Question

A copper rod 5050 cm long with insulated lateral surface has initial temperature u(x,0)=2xu ( x , 0 ) = 2 x, and at time t=0t = 0 its two ends are insulated. (a) Find u(x,t)u ( x , t ). (b) What will its temperature be at x=10x = 10 after 11 min? (c) After approximately how long will its temperature at x=10x = 10 be 45C45 ^ { \circ } \mathrm { C }?

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Answer for (a) :

According to Theorem 22, Heated rod with insulated ends we know that the boundary value problem

ut=k2ux2, (0<x<L,t>0);\frac{\partial u}{\partial t}=k\frac{\partial^2 u}{\partial x^2},~(0<x<L,t>0);

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

has the formal series solution

u(x,t)=a02+n=1anexp(n2π2kt/L2)cosnπxLu(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

an=2L0Lf(x)cosnπxLdx,(1)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)f(x).

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