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I have got this question in my exam and i was not able to solve it . The hint that i had gotten was to use Fourier transform and solve it . But i couldn't . . enter image description here

Can anyone help me . Thank you .

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Let $u$ and $\bar u$ be the solutions with initial values $f$ and $\bar f$ respectively. Then $\bar u-u$ is the solution with initial value $n^{-1}\sin(n\,\pi\,x/L)$. Computing this solution (by separation of variables) we get $$ \bar u(x,t)-u(x,t)=\frac1n\,e^{\tfrac{k\,\pi^2\,n^2}{L^2}t}\sin\Bigl(\frac{n\,\pi\,x}{L}\Bigr). $$ Then $$ \sup_{0\le x\le L}|\bar f(x)-f(x)|\to0\quad\text{as}\quad n\to\infty, $$ while for any $T>0$ (and assuming $k>0$) $$ \sup_{0\le x\le L,0\le t\le T}|\bar u(x,t)-u(x,t)|\to\infty\quad\text{as}\quad n\to\infty $$

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Sir , did u get the expression using seperation of variables and then using inverse fourier transform ? I am using fourier transfrom but its getting quite messy . –  Theorem Jul 25 '12 at 9:00
    
Looking for solutions of the form $T(t)X(X)$ leads (among others) to solutions of the form $e^{k\lambda^2t}\sin(\lambda x)$. Now choose an appropriate $\lambda$. –  Julián Aguirre Jul 25 '12 at 9:26
    
Thank you sir for ur great help. –  Theorem Jul 25 '12 at 10:11
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