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Numerical Method for Solving the Time-Fractional Dual-Phase-Lagging Heat Conduction Equation with the Temperature-Jump Boundary Condition

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Abstract

This article proposes a new nanoscale heat transfer model based on the Caputo type fractional dual-phase-lagging (DPL) heat conduction equation with the temperature-jump boundary condition. The model is proved to be well-posed. A finite difference scheme based on the L1 approximation for the Caputo derivative is then presented for solving the fractional DPL model. Unconditional stability and convergence of the scheme are proved by using the discrete energy method. Three numerical examples are given to verify the accuracy of the scheme. Results show the convergence order to be \(O(\tau ^{2-\alpha }+h^2)\) , which coincides with the theoretical analysis. A simple nanoscale semiconductor silicon device is illustrated to show the applicability of the model. It is seen from the numerical result that when \(\alpha =1\), the fractional DPL reduces to the conventional DPL and the obtained peak temperature is almost identical to those obtained in the literature. However, when \(\alpha <1\), the model predicts a higher peak temperature level than that when \(\alpha =1\). In particular, when \(\alpha = 0.7\) and 0.9, an oscillatory temperature at the beginning is observed. This indicates that the fractional DPL model can be an excellent candidate for analyzing the temperature instability appearing in electronic nano-semiconductor devices.

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Acknowledgements

We would like to express our gratitude to the editor and the anonymous reviewers for their many valuable comments and suggestions.

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Authors and Affiliations

  1. School of Mathematics, Southeast University, Nanjing, 210096, People’s Republic of China

    Cui-cui Ji & Zhi-zhong Sun

  2. Mathematics and Statistics, Louisiana Tech University, Ruston, LA, 71272, USA

    Weizhong Dai

  3. College of Mathematics and Statistics, Minnan Normal University, Zhangzhou, 363000, People’s Republic of China

    Weizhong Dai

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  1. Cui-cui Ji
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Correspondence to Zhi-zhong Sun.

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The research is supported by National Natural Science Foundation of China (No. 11671081) and by the Fundamental Research Funds for the Central Universities and the Research and Innovation Project for College Graduates of Jiangsu Province (Grant No. KYLX\(15_{-}0106\)) and the Scientific Research Foundation of Graduate School of Southeast University (Grant No. YBJJ1716) and by the China Scholarship Council (Grant No. 201706090099).

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Ji, Cc., Dai, W. & Sun, Zz. Numerical Method for Solving the Time-Fractional Dual-Phase-Lagging Heat Conduction Equation with the Temperature-Jump Boundary Condition. J Sci Comput 75, 1307–1336 (2018). https://doi.org/10.1007/s10915-017-0588-3

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