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I'm trying to solve a basic heat equation [tex]\frac{\partial T}{\partial t}=\alpha \frac{\partial^2 T}{\partial x^2}[/tex] I manage to get [tex]T=X(x)\tau(t)[/tex]
Then [tex]\tau(t)=A*e^{-\alpha*\lambda^2*t} and X(x)=C*sin(\lambda x) where \lambda=\pi/Ln n=1,2,3,...[/tex]
From here I don't know how or why I get to a Fourier Serie. Like this [tex] u(t,x) = \sum_{n = 1}^{+\infty} D_n \left(\sin \frac{n\pi x}{L}\right) e^{-\frac{n^2 \pi^2 kt}{L^2}} \\ and \\ D_n = \frac{2}{L} \int_0^L f(x) \sin \frac{n\pi x}{L} \, dx. [/tex]
Can someone explain why I have this Fourier Serie?
Thanks Link
Then [tex]\tau(t)=A*e^{-\alpha*\lambda^2*t} and X(x)=C*sin(\lambda x) where \lambda=\pi/Ln n=1,2,3,...[/tex]
From here I don't know how or why I get to a Fourier Serie. Like this [tex] u(t,x) = \sum_{n = 1}^{+\infty} D_n \left(\sin \frac{n\pi x}{L}\right) e^{-\frac{n^2 \pi^2 kt}{L^2}} \\ and \\ D_n = \frac{2}{L} \int_0^L f(x) \sin \frac{n\pi x}{L} \, dx. [/tex]
Can someone explain why I have this Fourier Serie?
Thanks Link