(Solved): 4. A thin plate with a hole in the centre, insulated from above and below, has dimensions as follo ...

4. A thin plate with a hole in the centre, insulated from above and below, has dimensions as follows: Assuming the origin is at the bottom left corner, the outer edges are kept at temperatures as follows: - \( u(0, y, t)=0 \) for \( 0 \leq y \leq 2 \), - \( u(2, y, t)=10 \) for \( 0 \leq y \leq 2 \), - \( u(x, 0, t)=\sin (\pi x)+5 x \) for \( 0 \leq x \leq 2 \), and - \( u(x, 2, t)=\sin (\pi x)+5 x \) for \( 0 \leq x \leq 2 \). 2 The circular boundary in the centre is insulated, meaning there is no temperature gradient in the normal direction of this boundary. Assume \( c=1 \), then the initial temperature is given by \[ u(x, y, 0)=\sin (\pi x) \cos (\pi y)+5 x . \] Using the PDE Modeller app in MATLAB, plot the temperature over the plate at \( t=20 \). Include in your submission: - a colour plot of the final temperatures, - a plot of the mesh used. - copies of the windows of each boundary condition entered, - a copy of the window where the PDE is specified, - a copy of the window containing the solve parameters used.