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Nonlocal theory on plane waves in higher order thermo-porous functionally graded semiconductor

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Abstract

The current study deals with reflected waves through a porous medium with viscoelastic properties. The considered continuum media is a semiconductor with elastic and thermal properties which varies exponentially along the depth of the medium. The thermal disturbance caused by elastic propagation is then encountered by the refined phase-lag heat conduction model. The scattering relation for the coupled waves has been computed. It is observed that the functionally graded, voids and non-local parameters reduce the amplitude ratios of reflected waves propagating through the medium. The solution in the form of amplitude ratios for reflected waves has been obtained analytically and represented graphically for a particular medium.

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Authors and Affiliations

  1. Department of Mathematics, PMAS Arid Agriculture University Rawalpindi, Rawalpindi, Pakistan

    Hashmat Ali

  2. Department of Mathematics, COMSATS University Islamabad, Islamabad Campus, Islamabad, Pakistan

    Asad Mahmood & Qazi Muhammad Zaigham Zia

  3. Department of Mathematics, COMSATS University Islamabad, Wah Campus, Islamabad, Pakistan

    Adnan Jahangir

  4. GST-Mathematics Division, Veritas University Abuja, Bwari-Abuja, Abuja, Nigeria

    Augustine Igwebuike Anya

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  1. Hashmat Ali
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  2. Asad Mahmood
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  3. Adnan Jahangir
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  4. Qazi Muhammad Zaigham Zia
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  5. Augustine Igwebuike Anya
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Appendices

