Cooling of a layered plate under mixed conditions

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.