11.3 Heat in a cylinder

A long cylinder of radius \(a\) at uniform temperature \(u_0\) has its surface held at \(0\) from \(t = 0\). The temperature depends on \(r\) and \(t\) only, and separation gives \[ u(r, t) = u_0\sum _{n = 1}^\infty c_n\,J_0\Bigl (\frac {j_{0n}r}a\Bigr )e^{-\kappa j_{0n}^2t/a^2}, \qquad c_n = \frac {2}{j_{0n}J_1(j_{0n})} , \] the coefficients of \(1\) from Theorem 11.2.1, using \(\int _0^1J_0(jr)\,r\,dr = J_1(j)/j\), which follows from \((rJ_1(r))' = rJ_0(r)\). For large times one term dominates, decaying at the rate \(\kappa j_{01}^2/a^2 = 5.78\kappa /a^2\); its coefficient \(c_1 = 1.602\) exceeds \(1\), so the one-term approximation overestimates at early times (Software 11.6.2).

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