Appendix-1

$$ L_{1} = N_{14} \left( {N_{1} N_{9} - N_{3}^{2} } \right) $$
$$ L_{2} = N_{14} \left\{ { - iN_{1} N_{2} + N_{4} N_{3} + N_{3} N_{8} - N_{2} N_{9} - N_{17} \left( {N_{1} N_{9} - N_{3}^{2} } \right)} \right\} $$
$$ \begin{aligned} L_{3} &= N_{11} \left( {N_{5} N_{9} - N_{3} N_{10} } \right) + N_{13} \left( {N_{3}^{2} - N_{1} N_{9} } \right) + N_{14} \left( {\omega^{2} N_{1} - N_{4} N_{8} + \omega^{2} N_{9} + iN_{2}^{2} } \right) \hfill \\ &\quad + \left( {N_{3}^{2} N_{14} - N_{1} N_{9} N_{14} } \right)\left( {C_{\alpha } - \pi } \right) + N_{14} N_{15} \left( {i\overline{A}_{4} N_{3} \cos \theta - N_{6} N_{9} } \right) \hfill \\ &\quad + N_{14} N_{16} \left( {N_{6} N_{3} - i\overline{A}_{4} N_{1} \cos \theta } \right) + N_{14} N_{17} \left( {N_{4} N_{3} - iN_{1} N_{2} + N_{3} N_{8} - N_{2} N_{9} } \right) \hfill \\ \end{aligned} $$
$$ \begin{aligned} L_{4} &= N_{11} \left( {\overline{A}_{3} \varepsilon_{2} N_{3} - iN_{5} N_{2} + N_{4} N_{10} } \right) + N_{12} \left( {N_{5} N_{9} - N_{3} N_{10} } \right) \hfill \\ &\quad + N_{13} \left( {iN_{1} N_{2} - N_{4} N_{3} - N_{3} N_{8} + N_{2} N_{9} } \right) - i\omega^{2} N_{2} N_{14} \left( {i + 1} \right) \hfill \\ &\quad - \left( {N_{4} N_{3} N_{14} + N_{3} N_{8} N_{14} - N_{2} N_{9} N_{14} + iN_{1} N_{2} N_{14} } \right)\left( {C_{\alpha } - \pi } \right) \hfill \\ &\quad + N_{14} N_{15} \left( {\varepsilon_{2} \overline{A}_{4} N_{3} - i\overline{A}_{4} N_{4} \cos \theta + iN_{6} N_{2} } \right) + \overline{A}_{4} N_{14} N_{16} \left( {iN_{2} \cos \theta - \varepsilon_{2} N_{1} } \right) \hfill \\ &\quad + N_{14} N_{16} \left( {i\overline{A}_{4} N_{2} \cos \theta - N_{6} N_{8} } \right) + N_{11} N_{17} \left( {N_{5} N_{9} - N_{3} N_{10} } \right) + N_{13} N_{17} \left( {N_{3}^{2} - N_{1} N_{9} } \right) \hfill \\ &\quad + N_{14} N_{17} \left( {\omega^{2} N_{1} + iN_{2}^{2} } \right) - N_{14} N_{17} \left( {\pi N_{3}^{2} + N_{4} N_{8} } \right) + N_{14} N_{17} \left( {\omega^{2} N_{9} + \pi N_{1} N_{9} } \right) \hfill \\ \end{aligned} $$
$$ \begin{aligned} L_{5}& = N_{11} \left( { - \varepsilon_{2} \overline{A}_{3} N_{4} + \omega^{2} N_{5} } \right) + N_{12} \left( {\varepsilon_{2} \overline{A}_{3} N_{3} - iN_{5} N_{2} + N_{4} N_{10} } \right) - N_{13} \left( {\omega^{2} N_{1} + iN_{2}^{2} } \right) \hfill \\ \quad + \left( \begin{gathered} N_{3} N_{10} N_{11} - N_{3}^{2} N_{13} + N_{1} N_{9} N_{13} - iN_{2}^{2} N_{14} + N_{4} N_{8} N_{14} - \omega^{2} N_{9} N_{14} \hfill \\ \quad- N_{5} N_{9} N_{11} - N_{1} \omega^{2} N_{14} \hfill \\ \end{gathered} \right)\left( {C_{\alpha } - \pi } \right) \hfill \\ &\quad + N_{13} \left( {N_{4} N_{8} - \omega^{2} N_{9} } \right) + \omega^{4} N_{14} + \left( {N_{3}^{2} N_{14} - N_{1} N_{9} N_{14} } \right)\left( {\pi C_{\alpha } + \varepsilon_{1} D_{\alpha } } \right) \hfill \\ &\quad + \overline{A}_{15} N_{15} \left( {N_{5} N_{9} - N_{3} N_{10} } \right) + N_{13} N_{15} \left( {N_{6} N_{9} - i\overline{A}_{4} N_{3} cos\theta } \right) + iN_{3} N_{14} N_{15} \cos \theta \left( {\pi \overline{A}_{4} - \varepsilon_{1} \overline{A}_{5} } \right) \hfill \\ &\quad - N_{14} N_{15} \left( {\varepsilon_{2} \overline{A}_{4} N_{4} + \omega^{2} N_{6} } \right) + \left( {N_{9} N_{14} N_{15} - N_{3} N_{14} N_{16} } \right)\left( {\varepsilon_{1} N_{7} - \pi N_{6} } \right) + \overline{A}_{15} N_{16} \left( {N_{1} N_{10} - N_{5} N_{3} } \right) \hfill \\ &\quad + N_{11} N_{16} \left( {N_{6} N_{10} - i\overline{A}_{4} N_{5} \cos \theta } \right) + N_{13} N_{16} \left( {i\overline{A}_{4} N_{1} \cos \theta - N_{6} N_{3} } \right) - i\overline{A}_{4} N_{14} N_{16} \cos \theta \left( {\omega^{2} + \pi N_{1} } \right) \hfill \\ &\quad + N_{14} N_{16} \left( {i\varepsilon_{1} \overline{A}_{5} N_{1} \cos \theta + \varepsilon_{2} \overline{A}_{4} N_{2} } \right) + N_{11} N_{17} \left( {\varepsilon_{2} \overline{A}_{3} N_{3} - iN_{5} N_{2} + N_{4} N_{10} } \right) + N_{12} N_{17} \left( {N_{5} N_{9} - N_{3} N_{10} } \right) \hfill \\ &\quad + N_{13} N_{17} \left( {iN_{1} N_{2} - N_{4} N_{3} - N_{3} N_{8} + N_{2} N_{9} } \right) + N_{14} N_{17} \left( {\pi N_{4} N_{3} - i\omega^{2} N_{2} - i\pi N_{1} N_{2} - \omega^{2} N_{2} + \pi N_{3} N_{8} - \pi N_{2} N_{9} } \right) \hfill \\ \end{aligned} $$
$$ \begin{aligned} L_{6} & = \left( \begin{gathered} - \varepsilon_{2} \overline{A}_{3} N_{3} N_{11} + iN_{5} N_{2} N_{11} - N_{4} N_{10} N_{11} - N_{5} N_{9} N_{12} + N_{3} N_{10} N_{12} \hfill \\ - iN_{1} N_{2} N_{13} + N_{4} N_{3} N_{13} + N_{3} N_{8} N_{13} - N_{2} N_{9} N_{13} + i\omega^{2} N_{2} N_{14} + \omega^{2} N_{2} N_{14} \hfill \\ \end{gathered} \right)\left( {C_{\alpha } - \pi } \right) \hfill \\ &\quad + N_{12} \left( {\omega^{2} N_{5} - \varepsilon_{2} \overline{A}_{3} N_{4} } \right) + \left( {iN_{1} N_{2} N_{14} - N_{4} N_{3} N_{14} - N_{3} N_{8} N_{14} + N_{2} N_{9} N_{14} } \right)\left( {\pi C_{a} + D_{a} \varepsilon_{1} } \right) \hfill \\ &\quad + N_{12} \left( {\omega^{2} N_{5} - \varepsilon_{2} \overline{A}_{3} N_{4} } \right) + \left( {iN_{1} N_{2} N_{14} - N_{4} N_{3} N_{14} - N_{3} N_{8} N_{14} + N_{2} N_{9} N_{14} } \right)\left( {\pi C_{a} + D_{a} \varepsilon_{1} } \right) \hfill \\ &\quad + \overline{A}_{15} N_{15} \left( {\overline{A}_{3} \varepsilon_{2} N_{3} - iN_{5} N_{2} + N_{4} N_{10} } \right) + N_{13} N_{15} \left( {i\overline{A}_{4} N_{4} \cos \theta - \varepsilon_{2} \overline{A}_{4} N_{3} - iN_{6} N_{2} } \right) \hfill \\ &\quad + \omega^{2} N_{2} N_{13} \left( {i + 1} \right) + \left( {\varepsilon_{2} N_{3} N_{14} N_{15} - iN_{4} N_{14} N_{15} \cos \theta + iN_{2} N_{14} N_{16} \cos \theta } \right)\left( {\pi \overline{A}_{4} - \overline{A}_{5} \varepsilon_{1} } \right) \hfill \\ &\quad + \overline{A}_{15} N_{16} \left( {N_{5} N_{8} - \varepsilon_{2} \overline{A}_{3} N_{1} } \right) + \left( {iN_{2} N_{14} N_{15} - N_{8} N_{14} N_{16} } \right)\left( {\pi N_{6} - \varepsilon_{1} N_{7} } \right) \hfill \\ &\quad - N_{16} \left( {\overline{A}_{15} N_{2} N_{10} + \overline{A}_{4} \varepsilon_{2} N_{5} N_{11} + \overline{A}_{3} \varepsilon_{2} N_{6} N_{11} } \right) + N_{12} N_{16} \left( {N_{6} N_{10} - i\overline{A}_{4} N_{5} \cos \theta } \right) \hfill \\ &\quad + N_{13} N_{16} \left( {\overline{A}_{4} \varepsilon_{2} N_{1} - i\overline{A}_{4} N_{2} \cos \theta + N_{6} N_{8} } \right) + \varepsilon_{2} N_{14} N_{16} \left( {\overline{A}_{5} \varepsilon_{1} N_{1} - \omega^{2} \overline{A}_{4} - \pi \overline{A}_{4} N_{1} } \right) \hfill \\ &\quad + N_{11} N_{17} \left( {\omega^{2} N_{5} - \overline{A}_{3} \varepsilon_{2} N_{4} + \pi N_{5} N_{9} - \pi N_{3} N_{10} } \right) + N_{12} N_{17} \left( {\overline{A}_{3} \varepsilon_{2} N_{3} - iN_{5} N_{2} + N_{4} N_{10} } \right) \hfill \\ &\quad + N_{13} N_{17} \left( {\pi N_{3}^{2} - \omega^{2} N_{1} - iN_{2}^{2} + N_{4} N_{8} - \omega^{2} N_{9} - \pi N_{1} N_{9} } \right) \hfill \\ &\quad + N_{14} N_{17} \left( {\omega^{4} + \pi \omega^{2} N_{1} + i\pi N_{2}^{2} - \pi N_{4} N_{8} + \pi \omega^{2} N_{9} } \right) \hfill \\ \end{aligned} $$
