Heat conduction in rectangular domain with periodic heat transfer coefficient

[bancux]

Dear all, I have difficulty in solving this problem (see the figure in attached thumbnails)

I have a rectangular shape with length/height of L and the thickness/width of $$\delta$$
Within the rectangular area, a heat conduction occurred. I would like to determine the temperature profile within the rectangular.

Based on the right hand side figure, I take a small cell with size of dx times dy. Applying the energy balance within the cell.

I have attached all the equation that I can think of, and also the boundary conditions. I have difficulty in solving the differential equation.

Thanks in advance.

note:

$$\lambda$$ is thermal conductivity
$$\alpha$$ is heat transfer coefficient, it is suppose to be dependent on x-location and time. I am going to simulate the heat transfer coefficient as a periodic function.

 

[bancux]

Still no answer...
Hmmm...