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We use this equation to describe the conduction of heat along the wire
and over in a position to put some mathematical meaning to it.
As time goes on, the value of the temperature changes
by the length of the time step times this expression.
So this expression has to be the rate of change of the temperature with respect to time.
Ṫ₈ if you will at times 0.
Of course, we're always dealing with estimates here even though I'm writing an equal sign.
But now that we know about central-difference formulas,
we have another interpretation of the term.
It's 1 mm squared delta x squared times the second derivative of our temperature.
Of course, I'm again cheating a little here.
We only know the temperatures of the different compartments.
We don't have any curve of which we could form the second derivative.
Now, we're going to write this equation in a professional fashion. Let's do this right.
Temperature depends both on space and time.
It changes along the wire that is both position, and it changes with time. We see temporal evolution.
What we need here is the rate of change with respect to time and the way to be writing this is this.
It's called the partial derivative of the temperature with respect to time.
It's not written with lower case letter d but with curly type of d.
This derivative of the temperature with respect to time equals 10 over 1 second times 1 mm².
And this one is the second partial derivative of temperature with respect to position written like this
Note how this is being written.
The curly d is squared not the complete expression in the numerator.
Even though in the denominator, everything is being squared.
They're going to discuss partial derivatives in the next segments so bare with me.
This equation is called the heat equation.
In this case, it describes the conduction of heat along the wire along one dimension,
and it's a partial differential equation.
PDE is the technical term. It contains partial derivatives.
That's why the differential equations we have seen so far are ordinary differential equations, ODEs.
PDEs, partial differential equations, are typical for problems in space or problems in space and time.
The constant involved here is often called alpha the thermal diffusivity.
The name already hints at this being a diffuse equation.
The heat diffuses along the wire.