Using Matlab, inally, in engineering, you will come across a standard differential equation called the Laplace Equation that, generally, will

represent a variety of two$-$dimensional models:

$2$

$2+2$

$2=$

where $=(,)$ and $$ is a constant value.

For example, in Electrical Engineering we have the Electrostatic Field Model:

$2$

$2+2$

$2=$

$$

with Voltage $=(,),$ is the constant volumetric charge density, and $$

is the constant permittivity of the material.

Also, in Mechanical Engineering we have the Fourier Conduction Model:

$2$

$2+2$

$2=2$

$$

with Temperature $=(,),$ is a constant thickness, $$ is a constant heat generation, and $$ is a constant thermal

conductivity of the material.

Regardless of the model the solution procedure is the same. We can present the two$-$dimensional solution as satisfying the following

numerical approximation

$,=+1,+1,+,+1+,1+$

$4,2<=<=1,2<=<=1$

Here, we expect constant values at the borders to the matrix:

$$ all columns of $$ at $=1$ will be a constant $($top row, $)$

$$ all columns of $$ at $=$ will be a constant $($bottom row, $)$

$$ all rows of $$ at $=1$ will be a constant $($left column, $)$

$$ all rows of $$ at $=$ will be a constant $($right column, $$

Now, because of the codependence of each $,$ we will need to iterate through this equation many times until the values in the matrix

stop changing. This is determined by putting the iterations in a while loop and iterating until the maximum value of an element$-$

directed relative difference is less than a predetermined value $$

MET $4076$ Sp$2023$ @ UC

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