1 and various values of t.