Eur. Phys. J. Plus (2016) 131: 289
https://doi.org/10.1140/epjp/i2016-16289-9

Regular Article

Eigenvalue approach on a two-dimensional thermal shock problem with weak, normal and strong conductivity

¹ Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box. 80203, 21589, Jeddah, Saudi Arabia
² Department of Mathematics, Faculty of Science and Arts - Khulais, University Of Jeddah, Jeddah, Saudi Arabia
³ Nonlinear Analysis and Applied Mathematics Research Group (NAAM), Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia
⁴ Department of mathematics, Faculty of Science, Sohag University, Sohag, Egypt

^* e-mail: faris.kau@hotmail.com

Received: 8 March 2016
Accepted: 7 July 2016
Published online: 29 August 2016

Abstract

The present paper is devoted to the study of a two-dimensional thermal shock problem with weak, normal and strong conductivity using the eigenvalue approach. The governing equations are taken in the context of the new consideration of heat conduction with fractional order generalized thermoelasticity with the Lord-Shulman model (LS model). The bounding surface of the half-space is taken to be traction free and subjected to a time-dependent thermal shock. The Laplace and the exponential Fourier transform techniques are used to obtain the analytical solutions in the transformed domain by the eigenvalue approach. Numerical computations have been done for copper-like material for weak, normal and strong conductivity and the results are presented graphically to estimate the effects of the fractional order parameter.