## Related questions with answers

Find the temperature u(x,t) in a bar of silver of length 10 cm and constant cross section of area 1 cm² (density 10.6 g/cm³, thermal conductivity 1.04 cal/(cm sec °C), specific heat 0.056 cal/(g °C) that is perfectly insulated laterally, with ends kept at temperatures 0°C and initial temperature f(x)°C, where f(x)=4-0.8|x-5|

Solution

VerifiedThe solution is represented via Fourier series

$\boxed{u(x,t)=\sum_{n=1}^{+\infty} B_n\sin\frac{n\pi x}{L}e^{-\lambda_n^2t},}$

where:

- $L$ is the length, $\lambda_n$ is the $n-th$ eigenvalue, computed as $\lambda_n=n c \pi/L$.
- $c^2$ is diffusivity, calculated as $c=\sqrt{K/(\sigma\rho)}$.
- $B_n$ are coefficients computed as $B_n=\frac{2}{L}\int\limits_0^L f(x)\sin(n\pi x/L)\dd{x}$.

Since $K=1.04\ \textrm{cal}/(\textrm{cm} \ \cdot\textrm{sec}\ \cdot \ ^\circ \textrm{C} )$, $\sigma=0.056\ \textrm{cal}/(\textrm{g}\ \cdot \ ^\circ \textrm{C})$, $\rho=10.0\ \textrm{g}/\textrm{cm}^3$ and $L=10\ \textrm{cm}$, we have

$\boxed{c\approx 1.32364,\quad \lambda_n\approx0.416\cdot n,\quad \lambda_n^2\approx 0.172918\cdot n^2.}$

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