(Solved): 6. A rectangular thin plate with a rectangular hole in the centre, insulated from above and below, ...

6. A rectangular thin plate with a rectangular hole in the centre, insulated from above and below, has dimensions as follows: The rectangular thin plate has corners at $(0,0)$ and $(3,4)$ the rectangular hole has corners at $(1,1)$ and $(2,3)$ The outer edges are kept at temperatures as follows: - $u(0,y,t)=0$ for $0?y?4$, - $u(3,y,t)=9$ for $0?y?4$, - $u(x,0,t)=\sin (?x)+x^2$ for $0?x?3$, and - $u(x,4,t)=\sin (?x)+x^2$ for $0?x?3$. The square boundary in the centre is insulated, meaning there is no temperature gradient in the normal direction of this boundary. The initial temperature is given by $u(x,y,0)=(\sin (?x))(\cos (?y))+x^2.$ Assume $c=1$. Using the PDE Modeller app in MATLAB, plot the temperature of the plate at $t=15$ (seconds). Include in your submission: - copies of the windows of each boundary condition entered, - a copy of the window where the PDE is specified, - a plot of the mesh used, - a copy of the window containing the solve parameters used, - a 2D colour plot of the final temperatures with contour lines and - a 3D colour plot of the final temperatures.