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Cell["1) Problema Dirichlet (bordi a temperatura fissata = 0)", "Title",
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Problem:   a bar of length L = 1, has initial temperature distribution at t=0 \
given by u(x,0) = 50 x (1-x), x in [0,1].  The two ends of the bar are kept \
at temperature = 0. Find the temperature distribution at t>0 . Assume the \
diffusivity  coefficient is \[Alpha] = 1.

To solve this problem we use the Fourier method. \
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2) Problema con condizione iniziale con un punto di non differenziabilita`\
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Problem:  We consider the same parameters as above, but now the initial \
condition is a sharp triangle: u(x,0) = x for x in [0,1/2] and (1-x) for x in \
[1/2, 1].
We can still use the Fourier method in the same way. Notice that the Fourier \
series captures well solutions with isolated points of non differentiability.\
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Problem:   a bar of length L = 1, has initial temperature distribution at t=0 \
given by u(x,0) = 50 x (1-x), x in [0,1].  The two ends of the bar are kept \
insulated (= Neumann boundary condition). Find the temperature distribution \
at t>0 . Assume the diffusivity  coefficient is \[Alpha] = 1.

To solve this problem we use the Fourier method. \
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Solution at different times. Notice that, despite the fact that the initial \
data did not satisfy the boundary condition (i.e., u_x was different from \
zero at the edges, the condition is satisfied as soon as t>0. This is due to \
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Fourier series will take care to restore them as t>0.   \
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Cell["5) problema finestra scaldata da un lato", "Title",
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Cell["\<\
Si consideri un problema in cui consideriamo la temperatura interna a una \
lastra di vetro (una finestra), inizialmente a temperatura uniforme u(x,t=0) \
= 0. Le condizioni di bordo sono le seguenti: da un lato (x = 0) la \
temperatura e` mantenuta a T1 = 0. Da un altro lato la temperatura viene \
gradualmente aumentata con un andamento descritto dalla funzione f(t), quindi \
u(L, t) = f(t), con f(t) una funzione arbitraria del tempo. \
\>", "Text",
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Cell["\<\
Abbiamo ridefinito u(x,t) = ub(x,t) + v(x,t). Ora, v(x,t) avra` cond. di \
bordo omogenee. Dobbiamo ora ridefinire condizione iniziale per v(x,t) e \
considerare il fatto che la PDE per v(x,t) ora diventa inomogenea.\
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Cell["\<\
Quindi per v[x,t] abbiamo PDE inomogenea con dato iniziale non banale. \
Spezziamo la soluzione in soluzione problema omogeneo con stesso dato \
iniziale + piu` soluzione eq. inomogenea con dato iniziale nullo.\
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