Question 1:
Consider a steel plate of size 15cm x 15cm. If two of the sides are held at 100 degree celsius and the other two sides at 0 degree celsius, find the steady state temperatures of interior points, assuming a grid size of 5cm x 5cm.

Solution
Using the five point difference formula:
\begin{align*} -4{ f }_{ 1 }+{ f }_{ 2 }+{ f }_{ 3 }+100+100\quad &=\quad 0 \\ -4{ f }_{ 2 }+{ f }_{ 1 }+100+0+{ f }_{ 4 }\quad &=\quad 0 \\ -4{ f }_{ 3 }+{ f }_{ 1 }+100+{ f }_{ 4 }+0\quad &=\quad 0 \\ -4{ f }_{ 4 }+{ f }_{ 2 }+{ f }_{ 3 }+0+0\quad &=\quad 0 \end{align*}
Solving the above system using Gauss Elimination method, we get:

$${ f }_{ 1 }=75,{ f }_{ 2 }=50,{ f }_{ 3 }=50{ f }_{ 4 }=25$$

Note: The linear system in the above problem is diagonally dominant. Therefore, it can be solved by Gauss-seidel method. When such an iteration is applied to Laplace’s equation, the iterative method is called Liebmann’s iterative method.

Question 2:
Solve the above problem using Liebmann’s iterative method.
Solution
Using the standard five point formula,
\begin{align*}{ f }_{ 1 }&=\frac { { f }_{ 1 }+{ f }_{ 3 }+200 }{ 4 } \\ { f }_{ 2 }&=\frac { { f }_{ 1 }+{ f }_{ 4 }+100 }{ 4 } \\ { f }_{ 3 }&=\frac { { f }_{ 1 }+{ f }_{ 4 }+100 }{ 4 } \\ { f }_{ 4 }&=\frac { { f }_{ 2 }+{ f }_{ 3 } }{ 4 } \end{align*}

To Find Initial Approximation
Using diagonal five point formula, assuming $${ f }_{ 4 }=0$$, we get:
$${ f }_{ 1 }=\frac { 100+{ f }_{ 4 }+100+100 }{ 4 } =75$$
Using standard five point difference formula:
\begin{align*} { f }_{ 2 }&=\frac { 1 }{ 4 } \left( 75+0+100+0 \right) =\quad 43.75 \\ { f }_{ 3 }&=\frac { 1 }{ 4 } \left( 100+75+0+0 \right) =\quad 43.75 \\ { f }_{ 4 }&=\frac { 1 }{ 4 } \left( 43.75+43.75+0+0 \right) =\quad 21.88 \end{align*}

Now we use Gauss-Siedel iteration method to find better approximation of the solution. Using this, we get the following values:

 $${ f }_{ 1 }$$ $${ f }_{ 2 }$$ $${ f }_{ 3 }$$ $${ f }_{ 4 }$$ 1st iteration 71.88 48.44 48.44 24.22 2nd iteration 74.22 49.61 49.61 24.81 3rd iteration 74.81 49.90 49.90 24.95 4th iteration 74.95 49.98 49.98 24.99

#### Written by Tessy C

Lecturer and Research Scholar in Mathematics.