Numerical solutions of the second-order one-dimensional hyperbolic telegraph equations are presented using the radial basis functions. The purpose of this paper is to propose a simple novel direct meshless scheme for solving hyperbolic telegraph equations. This is fulfilled by considering time variable as normal space variable. Under this scheme, there is no need to remove time-dependent variable during the whole solution process. Since the numerical solution accuracy depends on the condition of coefficient matrix derived from the radial basis function method. We propose a simple shifted domain method, which can avoid the full-coefficient interpolation matrix easily. Numerical experiments performed with the proposed numerical scheme for several second-order hyperbolic telegraph equations are presented with some discussions.

The telegraph equation, which has been used to describe phenomena in various fields, belongs to the hyperbolic partial differential equation scope. For example, the telegraph equation in (1 + 1) dimensions can model the vibrations of structures, the digital propagation and also has applications in the other fields [

Especially, there are several numerical methods concentrate on the second-order 1D linear hyperbolic telegraph equations. For example, Mohanty et al. [

Based on the above-mentioned investigations, we propose a direct meshless scheme with one-level approximation for the second-order 1D linear hyperbolic telegraph equations. This is fulfilled by considering time variable as normal space variable. There is no need to remove time-dependent variable during the whole solution process. Under this scheme, we can solve the hyperbolic telegraph equations in a direct way. The rest paper is organized as follows. The formulation of the direct radial basis function is briefly introduced with the methodology for the hyperbolic telegraph equations in Section 2. To cope with the full coefficient matrix derived from the radial basis function method, we propose a simple shifted domain method in Section 3. Section 4 presented some numerical examples to validate the applicability of the proposed direct meshless scheme. Finally, some conclusions are given in Section 5.

The general mathematical formulation of second-order linear hyperbolic telegraph equation in (1 + 1) dimensions is

in terms with the initial condition

and boundary conditions

Here, the coefficient

Almost all numerical techniques for

As is known to all, the traditional RBF methods are mostly used to solve 2D or higher dimensional problems. However, there is only one space variable

The

Actually, there is another definition of not

According to the definition of DRBF, the above-mentioned

We should seek for the unknown coefficients

The interpolation scheme upon which the

with

Here, we use

where

It should be noted that for a relatively large physical domain (with large

The procedure of the SDM is shown by the above-mentioned physical domain

This procedure is same as presented in the above-mentioned physical domain

where

For the other cases, the configuration of the shifted domain in the horizontal direction

In this section, three examples are considered to validate the DRBF. For fair comparison with the other numerical methods, we use the maximum absolute error (MAE), absolute error and root mean square error (RMSE). The RMSE is defined as [

where

In order to investigate the DRBF method with the shifted domain method, we consider the hyperbolic telegraph

and boundary condition

The corresponding coefficients are

The source term is

In this example, the physical domain

Methods | Time | RMSE | CPU time(s) |
---|---|---|---|

DRBF | 0.5 | 4.86E−07 | 0.76 |

DRBF | 1.0 | 6.60E−07 | 0.80 |

Reference [ |
0.5 | 2.33E−06 | 3.04 |

Reference [ |
1.0 | 4.37E−06 | 4.89 |

Reference [ |
0.5 | 8.75E−06 | 2.52 |

Reference [ |
1.0 | 5.07E−06 | 3.63 |

Reference [ |
0.5 | 7.95E−05 | 5.00 |

Reference [ |
1.0 | 1.46E−04 | 12.00 |

For different mesh sizes

In order to see the performance of the DRBF with different coefficients, we consider the hyperbolic telegraph

and boundary condition

The corresponding analytical/exact solution

with source term

For fair comparison with the other methods in [

Methods | MAE | |
---|---|---|

DRBF | 10,5 | 2.09E−07 |

DRBF | 20,10 | 5.97E−09 |

Reference [ |
10,5 | 2.10E−08 |

Reference [ |
20,10 | 3.70E−08 |

Reference [ |
10,5 | 3.40E−07 |

Reference [ |
20,10 | 4.20E−07 |

Reference [ |
10,5 | 2.00E−06 |

Reference [ |
20,10 | 2.40E−06 |

In order to see the difference between the exact and DRBF approximate solutions,

A new direct meshless scheme is presented for the second-order hyperbolic telegraph equations in (1 + 1) dimensions. The present numerical procedure, in which the time variable is considered as normal space variable, is based on the time-dependent radial basis function. There is no need to remove time-dependent variable during the whole solution process. Besides, a simple shifted domain method is proposed to cope with the solution accuracy related to the ill-conditioned coefficient matrix. From the numerical results in Section 4, we find that the proposed meshless method is superior to the other numerical methods. Besides, the direct meshless method can be extended to solve nonlinear problems with Newton iterative method considered. The DRBF with the shifted domain method is promising in dealing with the other types of time-dependent problems, fractional problems [