Thread: The heat equation View Single Post
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 Quote by Gregg I have forgotten about the function ## F(x,t) = \phi(x) G(t)## We have solutions of the form ##F_n(x,t) = (A_n \sin (n \pi x) + B_n \cos (n \pi x) ) c_3 e^{-\lambda t} ## Ah so we have ##F(x,0) = x^2, ## ##x\in [0,1] ## So this series ##F(x,0) = \sum_{n=0}^{\infty} \left( A_n \sin (n \pi x) + B_n \cos (n \pi x) \right)= x^2 ## ## x \in [0,1] ## So that's where Fourier comes in
You mean$$\phi_n(x) = \left( A_n \sin (2n \pi x) + B_n \cos (2n \pi x) \right)$$ and $$G_n(t) = e^{-4n^2\pi^2 t}$$so you are looking for a solution of your PDE like$$F(x,t) =a_0 + \sum_{n=0}^\infty \phi_n(x)G_n(t)= a_0 + \sum_{n=0}^\infty \left( A_n \sin (2n \pi x) + B_n \cos (2n \pi x) \right) e^{-4n^2\pi^2 t}$$It is only now that you can use the condition $$F(x,0)=x^2$$Also I have included a constant term ##a_0##. You need to work the case ##\lambda=0## to verify you get a constant term.