$$ \begin{aligned} L_{7} & = \left( \begin{gathered} \overline{A}_{3} \varepsilon_{2} N_{4} N_{11} - \omega^{2} N_{5} N_{11} - \overline{A}_{3} \varepsilon_{2} N_{3} N_{12} + iN_{5} N_{2} N_{12} - N_{4} N_{10} N_{12} + \omega^{2} N_{1} N_{13} \hfill \\ + iN_{2}^{2} N_{13} - N_{4} N_{8} N_{13} + \omega^{2} N_{9} N_{13} - \omega^{4} N_{14} \hfill \\ \end{gathered} \right)\left( {C_{a} - \pi } \right) \hfill \\ &\quad + \left( \begin{gathered} N_{1} N_{9} N_{13} - N_{3}^{2} N_{13} - \omega^{2} N_{1} N_{14} - iN_{2}^{2} N_{14} + N_{4} N_{8} N_{14} - \omega^{2} N_{9} N_{14} \hfill \\ - N_{5} N_{9} N_{11} + N_{3} N_{10} N_{11} \hfill \\ \end{gathered} \right)\left( {\pi C_{a} + D_{a} \varepsilon_{1} } \right) \hfill \\ &\quad + \overline{A}_{15} N_{15} \left( {\omega^{2} N_{5} - \overline{A}_{3} \varepsilon_{2} N_{4} + \pi N_{5} N_{9} - \pi N_{3} N_{10} } \right) + N_{13} N_{15} \left( {\overline{A}_{4} \varepsilon_{2} N_{4} + \omega^{2} N_{6} } \right) \hfill \\ &\quad + \left( {N_{9} N_{13} N_{15} - N_{3} N_{13} N_{16} - \omega^{2} N_{14} N_{15} + N_{10} N_{11} N_{16} } \right)\left( {\pi N_{6} - \varepsilon_{1} N_{7} } \right) - \omega^{4} N_{13} \hfill \\ &\quad + \left( \begin{gathered} \varepsilon_{2} N_{4} N_{14} N_{15} + iN_{5} N_{11} N_{16} \cos \theta + i\omega^{2} N_{14} N_{16} \cos \theta - \varepsilon_{2} N_{2} N_{14} N_{16} \hfill \\ + iN_{3} N_{13} N_{15} \cos \theta \hfill \\ \end{gathered} \right)\left( {\overline{A}_{5} \varepsilon_{1} - \pi \overline{A}_{4} } \right) \hfill \\ &\quad + \overline{A}_{15} N_{16} \left( {\overline{A}_{3} \varepsilon_{2} N_{2} - \pi N_{5} N_{3} } \right) - \varepsilon_{2} N_{12} N_{16} \left( {\overline{A}_{4} N_{5} + \overline{A}_{3} N_{6} } \right) + \left( \begin{gathered} i\overline{A}_{4} N_{13} N_{16} \cos \theta \hfill \\ + iN_{2} N_{13} N_{17} + \overline{A}_{15} N_{10} N_{16} \hfill \\ \end{gathered} \right)\left( {\omega^{2} + \pi N_{1} } \right) \hfill \\ &\quad - N_{13} N_{16} \left( {i\overline{A}_{5} \varepsilon_{1} N_{1} \cos \theta + \overline{A}_{4} \varepsilon_{2} N_{2} } \right) + \pi N_{11} N_{17} \left( {\overline{A}_{3} \varepsilon_{2} N_{3} - iN_{5} N_{2} } \right) + N_{4} N_{17} \left( {\pi N_{10} N_{11} - \overline{A}_{3} \varepsilon_{2} N_{12} } \right) \hfill \\ &\quad + N_{12} N_{17} \left( {\omega^{2} N_{5} + \pi N_{5} N_{9} - \pi N_{3} N_{10} } \right) + N_{13} N_{17} \left( \begin{gathered} \omega^{2} N_{2} - \pi N_{4} N_{3} \hfill \\ - \pi N_{3} N_{8} + \pi N_{2} N_{9} \hfill \\ \end{gathered} \right) \hfill \\ &\quad - \pi \omega^{2} N_{2} N_{14} N_{17} \left( {i + 1} \right) \hfill \\ \end{aligned} $$
$$ \begin{aligned} L_{8} & = \left( \begin{gathered} iN_{5} N_{2} N_{11} - \varepsilon_{2} \overline{A}_{3} N_{3} N_{11} - N_{4} N_{10} N_{11} - N_{5} N_{9} N_{12} + N_{3} N_{10} N_{12} \hfill \\ - iN_{1} N_{2} N_{13} + N_{4} N_{3} N_{13} + \omega^{2} N_{2} N_{14} + N_{3} N_{8} N_{13} - N_{2} N_{9} N_{13} + i\omega^{2} N_{2} N_{14} \hfill \\ \end{gathered} \right)\left( {\pi C_{a} + D_{a} \varepsilon_{1} } \right) \hfill \\ &\quad + \left( {\overline{A}_{3} \varepsilon_{2} N_{4} N_{12} - \omega^{2} N_{5} N_{12} - i\omega^{2} N_{2} N_{13} - \omega^{2} N_{2} N_{13} } \right)\left( {C_{a} - \pi } \right) - \overline{A}_{5} \varepsilon_{1} \varepsilon_{2} N_{1} N_{13} N_{16} \hfill \\ &\quad + \left( {\pi \overline{A}_{15} N_{15} + \pi N_{12} N_{17} } \right)\left( {\overline{A}_{3} \varepsilon_{2} N_{3} - iN_{5} N_{2} } \right) + \pi N_{15} \left( {\overline{A}_{15} N_{4} N_{10} - \overline{A}_{4} \varepsilon_{2} N_{3} N_{13} } \right) \hfill \\ &\quad + N_{13} N_{15} \left( {\overline{A}_{5} \varepsilon_{1} \varepsilon_{2} N_{3} + i\pi \overline{A}_{4} N_{4} \cos \theta - i\overline{A}_{5} \varepsilon_{1} N_{4} \cos \theta } \right) + \left( \begin{gathered} \overline{A}_{4} \varepsilon_{2} N_{13} N_{16} - \varepsilon_{2} \overline{A}_{3} \overline{A}_{15} N_{16} \hfill \\ - \omega^{2} N_{13} N_{17} \hfill \\ \end{gathered} \right)\left( {\omega^{2} + \pi N_{1} } \right) \hfill \\ &\quad + \pi \overline{A}_{15} N_{16} \left( {N_{5} N_{8} - N_{2} N_{10} } \right) - \left( {\varepsilon_{2} N_{5} N_{11} N_{16} + iN_{2} N_{13} N_{16} \cos \theta - \varepsilon_{2} \omega^{2} N_{14} N_{16} } \right)\left( {\pi \overline{A}_{4} - \overline{A}_{5} \varepsilon_{1} } \right) \hfill \\ &\quad + \left( {\overline{A}_{3} \varepsilon_{2} N_{11} N_{16} - iN_{2} N_{13} N_{15} - N_{10} N_{12} N_{16} - N_{8} N_{13} N_{16} } \right)\left( {\varepsilon_{1} N_{7} - \pi N_{6} } \right) + \pi N_{4} N_{10} N_{12} N_{17} \hfill \\ &\quad + \pi N_{11} N_{17} \left( {\omega^{2} N_{5} - \overline{A}_{3} \varepsilon_{2} N_{4} } \right) + \pi N_{13} N_{17} \left( {N_{4} N_{8} - iN_{2}^{2} } \right) + \pi \omega^{2} N_{17} \left( {\omega^{2} N_{14} - N_{9} N_{13} } \right) \hfill \\ \end{aligned} $$
$$ \begin{aligned} L_{9} & = \left( \begin{gathered} \overline{A}_{3} \varepsilon_{2} N_{4} N_{11} - \omega^{2} N_{5} N_{11} - \overline{A}_{3} \varepsilon_{2} N_{3} N_{12} + iN_{5} N_{2} N_{12} - N_{4} N_{10} N_{12} \hfill \\ + iN_{2}^{2} N_{13} - N_{4} N_{8} N_{13} + \omega^{2} N_{9} N_{13} - \omega^{4} N_{14} + \omega^{2} N_{1} N_{13} \hfill \\ \end{gathered} \right)\left( {\pi C_{a} + D_{a} \varepsilon_{1} } \right) \hfill \\ &\quad + w^{4} N_{13} \left( {C_{a} - \pi } \right) + \left( {\pi \overline{A}_{15} N_{15} + \pi N_{12} N_{17} } \right)\left( {\omega^{2} N_{5} - \overline{A}_{3} \varepsilon_{2} N_{4} } \right) \hfill \\ &\quad + \left( \begin{gathered} \varepsilon_{2} N_{4} N_{13} N_{15} - \varepsilon_{2} N_{5} N_{12} N_{16} \hfill \\ + i\omega^{2} N_{13} N_{16} \cos \theta - \varepsilon_{2} N_{2} N_{13} N_{16} \hfill \\ \end{gathered} \right)\left( {\pi \overline{A}_{4} - \overline{A}_{5} \varepsilon_{1} } \right) + \left( {\omega^{2} N_{13} N_{15} - \overline{A}_{3} \varepsilon_{2} N_{12} N_{16} } \right)\left( {\pi N_{6} - \varepsilon_{1} N_{7} } \right) \hfill \\ &\quad + \pi \overline{A}_{15} N_{16} \left( {\overline{A}_{3} \varepsilon_{2} N_{2} + \omega^{2} N_{10} } \right) + \pi \omega^{2} N_{2} N_{13} N_{17} \left( {i + 1} \right) \hfill \\ \end{aligned} $$
$$ \begin{aligned} L_{10} &= \left( {\varepsilon_{2} \overline{A}_{3} N_{4} N_{12} - \omega^{2} N_{5} N_{12} - i\omega^{2} N_{2} N_{13} - \omega^{2} N_{2} N_{13} } \right)\left( {\pi C_{a} + D_{a} \varepsilon_{1} } \right) \hfill \\ &\quad - \pi \omega^{2} \varepsilon_{2} N_{16} \left( {\overline{A}_{3} \overline{A}_{15} - \overline{A}_{4} N_{13} } \right) - \omega^{2} N_{13} \left( {\varepsilon_{1} \varepsilon_{2} \overline{A}_{5} N_{16} + \pi \omega^{2} N_{17} } \right) \hfill \\ \end{aligned} $$
$$ L_{11} = \omega^{4} N_{13} \left( {\pi C_{\alpha } + \varepsilon_{1} D_{\alpha } } \right) $$

