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PDE Solved Problem 01003

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Show that the solution of the    partial  differential equation  \[ \frac{\partial^2 u}{\partial x^2} = \frac{1}{k}\frac{\partial u}{\partial t} \]  which satisfies the conditions:

  1.   \(\frac{\partial u(x,t)}{\partial x} =0 \quad  \text{ for } x=0 \quad \text{ and }   x=a, \quad \text{ and all }t \),
  2.  \(u(x,t)\) is bounded for all \(-a\le x \le a\) as  \(t\to \infty\)
  3. \(u(x,t)\big|_{t=0}= |x| \),  for \(-a \le x\le a\),

         is given by  \[ y = \frac{a}{2}-\frac{4a}{\pi^2}\sum_{n=0}^\infty\frac{1}{(2n+1)^2} \cos \big(\tfrac{(2n+1)\pi x}{a}\big) e^{-[k(2n+1)^2\pi^2t]/a^2}.\]

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