We obtain the numerical solution of a Boussinesq system for twoway propagation of nonlinear dispersive waves by using the meshless method, based on collocation with radial basis functions. The system of nonlinear partial differential equation is discretized in space by approximating the solution using radial basis functions. The discretization leads to a system of coupled nonlinear ordinary differential equations. The equations are then solved by using the fourthorder RungeKutta method. A stability analysis is provided and then the accuracy of method is tested by comparing it with the exact solitary solutions of the Boussinesq system. In addition, the conserved quantities are calculated numerically and compared to an exact solution. The numerical results show excellent agreement with the analytical solution and the calculated conserved quantities.
Consider the initial and boundary value problem
This work studies the numerical solution of this system by means of radial basis functions (RBFs). The use of these types of basis functions has become very popular in recent times; see for instance the work of Buhmann [
We will examine the two cases for the Boussinesq type system. The first one we examine is the Bona and Chen system, which is important in the theory of dispersive waves,
The RBFs approach is one of the most effective meshless methods to solve numerically PDEs. The key feature of the RBFs is that its implementation does not require a mesh at all. The method employs approximants whose values depend only on the distance between some center point and a domain point, in an appropriate norm. Due to the use of the distance functions, we can control the convergence up to an order proportional to the spacing of two points where we evaluate such functions [
The functions
In terms of the RBFs, we can find the derivatives of the approximate solution of
If we substitute (
We solve the system over the real line
Additionally, the discretization of the boundary conditions, equations (
To solve the system (
The classical RK4 algorithm for a system of ODEs is given by
The RK4 scheme does not have stability issues as long as the time step
For the stability analysis nonlinearity in (
In this section, we use the proposed algorithm to calculate the numerical solution of the Boussinesq type system and its conserved quantities. These quantities were performed by means of the trapezoidal quadrature. The accuracy of the method is tested by computing the
The integrals involving the conserved quantities were calculated by using the Trapezoidal rule.
The first test problem was studied extensively by Chen in [
Error norms of
Method 



Amplitude 


Analytic  —  —  —  7.462309  37.43 
MQ  1.23 


7.462316  37.43 
G  2.66 


7.462310  37.43 
IQ  1.00 


7.462390  37.43 
IMQ  2.05 


7.462565  37.43 
Error norms of
Method 



Amplitude (PP) 



Analytic  —  —  —  4.91342  36.23  37.43 
MQ  1.23 


4.91343  36.23  37.43 
G  2.66 


4.91339  36.23  37.43 
IQ  1.00 


4.91575  36.23  37.43 
IMQ  2.05 


4.91481  36.23  37.43 
Conserved quantities.
Method 





Analytic  15.81139  0.0000  158.113  −22.5876975726 
MQ  15.81134 

158.113  −22.58776106781 
G  15.81139 

158.113  −22.58776062211 
IQ  15.81144 

158.168  −22.58432200892 
IMQ  15.81156 

158.152  −22.58512792633 
Comparison of
Soliton interaction of Boussinesq type system.
Evolution of
Evolution of
In this example we test our algorithm with a solution against the “full system” equations (
Error norms and conserved quantities for
Time 







MQ  
0.5 




1.7748239  1.6733200 
1.0 




1.7748238  1.6733198 
2.0 




1.7748235  1.6733196 
3.0 




1.7748233  1.6733193 
4.0 




1.7748231  1.6733195 
5.0 




1.7748232  1.6733192 
10.0 




1.7748237  1.6733193 
15.0 




1.7748246  1.6733198 
20.0 




1.7748260  1.6733205 
25.0 




1.7748285  1.6733212 
30.0 




1.7748236  1.6733103 
Error norms and conserved quantities for
Time 







G  
0.5 




1.7748239  1.6733200 
1.0 




1.7748238  1.6733198 
2.0 




1.7748235  1.6733196 
3.0 




1.7748233  1.6733193 
4.0 




1.7748231  1.6733195 
5.0 




1.7748232  1.6733192 
10.0 




1.7748237  1.6733193 
15.0 




1.7748246  1.6733198 
20.0 




1.7748260  1.6733205 
25.0 




1.7748285  1.6733212 
30.0 




1.7748236  1.6733103 
Error norms and conserved quantities for
Time 







IQ  
0.5 




1.7748239  1.6733200 
1.0 




1.7748238  1.6733198 
2.0 




1.7748235  1.6733196 
3.0 




1.7748233  1.6733193 
4.0 




1.7748231  1.6733195 
5.0 




1.7748232  1.6733192 
10.0 




1.7748237  1.6733193 
15.0 




1.7748246  1.6733198 
20.0 




1.7748260  1.6733205 
25.0 




1.7748285  1.6733212 
30.0 




1.7748236  1.6733103 
Error norms and conserved quantities for
Time 







IMQ  
0.5 




1.7748239  1.6733200 
1.0 




1.7748238  1.6733198 
2.0 




1.7748235  1.6733196 
3.0 




1.7748233  1.6733193 
4.0 




1.7748231  1.6733195 
5.0 




1.7748232  1.6733192 
10.0 




1.7748237  1.6733193 
15.0 




1.7748246  1.6733198 
20.0 




1.7748260  1.6733205 
25.0 




1.7748285  1.6733212 
30.0 




1.7748236  1.6733103 
In this last example, we present the interaction between two solitons. The initial conditions are taken to be
Solitons are permanent in form.
Solitons are localized within a region.
Solitons can interact with other solitons and emerge from the collision unchanged, except for a phase shift.
Convergence of mesh error.
We have solved numerically the Boussinesq type system of nonlinear PDE by means of RBFs. In general this algorithm solves the system accurately. In particular, this method has optimal results for multiquadric and Gaussian radial basis functions. The conserved quantities were calculated and show good agreement with the analytical results. In the final section we ran a simulation for interacting solitons that is in agreement with the known properties of solitons. In conclusion the meshless method is shown to give good results for a system of PDEs having high order derivatives and mixedpartial derivatives.
The authors declare that there is no conflict of interests regarding the publication of this paper.