Appendix-2

$$ \left. {\begin{array}{*{20}c} {\zeta_{i} = \frac{{\varepsilon_{1} \varsigma }}{{\varepsilon_{1} \mathchar'26\mkern-10mu\lambda \left( {\xi_{i}^{2} + \pi } \right)\gamma }}} \\ {\eta_{i} = \zeta_{i} \left( { - \frac{{\varepsilon_{1} }}{{\xi_{i}^{2} + \pi }}} \right)} \\ {\ell_{i} = \frac{{\overline{A}_{14} \left\{ {i\xi_{i} \left( {\sin \theta + \cos \theta } \right) + \varepsilon_{2} } \right\} - \overline{A}_{15} \zeta_{i} }}{{A_{11} \varepsilon_{2} \cos \theta - A_{12} - i\omega A_{13} + \omega^{2} - \xi_{i}^{2} \left( {A_{11} - \varepsilon^{2} \omega^{2} } \right)}}} \\ {\varpi_{i} = \frac{{ - \left\{ {\left( {\xi_{i}^{2} + i\xi_{i} \varepsilon_{2} \cos \theta - C_{\alpha } } \right)\zeta_{i} + D_{\alpha } \eta_{i} + \overline{A}_{5} } \right\}}}{{i\xi_{i} \overline{E}_{\alpha } \cos \theta }}} \\ \end{array} } \right\} $$

where;

