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Analytical solution of time fractional Cattaneo heat equation for finite slab under pulse heat flux

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Abstract

A fractional Cattaneo model is derived for studying the heat transfer in a finite slab irradiated by a short pulse laser. The analytical solutions for the fractional Cattaneo model, the classical Cattaneo-Vernotte model, and the Fourier model are obtained with finite Fourier and Laplace transforms. The effects of the fractional order parameter and the relaxation time on the temperature fields in the finite slab are investigated. The results show that the larger the fractional order parameter, the slower the thermal wave. Moreover, the higher the relaxation time, the slower the heat flux propagates. By comparing the fractional order Cattaneo model with the classical Cattaneo-Vernotte and Fourier models, it can be found that the heat flux predicted using the fractional Cattaneo model always transports from the high temperature to the low one, which is in accord with the second law of thermodynamics. However, the classical Cattaneo-Vernotte model shows that the unphysical heat flux sometimes transports from the low temperature to the high one.

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Authors and Affiliations

  1. School of Port and Transportation Engineering, Zhejiang Ocean University, Zhoushan, 316022, Zhejiang Province, China

    Guangying Xu & Jinbao Wang

Authors
  1. Guangying Xu
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  2. Jinbao Wang
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Correspondence to Jinbao Wang.

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Citation: XU, G. Y. and WANG, J. B. Analytical solution of time fractional Cattaneo heat equation for finite slab under pulse heat flux. Applied Mathematics and Mechanics (English Edition), 39(10), 1465–1476 (2018) https://doi.org/10.1007/s10483-018-2375-8

Project supported by the National Natural Science Foundation of China (No. 11372281), the Science and Technology Plan Project of Zhoushan (No. 2016C41009), and the Innovative Team Project of Zhejiang Ocean University

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Xu, G., Wang, J. Analytical solution of time fractional Cattaneo heat equation for finite slab under pulse heat flux. Appl. Math. Mech.-Engl. Ed. 39, 1465–1476 (2018). https://doi.org/10.1007/s10483-018-2375-8

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  • DOI: https://doi.org/10.1007/s10483-018-2375-8

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Chinese Library Classification

2010 Mathematics Subject Classification

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