On the use of an accurate implicit spectral approach for the telegraph equation in propagation of electrical signals

Abstract

This paper focuses on solving numerically the telegrapher problems in the two-dimensional case, which is a well-known second-order differential equation using a relatively innovative methodology and approach. In terms of approach, we have applied a new robust true mesh-free local approach combining the spectral method and and the Newmark scheme for space and time approximations respectively. Unlike several meshless methods, the proposed mesh-free can be based only on a unidimensional study which will be extended to the higher-dimensional problems by a Kronecker product. In this way, the shape functions relative to this approach can be easily calculated with reasonably both computer space and time. The comparison between the obtained results and with available ones in the literature shows that the Mesh-free Spectral with Newmark Method (MS-NM) approach is a suitable and competitive approach for resolving time-space differential equations.

Appendix A. Implementation details and technique to compute the cited errors

For the readers interested in the calculation of the errors, we specify in this section the details of the implementation in Matlab and the use of the errors in order to verify the numerical precision and to have a clearer view on the results. We provide specific notes concerning the relative error $L_{\text {re}}$, the relative norm $L_{2}$ of the error as well as the maximum absolute errors and root mean square errors $L_{\text {rms}}$ with the help of the following formulas:

$$\begin{aligned} \left\{ \begin{array}{*{5}{l}} L_{\propto } &{}=&{} max \ |v(x_{j},y_{j},T)-u(x_{j},y_{j},T)|&{};&{} {1\le j\le N-1},\\ \\ L_{2} &{}=&{} \left( \sum \limits _{j=1}^{N} v(x_{j},y_{j},T)-u(x_{j},y_{j},T)\right) ^{\frac{1}{2}}&{};&{} {1\le j\le N-1},\\ \\ L_{\text {rms}} &{}=&{} \left( \dfrac{1}{N} \sum \limits _{j=1}^{N} \left( v(x_{j},y_{j},T)-u(x_{j},y_{j},T)\right) ^2\right) ^{\frac{1}{2}}&{};&{} {1\le j\le N-1}, \\ \\ L_{\text {re}} &{}=&{} \left( \dfrac{\sum \nolimits _{j=1}^{N} \left( v(x_{j},y_{j},T)-u(x_{j},y_{j},T)\right) ^2}{\sum _{j=1}^{N} u^2(x_{j},y_{j},T)} \right) ^{\frac{1}{2}}&{};&{} {1\le j\le N-1}. \end{array} \right. \end{aligned}$$

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