1. Introduction
A disturbance of the ocean surface generally resulting from deepsea earthquakes shifting the sea floor and generating tsunami waves and oceanic acoustic fields has interested scientists for a long time [
1,
2]. Tsunamis are nearshore propagating waves with long wavelengths and enormous amplitudes. The possibility of migration of these waves into the coast and devastation of property is substantial. Wave trains and wave forms with leading elevated or depressed waves have been previously observed. With respect to human catastrophes, the wavelength and amplitude ranges of these kinds of wave are considerable. Climate change and global warming are examples of these great natural disasters. Flooding, heat waves, early spring arrival, sealevel rise, glacier melting, coral reef bleaching, and disease contagions are the presentday results of climate change [
3,
4,
5,
6,
7]. Nevertheless, these giant waves can constitute alternative energy resources for nearfuture applications if the essential technology is implemented [
8,
9,
10,
11,
12].
Nonlinear partial differential equations (NPDEs) govern many natural phenomena arising in mathematical physics and engineering sciences [
13,
14,
15]. Nonlinear waves are an important scientific research field. In recent decades, numerous scientists developed various mathematical models, such as the Korteweg–de Vries (KdV) [
16], regularizedlong wave (RLW) [
17], and Rosenau [
18] equations, to describe wave behavior. Indeed, the wave–wall and wave–wave interactions in compact discrete systems dynamics cannot be appropriately accomplished by the KdV model. To tackle this issue, Rosenau [
19,
20] introduced the following socalled Rosenau model:
which commonly represents the dense discrete system and simulates the longchain transmission model via an LC flow in the computer and radio fields. The symbol
$u=u(x,t)$ represents the wave velocity and the term
${u}_{xxt}$ in the Rosenau model (
1) is used to take into account nonlinear waves. Park [
21] proved the uniqueness and existence of the solution to (
1).
For further analysis of nonlinear waves, one term
${u}_{xxt}$ needs to be involved in the Rosenau Equation (
1). The obtained model is typically known as the following Rosenau–RLW model [
22,
23,
24]:
Following [
25,
26], the Rosenau–RLW model can be developed in the generalized Rosenau–RLW model as:
in which
$p\ge 1$ is a positive integer.
The KdV equation was modified by Kawahara [
27] using solitary waves to balance the nonlinear effect via the higherorder dispersion effect. Hence, Kawahara [
27] introduced a generalized nonlinear dispersive relationship through the addition of a fifthorder term to this model. He took into account the effects of higherorder dispersion by approximating his model in the following form:
The Kawaharatype equation was proposed for the shallow water wave theory with surface tension [
27]. If the third nonlinear term on the lefthand side of the equation is substituted by
${u}^{2}{u}_{x}$, then the Equation (
4) is called as the modified Kawahara equation.
In order to take nonlinear waves into consideration, Pan and He [
28] derived the Rosenau–Kawahara equation with the addition of the viscous terms
${u}_{xxxxx}$ and
$+{u}_{xxx}$ and obtained the generalized form of the Rosenau–RLW model (
3). They investigated the solitary and periodic solutions of the following equation:
In this paper, we focus on finding the approximate solutions of the initial boundary value problem (IBVP) for the onedimensional (1D) generalized Rosenau–Kawahara–RLW model as
where the initial and boundary conditions (abbreviated as IC and BCs, respectively) are prescribed as
in which constants
$\alpha ,\beta ,\gamma $,
$\eta $,
$\sigma $ and
$\mu $ represent nonnegative constants,
$p\ge 2$ denotes a positive integer,
$g\left(x\right)$ is prescribed continuous function, and
$u=u(x,t)$ is a realvalued function.
When
$\gamma =\sigma =0$, Equation (
6) converts to the generalized Rosenau–RLW model. For the case of
$\alpha =\mu =\eta =1$,
$\gamma =\sigma =0$ and
$\beta =2,$ Equation (
6) becomes to the usual Rosenau–RLW model. For the special case
$\alpha =\gamma =\eta =\sigma =1$,
$\mu =0$ and
$\beta =2,$ Equation (
6) corresponds to the usual Rosenau–Kawahara model and for
$\mu =0$, Equation (
6) becomes the generalized Rosenau–Kawahara model.
Lemma 1 (See [
28]).
Let ${u}_{0}\left(x\right)\in {C}_{0}^{7}\left([a,b]\right)$. Then, the IBVP (6)–(8) satisfies the following energy conservative property:where ${C}_{0}^{7}\left([a,b]\right)$ represents the set of functions that are seventh order continuous differentiable in the spatial interval $[a,b]$ and have compact supports inside $(a,b)$. Over the last few years, some analytical and numerical approaches have been adopted to obtain the solution of the IBVP (
6)–(
8). Jin [
29] applied the homotopy perturbation and variational iteration methods. Korkmaz and Dag [
30] used the cosine expansion and Lagrange interpolation polynomials based on differential quadrature. Zuo [
31] adopted the tanh ansatz and sech ansatz techniques to obtain exact bright and dark 1soliton solutions. Pan and He [
28] proposed a threelevel linearly implicit finite difference (FD) approach. Later, He [
32] derived the exact solitary wave solution with power law nonlinearity and advanced a threelevel linearly implicit difference approach. Wang and Dai [
33] developed a threelevel conservative fourthorder FD approach, while Gazi et al. [
34] employed a septic Bspline collocation finite element (FE) technique.
