Lecture 14 : Introduction to the Fourier Method
I. Heat Equation with Dirichlet BC
* Recall the heat equation with Zero Dirichlet BC
JUf= Kume , 042cL IT Ufo,t)=U(IT,f) = 0
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Luca,o)= now
* Note that this equation has the ” special” family of solutions :
Ucaf)= e-“”t sirloin] NEIN.
Because the equation is linear & homogeneous N
we can add solutions Ubyt)= It an Unlsyt n=I
This brings us to figuring out how to N
write hocus = 2T an Sircnx) . n=1
Note: In general we need an infinite series.
B . Finding coefficients
one can find the coefficients
by integrating
Sihlnxsinlmx)=¡ê(cos ( (n- m )x ) – coslcntmixs)
J’T /11-12 , n=M o sinlnxcoscmx)dk=2o , ntm
*This means £¤SotSinCnn)Uolx)dx=an .
* Given Uocx) w e c a n compute these numbers . Then the issue is converged !
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