The numerical solution of nonlinear generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations via the meshless method of integrated radial basis functions

Abstract

In this paper, a numerical technique is proposed for solving the nonlinear generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations. The used numerical method is based on the integrated radial basis functions (IRBFs). First, the time derivative has been approximated using a finite difference scheme. Then, the IRBF method is developed to approximate the spatial derivatives. The two-dimensional version of these equations is solved using the presented method on different computational geometries such as the rectangular, triangular, circular and butterfly domains and also other irregular regions. The aim of this paper is to show that the integrated radial basis function method is also suitable for solving nonlinear partial differential equations. Numerical examples confirm the efficiency of the proposed scheme.

Appendix

Since we used the IRBF method, only the time variable is discretized by the following difference scheme:

$$\begin{aligned}&\frac{U^{n+1} - U^{n}}{\hbox {d}t} - \frac{\Delta U^{n+1} - \Delta U^{n}}{\hbox {d}t} - \frac{\Delta U^{n+1} + \Delta U^{n}}{2}\nonumber \\&\quad + \frac{\nabla \cdot U^{n+1} + \nabla \cdot U^{n}}{2} -\nabla \cdot (F(U^n))=0, \end{aligned}$$

(5.1)

we replace $U^{n+1}$ in Eq. (5.1) with Taylor series expansions as follows:

$$\begin{aligned} U^{n+1}=U^{n}+\hbox {d}t U_{t}^{n} + \frac{\hbox {d}t^2}{2!} U_{tt}^{n}+\frac{\hbox {d}t^3}{3!} U_{ttt}^{n} +\cdots . \end{aligned}$$

(5.2)

Substituting Eq. (5.2) into Eq. (5.1), we have

$$\begin{aligned} \begin{aligned}&\frac{1}{\hbox {d}t}\left( U^{n}+\hbox {d}t U_{t}^{n} + \frac{\hbox {d}t^2}{2!} U_{tt}^{n}+\frac{\hbox {d}t^3}{3!} U_{ttt}^{n} +\dots -U^{n}\right) \\&\quad -\frac{1}{\hbox {d}t}\left( U_{xx}^{n}+\hbox {d}t U_{txx}^{n} + \frac{\hbox {d}t^2}{2!}U_{ttxx}^{n}\right. \\&\quad \left. +\frac{\hbox {d}t^3}{3!} U_{tttxx}^{n} +\dots -U_{xx}^{n}\right) \\&\quad -\frac{1}{2}\left( U_{xx}^{n}+\hbox {d}t U_{txx}^{n} + \frac{\hbox {d}t^2}{2!} U_{ttxx}^{n}+\frac{\hbox {d}t^3}{3!} U_{tttxx}^{n} +\dots +U_{xx}^{n}\right) \\&\quad +\frac{1}{2}\left( U_{x}^{n}+\hbox {d}t U_{tx}^{n} + \frac{\hbox {d}t^2}{2!} U_{ttx}^{n}\right. \\&\quad \left. +\frac{\hbox {d}t^3}{3!} U_{tttx}^{n} +\dots +U_{x}^{n}\right) -\nabla .\left( F\left( U^n\right) \right) =0. \end{aligned} \end{aligned}$$

(5.3)

Rearranging Eq. (5.3), we obtain

$$\begin{aligned}&U_{t}^{n}-U_{txx}^{n}-U_{xx}^n+U_{x}^{n}-(F(U^{n}))_x\\&\quad =\frac{\hbox {d}t}{2!} U_{tt}^{n}+U_{txx}^{n}+\frac{\hbox {d}t}{2} U_{txx}^{n}+\frac{\hbox {d}t}{2} U_{tx}^{n}+\cdots , \end{aligned}$$

where

$$\begin{aligned} \hbox {LHS}:= U^{n}-U_{txx}^{n}-U_{xx}^n+U_{x}^{n}-(F(U^{n}))_x, \end{aligned}$$

and

$$\begin{aligned} \hbox {RHS}:=\frac{\hbox {d}t}{2!} U_{tt}^{n}+\frac{\hbox {d}t}{2} U_{tx}^{n}+U_{txx}^{n}+\frac{\hbox {d}t}{2} U_{txx}^{n}+\frac{\hbox {d}t}{2!} U_{ttxx}^{n}+\cdots . \end{aligned}$$

Substituting $U_{tt}^{n}-U_{ttxx}^{n}=U_{txx}^{n}-U_{tx}^{n}$ in RHS, we have

$$\begin{aligned}&\hbox {RHS}:=\frac{\hbox {d}t}{2}\left( U_{txx}^{n}-U_{tx}^{n}\right) \\&\qquad +\,\frac{\hbox {d}t}{2} U_{tx}^{n}+\left( 1+\frac{\hbox {d}t}{2}\right) U_{txx}^{n} +\dots =(1+\hbox {d}t)U_{txx}+\cdots . \end{aligned}$$

The lowest order partial derivative in RHS is $\frac{\partial ^2 u}{\partial x^2}$. This term represents the dissipative aspect of the physical viscosity on the flow. Thus, the obtained solution by the difference scheme is dissipative in nature.