Meshfree (meshless) methods have drawn considerable interest from the scientific community in recent decades. Unlike conventional meshdependent techniques (such as the FE, FD, and spectral techniques), these methods are independent of predefined grids and alignment for discretizing the domain. They use merely a group of scattered nodes provided by the initial data in order to cover the interior and the boundaries of the domain. They are also independent of the problem’s geometry. The radial basis function (RBF) is one type of these methods. The RBF method utilizes a univariate function with an Euclidean norm, which converts a multidimensional problem into one that is virtually onedimensional. Meshless RBFs have recently been widely utilized as a potential choice for solving PDEs in different applications [
35]. The meshless characteristic of RBFbased methods provides flexibility with respect to the problem geometry, simplicity of multidimensional application, and a high convergence order. The RBF method may be either local or global, each of which has advantages and disadvantages. In global methods, all the nodal points in the domain of the problem are used, and implementation is simple. Smallscale problems can be easily solved by global methods, although illconditioned interpolation matrices are often encountered in these techniques. On the other hand, the local RBF techniques use only nodes in every subdomain’s influence area around each spatial point. This mitigates the original illconditioning problem and the computational cost. Some authors have tried localized RBFbased strategies, such as the localized RBFgenerated FD (LRBFFD) [
36,
37] and the localized RBF partition of unity (LRBFPU) [
38,
39], which produce wellconditioned systems.
The major objective of this work is to implement the meshfree LRBFFD strategy for computing the solitary wave propagation of the generalized Rosenau–Kawahara–RLW model. The major advantages of the proposed meshfree (meshless) technique and the related generalization over surfaces are that they are independent from a background mesh or cell for approximation and are easy to implement on different irregular domains in multidimensional spaces. The meshless LRBFFD is the hybridization of the meshless concept with the FD technique. Nonetheless, this approach does not require meshing over the stencil nodes (the local subdomain or the subdomain), unlike the FD method. This process is performed for all grid points within the computational region. In addition, the grid points in each stencil can be readily increased for improving accuracy.
The outline of this paper has been organized as follows.
Section 2 introduces the LRBFFD strategy and the meshfree scheme of lines is applied to discretize the spatial variable of the generalized Rosenau–Kawahara–RLW model. Consequently, a nonlinear system of ODEs is derived that can be solved using either a numerical time stepping method or a direct solution in the time dimension. Some numerical tests are given in
Section 3 to verify the numerical accuracy and performance of the LRBFFD. In addition, it is shown that the computational efficiency of the proposed method is sufficiently superior to one exhibited by the other schemes in the existing literature. Finally,
Section 4 presents the concluding remarks.
3. Numerical Experiments
This section introduces three numerical examples on the generalized Rosenau–Kawahara–RLW model to measure the accuracy and the performance of the LRBFFD technique. For this purpose, we calculate the
${L}_{\infty}$,
${L}_{2}$, and
${L}_{\mathrm{rms}}$ norm errors as
Here,
${U}_{i}$ and
${u}_{i}$ denote the numerical and exact solutions, respectively. The numerical examples use the multiquadric RBF (MQ)
$\begin{array}{c}\varphi \left(r\right)=\sqrt{1+{\epsilon}^{2}{r}^{2}}\end{array}$ as the basis function with a shape parameter
$\epsilon $. The accuracy and flatness of the function heavily depend on this parameter
$\epsilon $, but there is no agreement on the best value. The LRBFFD method places great importance on the selection of
$\epsilon $. To determine the optimal shape parameter
$\epsilon $, we utilize Algorithm 2 from Sarra’s method [
41].
Algorithm 2: Optimal shape parameter [41]. 

The MATLAB R2016a environment on a Windows 10 desktop computer with 4 GB RAM was used for numerical computations. The condest command in MATLAB can be used to obtain the condition number (CN) of the coefficient matrix.
Example 1. Let us study the generalized Rosenau–Kawahara–RLW model (6) associated with $\alpha =\beta =\mu =\eta =\sigma =1$ and $p=2$ on the space interval $[40,200]$ so that exact solitary wave solution is Hereafter, we study this example based on the LRBFFD collocation technique for different values of
$\delta t$,
N,
${n}_{i}$ and
T.
Table 1 reports the errors of numerical solutions
${L}_{\infty}$,
${L}_{2}$, and
${L}_{\mathrm{rms}}$ norms, CN and computational times (in seconds) at different values of stencil sizes
${n}_{i}$ with
$\delta t=1/1000$ when
$T=2$.
Table 2 compares the errors of numerical solutions using
${L}_{\infty}$ and
${L}_{2}$ norms with techniques described in [
28,
33] when
$T=10$ by taking
$\delta t=0.005$.
