Journal of Applied Mathematics and Stochastic Analysis
Volume 13 (2000), Issue 2, Pages 197-206

Cooling of a layered plate under mixed conditions

F. D. Zaman1 and R. Al-Khairy2

1King Fahd University of Petroleum and Minerals, Department of Mathematical Sciences, Dhahran 31261, Saudi Arabia
2Dammam College of Girls, Department of Mathematics, Dammam, Saudi Arabia

Received 1 June 1998; Revised 1 November 1999

Copyright © 2000 F. D. Zaman and R. Al-Khairy. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We consider the temperature distribution in an infinite plate composed of two dissimilar materials. We suppose that half of the upper surface (y=h,<x<0) satisfies the general boundary condition of the Neumann type, while the other half (y=h,0<x<) satisfies the general boundary condition of the Dirichlet type. Such a plate is allowed to cool down on the lower surface with the help of a fluid medium which moves with a uniform speed v and which cools the plate at rate Ω. The resulting mixed boundary value problem is reduced to a functional equation of the Wiener-Hopf type by use of the Fourier transform. We then seek the solution using the analytic continuation and an extended form of the Liouville theorem. The temperature distribution in the two layers can then be written in a closed form by use of the inversion integral.