Cartesian and Polar
1D Heat Equation
Cartesian Coordinates
The one-dimensional heat equation describes heat distribution in one dimension. Even though a rod is three-dimensional, it’s properties allow us to model heat distribution on a rod with the one-dimensional heat equation.
​
​
The graphs below show the heat distribution on a long rod. The initial heat distribution is the boxcar function with heat of 1 from x=0 to x=1.
​
​
​
​
​
​
​
​
​
One can plot the change of temperature over time. The following graphic shows its temporal change on the x-axis and its heat distribution on the y-axis.
It is the same equation as the equation above.
If you wonder how we calculated the graphics above. Check out our Mathematica code.
The Equation:
%3Du_t.png){kind=small}
This is the thermal diffusivity, this is material specific. It measures the rate of transfer of heat of a material from the hot end to the cold end
Where k is the thermal conductivity, p is the density, and cp is the specific heat capacity.
The curvature of the heat in the plate, this takes the curvature at all the coordinates in the plate.
This is the rate of change of temperature of the plate over time
Just U(t,x) would mean the actual temperature of the plate over time
Why the Second Derivative?
The heat equation is based on curvature (and not the linear component) meaning that it must be based on the second derivative, which is curvature of the values.
If we were to visualize a linear temperature profile in theory then this would mean equilibrium, which means that there would be not heat transfer or diffusion. This explains why we must take the second derivative - the curvature - in order to see how heat will transfer until it is in equilibrium.
Initial Conditions
The initial conditions are what the temperature of the plate looks like at t = 0. In this case, the boxcar function is the initial condition.
​
in case you don't know what B(x) means:
​
​
​
​
This will mean that the rod initially has this temperature profile - showing which parts are hot and cold
Boundary Conditions
The x in the boundary condition will give where the boundary is enacted. For an example, if it is a rod of lentgh 1 and the boundary condition has x=1 then the condition is true for the right side of the rod (at x=1).
When the function is a constant this means that it is being held at a constant temperature. When the function is positive it means that the plate is being heated at that location and when the function is negative it means that the plate is being cooled at this location
When the function is any combination of positive and negative values (often sine or cosine) the plate is being cooled or heated the value of the function along that line.
When the boundary conditions contains either Ut or U the temperature change is dependent on the temperature of the plate itself.
Having Ut is altering with the slope of the temperature, therefore if there is a lot of differences on the plate there will be more heat transfer at the boundary.
​
Having U depends on the actual temperature meaning that the more extreme the temperature values the greater the exchange at the boundary.
​
These transfer into having variable input and output.
​
​
​
This boundary condition means that its temperature gradient is 0 at x=1. As heat flows down its temperature gradient, this means that the rod is insulated at x=1. No heat can leave the rod at x=1.
Example 1
This is what it looks like.​
If you want to know how to calculate the solution, feel free to look at this Mathematica code.
Example 2
Describe the following PDE and interpret its boundary conditions.
​
​
​
​
​
​
This PDE is a 1D heat equation. It is a model for a rod of length 1 from x=0 to x=1. It is in a cold bath on its left and insulated on its right. Let's look at the meaning of the different parts:
One-dimensional heat equation
​
​
Length 1, start at 0, goes until 1
​
​
The left side of the rod, at x=0, is in a cold bath of temperature 0.
​
​
The right side of the rod, at x=1, is insulated.