Analytical and Numerical Investigation for the DMBBM Equation

: The nonlinear dispersive modiﬁed Benjamin-Bona-Mahony (DMBBM) equation is solved numerically using adaptive moving mesh PDEs (MMPDEs) method. Indeed, the exact solution of the DMBBM equation is obtained by using the extended Jacobian elliptic function expansion method. The current methods give a wider applicability for handling nonlinear wave equations in engineering and mathematical physics. The adaptive moving mesh method is compared with exact solution by numerical examples, where the explicit solutions are known. The numerical results verify the accuracy of the proposed method.

method [Ren and Zhang (2006) ;Zhang, Wang, Wang et al. (2006)], extended tanh-method [Fan (2000); Wazwaz (2007)], Riccati-Bernoulli sub-ODE method [Yang, Deng and Wei (2005); Abdelrahman and Sohaly (2018)], modified trial equation method [Bulut, Yel and Baskonus (2017); Kocak, Bulut and Yel (2014)] and so on, have been proposed for obtaining exact solutions to the NLPDEs. This paper is concerned with the (1+1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony (DMBBM) equation [Yusufoglu (2008)], in the form where α is a non-zero constant. This equation describes an approximation for surface long waves in nonlinear dispersive media. Moreover, it characterizes the hydro magnetic waves in cold plasma, acoustic gravity waves in compressible fluids and acoustic waves in inharmonic crystals [Yusufoglu (2008); Khan, Akbar and Islam (2014)]. We obtain the numerical solution of PDEs by approximating the PDEs on a mesh. We divide the domain into a finite set of nodes to obtain a uniform mesh. It is well known that some solutions have regions with rapid variations such as steep fronts, shock-like structures.
To resolve this kind of region, we need to use a massive number of points, which means the step size, h = x i − x i−1 , should be small enough for the entire domain. Computationally, this is intensive and expensive. Therefore, we attempt to find an alternative approach to have an adaptive mesh that manually puts more points in regions in which the solution changes rapidly and fewer points elsewhere. Unfortunately, this is useful only in time-independent problems. So, if the solution moves as the time moves, we must redistribute the mesh in each time. Overcoming this kind of issue, we use the adaptive moving mesh methods, which focus on the regions with rapid variations and moves the mesh as the solution moves with time. The adaptive mesh intends to have an adequate quality of accuracy and efficiency without using an excessive number of mesh points, as compared to a different method such as uniform mesh method. Dehghan et al. [Dehghan, Abbaszadeh and Mohebbi (2015b)] solved the two-dimensional nonlinear generalized Benjamin-Bona-Mahony-Burgers, using interpolating element-free Galerkin technique. Indeed, Dehghan et al. [Dehghan, Abbaszadeh and Mohebbi (2014a)] considered the three-dimensional nonlinear generalized Benjamin-Bona-Mahony-Burgers equation equation via the meshless method of radial basis functions. The results given in Dehghan et al. Mohebbi (2015b, 2014a)] are so efficient and robust. In this paper, we apply the extended Jacobian elliptic function expansion method [Chen, Xu, Liu et al. (2003)] to construct the exact solutions for the (1+1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony (DMBBM) equation. The numerical part of this article will be investigated by using the r-adaptive moving mesh method [Huang and Russell (2010); Budd, Huang and Russell (2009);Liseikin (2009)], which works with a monitor function, sometimes referred to as a mesh density function, and moving mesh partial differential equations (MMPDEs). The procedure of this method is embodied in moving a fixed number of grid nodes into regions in which the error of the solution is high. The monitor function fluctuates/moves according to the movement of a particular solution, such as a solution with large curvatures. On other words, the mesh density function is distributed between the grid elements. It is important to mention that an appropriate selection for the monitor function powerfully contributes to a great performance of this method. Since the r-adaptive moving mesh method refines the mesh, the internal layers are resolved more accurately. It is notable to notice that the MMPDEs can be rewritten by a diffusion equation [Huang and Russell (2010); Budd, Huang and Russell (2009)]. As a result, the MMPDEs can be simply executed within the finite difference techniques. A massive number of natural problems (for instance, problems occurred in fluid mechanics [Tang (2005); Yibao, Jeong and Kim (2014)], heat transfer [Ceniceros and Hou (2001)] and meteorological [Walsh (2010); Budd, Cullen and Walsh (2013)]) has been successfully solved by applying the r-adaptive techniques. The organization of this article is given as follows. In Section 2, we give some new exact solutions to the DMBBM equation, using extended Jacobian elliptic function expansion method. In Section 3, we introduce the numerical solution for the DMBBM equation, namely on a fixed mesh or on an adaptive mesh. Moreover, we give a comparison between exact and numerical solutions. Section 4 is devoted to conclude this work.