$$ \varsigma = \left[ \begin{gathered} i\xi_{i} \overline{E}_{\alpha } \sin \theta \left\{ \begin{gathered} \left( {\xi_{i}^{2} \left( {\overline{A}_{1} - 1} \right)\sin \theta \cos \theta + i\xi_{i} \overline{A}_{1} \varepsilon_{2} \sin \theta } \right)\left( {\overline{A}_{3} \varepsilon_{2} + i\xi_{i} \overline{A}_{2} \cos \theta } \right) \hfill \\ + \left( {\xi_{i}^{2} \left( {\cos^{2} \theta + \overline{A}_{1} \sin^{2} \theta - \varepsilon^{2} \omega^{2} } \right) + \omega^{2} - i\xi_{i} \overline{A}_{1} \varepsilon_{2} \cos \theta } \right)\left( {i\xi_{i} \overline{A}_{2} \sin \theta } \right) \hfill \\ \end{gathered} \right\} \hfill \\ - \left( {i\xi_{i} \overline{E}_{\alpha } \cos \theta } \right)\left\{ \begin{gathered} \left( {\xi_{i}^{2} \left( {\sin^{2} \theta + \overline{A}_{1} \cos^{2} \theta - \varepsilon^{2} \omega^{2} } \right) + \omega^{2} - i\xi_{i} \overline{A}_{1} \varepsilon_{2} \cos \theta } \right) \hfill \\ \left( {\overline{A}_{3} \varepsilon_{2} + i\xi_{i} \overline{A}_{2} \cos \theta } \right) + \left( \begin{gathered} \xi_{i}^{2} \left( {1 - \overline{A}_{1} } \right)\sin \theta \cos \theta \hfill \\ + i\xi_{i} \varepsilon_{2} \left( {1 - 2\overline{A}_{1} } \right)\sin \theta \hfill \\ \end{gathered} \right)\left( {i\xi_{i} \overline{A}_{2} \sin \theta } \right) \hfill \\ \end{gathered} \right\} \hfill \\ \end{gathered} \right] $$
$$ \mathchar'26\mkern-10mu\lambda = \left[ \begin{gathered} i\xi_{i} \overline{E}_{\alpha } \cos \theta \left\{ {\left( { - i\xi_{i} \overline{A}_{4} \sin \theta } \right)\left( {\overline{A}_{3} \varepsilon_{2} + i\xi_{i} \overline{A}_{2} \cos \theta } \right) + \overline{A}_{4} \left( {\varepsilon_{2} + i\xi_{i} \cos \theta } \right)\left( {i\xi_{i} \overline{A}_{2} \sin \theta } \right)} \right\} \hfill \\ - \left( {\xi_{i}^{2} + i\xi_{i} \varepsilon_{2} \cos \theta - C_{\alpha } } \right)\left\{ \begin{gathered} \left( {\xi_{i}^{2} \left( {\overline{A}_{1} - 1} \right)\sin \theta \cos \theta + i\xi_{i} \overline{A}_{1} \varepsilon_{2} \sin \theta } \right)\left( {\overline{A}_{3} \varepsilon_{2} + i\xi_{i} \overline{A}_{2} \cos \theta } \right) \hfill \\ + \left( \begin{gathered} \xi_{i}^{2} \left( {\cos^{2} \theta + \overline{A}_{1} \sin^{2} \theta - \varepsilon^{2} \omega^{2} } \right) + \omega^{2} \hfill \\ - i\xi_{i} \overline{A}_{1} \varepsilon_{2} \cos \theta \hfill \\ \end{gathered} \right)\left( {i\xi_{i} \overline{A}_{2} \sin \theta } \right) \hfill \\ \end{gathered} \right\} \hfill \\ \end{gathered} \right] $$
$$\upgamma = \left[ \begin{gathered} \left( {\xi_{i}^{2} + i\xi_{i} \varepsilon_{2} cos\theta - C_{\alpha } } \right)\left\{ \begin{gathered} \left( {i\xi_{i} \overline{A}_{5} sin\theta } \right)\left( {\overline{A}_{3} \varepsilon_{2} + i\xi_{i} \overline{A}_{2} cos\theta } \right) \hfill \\ + \overline{A}_{5} \left( {i\xi_{i} \overline{A}_{2} sin\theta } \right)\left( {\varepsilon_{2} + i\xi_{i} cos\theta } \right) \hfill \\ \end{gathered} \right\} \hfill \\ - D_{\alpha } \left\{ \begin{gathered} \left( {\xi_{i}^{2} \left( {\overline{A}_{1} - 1} \right)sin\theta cos\theta + i\xi_{i} \overline{A}_{1} \varepsilon_{2} sin\theta } \right)\left( {\overline{A}_{3} \varepsilon_{2} + i\xi_{i} \overline{A}_{2} cos\theta } \right) \hfill \\ + \left( {\xi_{i}^{2} \left( {cos^{2} \theta + \overline{A}_{1} sin^{2} \theta - \varepsilon^{2} \omega^{2} } \right) + \omega^{2} - i\xi_{i} \overline{A}_{1} \varepsilon_{2} cos\theta } \right)\left( {i\xi_{i} \overline{A}_{2} sin\theta } \right) \hfill \\ \end{gathered} \right\} \hfill \\ \end{gathered} \right] $$

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Ali, H., Mahmood, A., Jahangir, A. et al. Nonlocal theory on plane waves in higher order thermo-porous functionally graded semiconductor. Indian J Phys (2024). https://doi.org/10.1007/s12648-024-03233-0

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