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(Solved): 4. Set up a numerical procedure to enable the partial differential equation \[ \frac{\partial^{2} ...



4. Set up a numerical procedure to enable the partial differential equation
\[
\frac{\partial^{2} T}{\partial x^{2}}+\frac{\p

4. Set up a numerical procedure to enable the partial differential equation \[ \frac{\partial^{2} T}{\partial x^{2}}+\frac{\partial^{2} T}{\partial y^{2}}=x y \] to be solved in the region (see sketch) defined by the following boundaries: (a) the \( \mathrm{x} \) axis from \( \mathrm{x}=0 \) to \( \mathrm{x}=3 \) (b) the lines \( \mathrm{y}=3 \) and \( \mathrm{x}=3 \). The Boundary Temperatures are as follows: (a) \( \mathrm{T}=3 \mathrm{x} \) along the upper edge \( \mathrm{y}=3 \) (b) \( T=9-y^{2} \) along the \( y \) axis, (c) \( T=y^{2} \) along \( \mathrm{x}=3 \); and (d) \( \frac{\partial T}{\partial y}=-T \) along the \( \mathrm{x} \) axis. Write your solution out in matrix format 20 Marks


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