The extended Jacobian elliptic function expansion method
Using the traveling wave transformation where µ is a real constant, Eq. (1) transforms into the following ODEs: Integrating with respect to ξ and taking the constant of the integration by zero, we obtain the following ODE: According to the extended Jacobian elliptic function expansion method [Chen, Xu, Liu et al. (2003)], the solution of Eq. (4) can be expressed in the following form: where a 0 , a 1 and b 1 are constants determined later. From Eq. (5) we have Substituting Eqs. (5)-(7) into Eq. (4) and equating all the coefficients of sn 3 , sn 2 cn, sn 2 , sncn, sn, cn, sn 0 to zero, we obtain that: Solving the above system of equations yields the following cases: Case 1.
Then, the first family of equation is As long as m → 1, Eq. (15) is degenerated as follows: Case 2.
As long as m → 1, Eq. (19) is degenerated on the following form: Case 3.
As long as m → 1, Eq. (19) is degenerated as the following form: Case 4.
Then, the fourth family of equation is As long as m → 1, Eq. (21) is degenerated as follows:

Numerical results
This section is mainly devoted to seek the numerical solution of Eq. (1) Here, a and b are supposed to be the boundaries of the spatial domain and T e denotes a particular time. As mentioned previously, the adaptive moving mesh and fixed mesh techniques are employed to obtain the numerical results of Eq.
(1). The corresponding boundary conditions used in these methods are given as follows: Furthermore, the corresponding initial condition is selected by calculating the exact solution (Eq. (22)) at t = 0.

Numerical results of Eq. (1) on a fixed mesh
The physical domain is uniformly partitioned into N + 1 points by assuming that where h = (b − a)/N. We also use the centred finite differences so as to discretise the derivatives appeared in Eq.

Numerical results of Eq. (1) on an adaptive mesh
This subsection mostly focuses on finding the numerical results of Eq. (1) by employing the adaptive moving mesh methods. The coordinate transformation is taken by where x and ξ are assumed to be the spatial and computational coordinates, respectively. Consequently, the solution u is obviously represented as follows: u(x, t) = u(x(ξ, t), t).
Figs. 2(a)-2(c) present the time evolution of u(x, t), x(ξ, t), andq(x, t), respectively, using the adaptive moving mesh with MMPDE7 Eq. (27) and the arc-length monitor function function Eq. (28), respectively. Time is chosen to move between 0 and 5. Remark that the monitor function, appeared in Fig. 4(c), takes 1 everywhere except the area of the steep front. Therefore, it controls the movement of the mesh points, illustrated in Fig. 4(d), so that the region of the steep front takes more points than elsewhere. Tab. 1 presents how the numerical solutions obtained using the above schemes are convergent to the exact solution. The error is measured by obtaining the numerical results for above schemes at t = 5. Figs. 5(a) and 5(b) demonstrate a brief description of the errors' columns, presented in Tab. 1. Notice that the adaptive moving mesh solution is outstandingly accurate and concurrent to the exact solution than that obtained using the uniform mesh scheme. For the CPU time taken, the uniform mesh and adaptive moving mesh schemes take a very similar time to arrive at t = 5 for the same value of ∆x. Finally, we can surely assume that the technique of the adaptive moving mesh is more accurate and convergent than that of the uniform mesh scheme.

Conclusions
In this article, we have solved the DMBBM equation numerically using adaptive moving mesh PDEs (MMPDEs) method. Moreover, it has shown that the extended Jacobian elliptic function expansion technique can be used for building the wave solutions for the DMBBM equation. The main advantage of this method is that it gives results in terms of the Jacobi elliptic function which are more general. Indeed, these solutions may have significant applications in physics and applied mathematics. The comparison between the exact and numerical solutions are also investigated. The results of this work show how the adaptive moving mesh PDEs (MMPDEs) method is very powerful, robust and vital. Another interesting aspect of these results is that the proposed two methods used can be applied to other nonlinear equation in applied science.