## Related questions with answers

A copper rod $50$ cm long with insulated lateral surface has initial temperature $u ( x , 0 ) = 2 x$, and at time $t = 0$ its two ends are insulated. (a) Find $u ( x , t )$. (b) What will its temperature be at $x = 10$ after $1$ min? (c) After approximately how long will its temperature at $x = 10$ be $45 ^ { \circ } \mathrm { C }$?

Solution

Verified### Answer for (a) :

According to Theorem $2$, Heated rod with insulated ends we know that the boundary value problem

$\frac{\partial u}{\partial t}=k\frac{\partial^2 u}{\partial x^2},~(0<x<L,t>0);$

$u_x(0,t)=u_x(L,t)=0,~u(x,0)=f(x)$

has the formal series solution

$u(x,t)=\frac{a_0}{2}+\sum_{n=1}^{\infty} a_n \exp\left(-n^2\pi^2 k t/L^2\right)\cos \frac{n\pi x}{L}$

where

$a_n=\frac{2}{L}\int_0^L f(x)\cos \frac{n\pi x}{L}dx,\tag{1}$

Fourier cosine coefficient corresponding to the function $f(x)$.

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