Can anyone give me hints of how to solve this problem?
Problem: Solve the heat equation du/dt=d^(2)u/dx^(2), 0<x<4,t>0, with the conditions: u(0,t)=u(4,t)=0 for all t>=0, and u(x,0)=f(x), where f(x)=x for 0<=x<=2 and f(x)=4-x for 2<=x<=4.
Can anyone give me hints of how to solve this problem?
Problem: Solve the heat equation du/dt=d^(2)u/dx^(2), 0<x<4,t>0, with the conditions: u(0,t)=u(4,t)=0 for all t>=0, and u(x,0)=f(x), where f(x)=x for 0<=x<=2 and f(x)=4-x for 2<=x<=4.
You have:
Then this reduces to finding the Fourier Sine series of right? Or rather the odd-extension of the function which I've drawn in the first plot below. Suppose that's all you had to do, find . Can you do that? Then we can solve the PDE and it should look like the second plot.
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No. Look carefully at the plot of the odd extension. The leg from -4 to -2 is not defined by 4-x. Make sure you have the odd extension defined properly, then integrate over the entire interval (-4,4) taking the proper leg of the graph in each interval correctly. You want:
Right? I know it's a mess. Just take it a little at a time. You know since it's an odd function, the cosine terms will drop out right? I'll do one anyway on the interval (2,4):
You need to calculate these correctly over the appropriate interval determined by the value of f(x) so that means the part actually has three terms right? Try calculating the coefficient on each of the three intervals. Also, since it's odd, probably a way to combine the integrals to make the calculations easier. I just did them all in Mathematica.