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<p>The telegraph equation plays a significant role in modeling wave motion affected by damping and dispersion. Herein, a robust space-time Tau spectral method is developed based on the use of shifted Lucas polynomials as basis functions in both spatial and temporal dimensions. Explicit formulas for these polynomials are derived and employed to construct the proposed numerical algorithm. The main idea is to apply the Tau framework so that the governing equation, together with its boundary and initial conditions, is transformed into an equivalent system of algebraic equations by means of suitable tensor product properties. The convergence and stability of the method are analyzed within the setting of a Hilbert space. Several numerical examples are included to assess the performance of the technique. The obtained results show that the method provides very high accuracy and efficient computation, exhibiting the expected spectral (exponential) convergence rate. For moderate polynomial degrees, the method achieves errors at or near machine precision, demonstrating its effectiveness for the telegraph equation.</p>

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Shifted Lucas polynomial-driven Tau approach for the telegraph equation: theoretical and numerical insights into accuracy and stability

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年5月13日

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  • M. H. Salama,
  • H. A. Zedan,
  • W. M. Abd-Elhameed,
  • Y. H. Youssri

摘要

The telegraph equation plays a significant role in modeling wave motion affected by damping and dispersion. Herein, a robust space-time Tau spectral method is developed based on the use of shifted Lucas polynomials as basis functions in both spatial and temporal dimensions. Explicit formulas for these polynomials are derived and employed to construct the proposed numerical algorithm. The main idea is to apply the Tau framework so that the governing equation, together with its boundary and initial conditions, is transformed into an equivalent system of algebraic equations by means of suitable tensor product properties. The convergence and stability of the method are analyzed within the setting of a Hilbert space. Several numerical examples are included to assess the performance of the technique. The obtained results show that the method provides very high accuracy and efficient computation, exhibiting the expected spectral (exponential) convergence rate. For moderate polynomial degrees, the method achieves errors at or near machine precision, demonstrating its effectiveness for the telegraph equation.