The present paper proposes a new unconditionally stable method to solve telegraph equation by using associated Hermite (AH) orthogonal functions. Unlike other numerical approaches, the time variables in the given equation can be handled analytically by AH basis functions. By using the Galerkin’s method, one can eliminate the time variables from calculations, which results in a series of implicit equations. And the coefficients of results for all orders can then be obtained by the expanded equations and the numerical results can be reconstructed during the computing process. The precision and stability of the proposed method are proved by some examples, which show the numerical solution acquired is acceptable when compared with some existing methods.

In this work, the following

During past years, much literatures have paid attention to the analysis and development of telegraph equation, see, for example, [

To construct an unconditional stability method to solve telegraph equation, Mohanty et al. [

In this study, we attempt to solve the telegraph equation using associated Hermite (AH) orthogonal functions. The Hermite functions, which were widely studied in the Hermite spectral method (HSM), are constructed by Hermite basis functions with a translated and scaled weighting function [

The fundamental aim of the literature is eliminating of the time variables in telegraph equation and then construct a new unconditionally stable method, the accuracy of which is absolutely independent of temporal step. In our presented method, the time variables in (

AH functions can be expressed as

The Hermite polynomials satisfy the following recursive relationship [

By introducing a time-translating parameter, AH functions could be transformed to a causal form:

Combining (

Combining (

For the second partial derivative of

The partial differential with respect to

Rewriting the telegraph equation (

In (

Discretize (

From (

Rewriting (

The application of presented method to telegraph equation based on a concealment condition that the initial valve of the equation should be 0; that is

Rewriting (

To solve (

Contrary to the other numerical approach, this presented method has an implicit relationship in each variables, which can be reflected in the sparse matrix [

In this part, the following four examples of the telegraph equation with exact solution have been solved by the proposed scheme. To measure the accuracy and versatility of this proposed scheme, the following

We consider the telegraph equation:

The exact solution of this equation is

Comparison of RMS error in Example

Methods | | | | | |
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Method in [ | 1/8 | | | | |

1/16 | | | | | |

1/32 | | | | | |

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Method in [ | 1/8 | | | | |

1/16 | | | | | |

1/32 | | | | | |

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Method in [ | 1/8 | | | | |

1/16 | | | | | |

1/32 | | | | | |

| |||||

Proposed method | 1/8 | | | | |

1/16 | | | | | |

1/32 | | | | |

Comparison of RMS error in Example

Methods | | | | | |
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Method in [ | | | | | |

| | | | | |

| | | | | |

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Method in [ | | | | | |

| | | | | |

| | | | | |

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Method in [ | | | | | |

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Proposed method | | | | | |

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Comparison of analytical and numerical result of Example

In this case, the telegraph equation (

We have

Errors in numerical result of Example

Methods | | | ||||
---|---|---|---|---|---|---|

| | RMS error | | | RMS error | |

Method in [ | | | | | | |

Method in [ | | | | | | |

Method in [ | | | — | | | — |

Proposed method | | | | | | |

RMS errors of Example

| Linear RBF [ | Cubic RBF [ | Method in [ | Proposed method |
---|---|---|---|---|

| | | | |

| | | | |

| | | | |

Space-time graph of numerical result up to

In this case, we consider telegraph equation (

In this problem, we have

Errors in numerical result of Example

Methods | Time | | | RMS error |
---|---|---|---|---|

Method in [ | | | | — |

Method in [ | | | | — |

Method in [ | | | | |

Method in [ | | | | |

Proposed method | | | | |

Proposed method | | | | |

Space-time graphs of numerical solution of Example

We consider telegraph equation with

The analytical solution of this equation is

Methods | Time | | | RMS error | CPU time (s) |
---|---|---|---|---|---|

Method in [ | | 1.4386 × 10^{−4} | 1.8479 × 10^{−5} | 1.4315 × 10^{−5} | 0 |

Method in [ | | 7.5545 × 10^{−5} | 1.0455 × 10^{−5} | 7.5170 × 10^{−6} | 2 |

Method in [ | | 4.5526 × 10^{−5} | 5.9153 × 10^{−5} | — | 0.43 |

Method in [ | | 3.0161 × 10^{−6} | 5.2032 × 10^{−6} | — | 1.46 |

Proposed method | | 1.7961 × 10^{−6} | 3.4326 × 10^{−7} | 1.7784 × 10^{−7} | 0.34 |

Proposed method | | 1.4776 × 10^{−6} | 2.2537 × 10^{−7} | 1.4631 × 10^{−7} | 0.35 |

Space-time graph of numerical result up to

In the literature, we have constructed a new unconditionally stable method to solve the telegraph equation. Since the time variables have been eliminated from computations, the convergence and precision of the presented method are independent of the temporal step. To demonstrate the stability and accuracy of this proposed method, four numerical examples are conducted and compared with the numerical results available in previous literatures. The comparison numerical results reveal the unconditional stability and high precision of the presented scheme in the current literature.

The paper has provided a new mentality for solving

The authors declare that they have no competing interests regarding the publication of this paper.

^{2}+

^{4}) for solving the second-order 1D linear hyperbolic equation