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An accurate localized meshfree collocation technique for the telegraph equation in propagation of electrical signals

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Abstract

This paper presents an accurate localized meshfree collocation technique for the approximate solution of the second-order two-dimensional telegraph model. This model is an useful description of the propagation of electrical signals in a transmission line as well as wave phenomena. The proposed algorithm approximates the unknown solution in two steps. First, the discretization of time variable is accomplished by the Crank–Nicolson finite difference. Additionally, the unconditional stability and the convergence of the temporal semi-discretization approach are analysed with the help of the energy method in an appropriate Sobolev space. Second, the discretization of the spatial variable and its partial derivatives is obtained by the localized radial basis function partition of unity collocation method. The global collocation methods pose a considerable computational burden due to the calculation of the dense algebraic system. With the proposed approach, the domain is decomposed into several subdomains via a kernel approximation on every local domain. Therefore, it is possible to make the algebraic system more sparse and, consequently, to achieve a small condition number and a limited computational cost. Three numerical examples support the theoretical study and highlight the effectiveness of the method.

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Acknowledgements

The authors are very grateful to the editors and the anonymous referees for useful comments and suggestions that contributed to improving the quality of the paper. The authors are also thankful to Prof. Dr. Alexandra Galhano from the Institute of Engineering, Polytechnic of Porto, for the discussions and her helpful comments.

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Authors and Affiliations

  1. School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran

    O. Nikan

  2. Department of Applied Mathematics, Xi’an Jiaotong-Liverpool University, Suzhou, 215123, China

    Z. Avazzadeh

  3. Institute of Engineering, Polytechnic of Porto, Rua Dr. António B. de Almeida 431, 4249-015, Porto, Portugal

    J. A. Tenreiro Machado

  4. Department of Mathematics, Velayat University, Iranshahr, Iran

    M. N. Rasoulizadeh

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  1. O. Nikan
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Correspondence to Z. Avazzadeh.

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Nikan, O., Avazzadeh, Z., Machado, J.A.T. et al. An accurate localized meshfree collocation technique for the telegraph equation in propagation of electrical signals. Engineering with Computers 39, 2327–2344 (2023). https://doi.org/10.1007/s00366-022-01630-9

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