Table 3 compares the
${L}_{\infty}$ and
${L}_{2}$ norm errors with methods introduced in [
28,
33] by various values of
$\delta t$ and
N when
$T=40.$ Based on comparisons in
Table 2 and
Table 3, we can observe that the proposed strategy is slightly better than the techniques introduced in [
28,
33].
Table 4 lists the conservative invariant
E over spatial interval
$[40,200]$ at various total times
T. It can be seen that the method is conservative perfectly (up to 5 decimals) for energy during the longterm time evolution of the solitary wave.
Figure 2 shows the numerical solution and corresponding maximum norm errors when
$\delta t=1/1000$,
$N=600$ and
${n}_{i}=581$ over spatial interval
$[40,200]$.
Figure 3 displays the longtime behavior of numerical solutions with
$N=500,\phantom{\rule{0.166667em}{0ex}}{n}_{i}=467$ and
$\delta t=1/1000$ at several total times
$T\in \{0,20,30,40,60\}$ over spatial interval
$[40,200]$. As seen in
Figure 3, the single solitons move to the rightside with the preserved amplitude and shape. Finally,
Figure 4 depicts the maximum norm errors
${L}_{\infty}$ at various total times
$T\in \{0,20,30,40,60\}$ with
$N=350,\phantom{\rule{0.166667em}{0ex}}{n}_{i}=321$ and
$\delta t=0.01$ over spatial interval
$[40,200]$.
Example 2. Consider the generalized Rosenau–Kawahara–RLW model (6) associated with $\alpha =\beta =\gamma =\eta =\sigma =1$, $\mu =2$ and $p=4$ on the space interval $[40,240]$ so that the exact solitary wave solution is This example is simulated by using the LRBFFD collocation scheme for different values of
$\delta t$,
N,
${n}_{i}$ and
T.
Table 5 presents the errors of numerical solutions having
${L}_{\infty}$,
${L}_{2}$, and
${L}_{\mathrm{rms}}$ norms, CN and computational times (in seconds) at several values of stencil sizes
${n}_{i}$ with
$\delta t=1/1000$ when
$T=5$.
Table 6 includes the errors of approximate solutions by making use of
${L}_{\infty}$ and
${L}_{2}$ norms with techniques described in [
28,
33] by taking
$\delta t=0.005$ at total time
$T=10$. In view of
Table 5, we can observe that the numerical accuracy of the LRBFFD is clearly better than the technique described in [
28,
33].
Table 7 lists the conservative invariant
E at several total times
T over spatial interval
$[40,240]$. One can observe that
E is conserved (up to 8 decimals) and the method can be well applied to investigate the solitary wave over a long time.
Figure 5 represents the longtime behavior of numerical solutions with
$N=500,\phantom{\rule{0.166667em}{0ex}}{n}_{i}=437$ and
$\delta t=1/1000$ at several total times
$T\in \{0,20,30,40,60\}$ over spatial interval
$[40,240]$. As observed in
Figure 5, the single solitons move to the right side with the preserved amplitude and shape.
Figure 6 shows the maximum norm errors
${L}_{\infty}$ at several total times
$T\in \{20,30,40,60\}$ with
$N=1200,\phantom{\rule{0.166667em}{0ex}}{n}_{i}=1153$ and
$\delta t=0.1$ over spatial interval
$[40,240]$.
Figure 7 depicts the numerical solution and corresponding maximum norm errors
${L}_{\infty}$ when
$\delta t=1/1000$,
$N=450$ and
${n}_{i}=379$ over spatial interval
$[40,240]$. Finally,
Figure 8 depicts the relevant matrix’s sparsity structures
$\mathbf{M}$ with
$N=110$ in the case of
${n}_{i}\in \{11,15\}$.
Example 3. Consider the Kawaharatype Equation (6) with parameters as $\alpha =\beta =\gamma =\sigma =1$, $\eta =\mu =0$ and $p=1$ on the space interval $[20,30]$ so that the exact solitary wave solution is The LRBFFD collocation method is adopted for solving this problem for different values of
$\delta t$,
N,
${n}_{i}$ and
T.
Table 8 presents the errors of numerical solutions by means of
${L}_{\infty}$,
${L}_{2}$ and
${L}_{\mathrm{rms}}$ norms, and computational times (in seconds) when
$T=1$ at several values of stencil sizes
${n}_{i}$ with
$\delta t=0.01$.
Table 9 represents the errors of approximate solutions based on
${L}_{\infty}$ and
${L}_{2}$ norms with techniques described in [
42,
43] at several values of time step
$\delta t$ for
$N=250$,
${n}_{i}=217$ and
$c=1.56$ over spatial interval
$[20,30].$ In view of
Table 8, we can see that the results by the proposed method show improvement over the techniques presented in [
42,
43]. Finally,
Figure 9 depicts the numerical solution and corresponding maximum norm errors when
$\delta t=0.1$,
$N=250$ and
${n}_{i}=235$ over spatial interval
$[20,30]$.