For my high-school maths exploration, I have chosen to simulate heat flow in a one-dimensional rod using GNU Octave. I will then solve the heat equation and judge the accuracy of the heat equation in modelling the data.
For judging the accuracy of the model, I was thinking about using RMSE, but I can't find any formulae which work for three variables (temperature, time, length). Are there any other formulae of RMSE that can be suited to this purpose?
Alternatively, are there better statistical analysis tools that will help me judge the accuracy of the model?
As heat is continually applied to the rod the temperature will continue to rise if there are no other losses.
I recommend you simulate a rod placed between two temperature sources and calculate the temperature distribution within the rod.
For example, if one end of the rod is subjected to 5-C and the other end of the rod is subjected to 125-C. Then you can simulate the temperature of discrete points along the rod.
This can be done by hand.
Let's say there are five evenly distributed points along the rod. $Point_1$ is at 5-C and $Point_5$ is at 125-C.
Thus: $$Point_1 = 5$$ $$Point_2 = (Point_1 + Point_3) / 2$$ $$Point_3 = (Point_2 + Point_4) / 2$$ $$Point_4 = (Point_3 + Point_5) / 2$$ $$Point_5 = 125$$
Assume $0$ at $Point_2$, $Point_3$, and $Point_4$.
This is solved iteratively by solving for $Point_2$ (which will then be 2.5-C). Then solve for $Point_3$ (which will then be 1.25-C). Then solve for $Point_4$ (which will then be 63.125-C).
Then repeat the process until the values don't change anymore. (In complex calculations the iteration is often stopped when the changes for each calculation fall within a minimum level of acceptable error. Like, 2% of the previous calculation (your choice).)
A programmable calculator does this easily and can present the results continually.