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11/18/2015
@1:26 PM
Partial Differential Equations
This semester I took MATH2100 which focused on ordinary differential equations (ODEs) in the first half of semester and partial differential equations (PDEs) in the second half of semester. One of the topics covered in PDEs was the one dimensional heat equation.
For some context, consider a uniform rod with length L. The temperature in the rod would be evenly distributed assuming the environment temperature stays the same and there is no heat loss from the rod. This can be represented in Figure 1 below where the x-axis is a point on the rod between 0 and L and the y-axis is the temperature at the corresponding x point.
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| Figure 1 Temperature of a rod with no heating |
Now what happens when I add a heat source like a candle at the half way point of the rod (L/2)? The temperature on the rod at the point of heating would be higher and decrease as you move further away from it. Note that temperature should be symmetrical about L/2.
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| Figure 2 Temperature of a rod with heating |
Over time, the heat will be able to distribute itself further away from the point of heating like at t = 5 and at t = 20. Eventually with sufficient time the temperature of the rod will become uniform again like in Figure 1.
This behaviour is known as heat flow and can be described mathematically! First, we need to think about the variables which affect the temperature in the rod. There are three:
- Heating source(s) and their location,
- Distance along the rod and,
- Time
We can let the function u(x,t) describe the temperature along the rod where x is a point on the rod and t denote time.
The one dimensional (1-D) heat equation is given as:
Where the subscripts are the partial derivatives of the function u (and hence the name partial differential equations).
Now we can try a possible solution:
Taking the partial derivatives:
Substitute into the 1-D heat equation:
Now we have found the mathematical function which describes the behaviour of heat flow. There are infinite solutions since A and b can be any constant and it will still work!
Unfortunately that is incorrect :(
Although the function we have found satisfies the PDE, we still need additional information. Like ODEs we need to know the initial conditions (IC) of the rod (i.e. at t = 0) and what's happening at the ends of the rod (i.e. at x = 0 and x = L), called the boundary conditions (BC) .
There are a few BCs we can consider:
- The ends are insulated
- There are ice cubes at the ends of the rod which means temperate is zero at the ends
- Or a mixture of the two
If we consider the ICs, BCs and the 1-D heat equation, there will only be one solution which satisfies all three. So for different scenarios of heat flow, there will only be one function which can model its behaviour. Personally, I find it remarkable how there can only be one solution to model heat flow for a given situation.
Another important thing to note is heat flow looks really similar to the Gaussian distribution and the heat equation can intuitively model Brownian motion. Perhaps a post for another time.
W.T
