Question $2$

Given the information below, determine the value of $u$ at global nodes $5$ and $8.$ Show detailed calculations. Note: Provide clear screenshots of the entire screen when using online calculators. The date and time on your screen must be visible in the screenshot.

$2$D Partial differential equation

$\frac{de{l}^{2}u}{del{x}^2}+\frac{de{l}^{2}u}{del{y}^2}=0$

Boundary conditions:

$u(x,0)=15-10x$ for $0x1\mathrm{.}5$

$u(x,4\mathrm{.}5)=10$ for $0\mathrm{.}75x<mn>1</mn><mn>.</mn><mn>5</mn>$

$u(0,y)=15$ and $u(1\mathrm{.}5,y)=0$ for $0y4\mathrm{.}5$

$3$ of $5$

APM$4814-$ Major Test $2-27/08/2024$

$\backslash$table$[[$FE no$.,$Element Coefficient matrix$],[1,0\mathrm{.}25,0,-0\mathrm{.}25],[0,1,-1],[-0\mathrm{.}25,-1,1\mathrm{.}25],[2,1,-1,0],[-1,1,25,-0\mathrm{.}25],[0,-0\mathrm{.}25,0\mathrm{.}25],[3,0\mathrm{.}25,0,-0\mathrm{.}25],[0,1,-1],[-0\mathrm{.}25,-1,1\mathrm{.}25],[1,-1,0],[,-1,1\mathrm{.}25,-0\mathrm{.}25],[,0,-0\mathrm{.}25,0\mathrm{.}25],[5,,,],[,1,-1],[-0\mathrm{.}25,-1,1\mathrm{.}25],[,1,-1,0],[,-1,1\mathrm{.}25,-0\mathrm{.}25]]$

$\backslash$table$[[$FE no$.,$Element coefficient matrix$],[7,-0\mathrm{.}25,0,-0\mathrm{.}25],[0,1,-1],[-0\mathrm{.}25,-1,1\mathrm{.}25],[8,1,-1,0],[-1,1\mathrm{.}25,-0\mathrm{.}25],[0,-0\mathrm{.}25,0\mathrm{.}25],[9,1,-1,0],[-1,1\mathrm{.}25,-0\mathrm{.}25],[0,-0\mathrm{.}25,0\mathrm{.}25],[0\mathrm{.}25,0,-0\mathrm{.}25],[,0,1,-1],[,-0\mathrm{.}25,-1,1\mathrm{.}25],[,1,-1,0],[,-1,1\mathrm{.}25,-0\mathrm{.}25],[,0,-0\mathrm{.}25,0\mathrm{.}25]]$

Global coefficient matrix

$\backslash$table$[[$Node$,1,2,3,4,5,6,7,8,9,10,11],[1,1\mathrm{.}25,-1,0,-0\mathrm{.}25,0,0,0,0,0,0,0],[2,-1,2\mathrm{.}5,-1,0,-0\mathrm{.}5,0,0,0,0,0,0],[3,0,-1,1\mathrm{.}25,0,0,-0\mathrm{.}25,0,0,0,0,0],[4,-0\mathrm{.}25,0,0,2\mathrm{.}5,-2,0,-0\mathrm{.}25,0,0,0,0],[5,0,-0\mathrm{.}5,0,-2,4\mathrm{.}5,-2,0,-0\mathrm{.}5,0,0,0],[6,0,0,-0\mathrm{.}25,0,-2,2\mathrm{.}5,0,0,-0\mathrm{.}25,0,0],[7,0,0,0,-0\mathrm{.}25,0,0,2\mathrm{.}5,0\mathrm{.}25,0,0,0],[8,0,0,0,0,-0\mathrm{.}5,0,0\mathrm{.}25,,,,],[9,0,0,0,0,0,-0\mathrm{.}25,0,,3\mathrm{.}5,0,-0\mathrm{.}25],[10,0,0,0,0,0,0,0,,0,0\mathrm{.}25,-1],[11,0,0,0,0,0,0,0,,-0\mathrm{.}25,-1,2\mathrm{.}25]]$

$[45$ marks$]$

Question $2$

Given the information below, determine the value of $u$ at global nodes $5$ and $8.$ Show detailed calculations. Note: Provide clear screenshots of the entire screen when using online calculators. The date and time on your screen must be visible in the screenshot.

$2$D Partial differential equation

$\frac{de{l}^{2}u}{del{x}^2}+\frac{de{l}^{2}u}{del{y}^2}=0$

Boundary conditions:

$u(x,0)=15-10x$ for $0x1\mathrm{.}5$

$u(x,4\mathrm{.}5)=10$ for $0\mathrm{.}75x<mn>1</mn><mn>.</mn><mn>5</mn>$

$u(0,y)=15$ and $u(1\mathrm{.}5,y)=0$ for $0y4\mathrm{.}5$

$3$ of $5$