Solve with boundary conditions u(c, t) 200, u(r, 0) 0. With these imposed conditions

Chapter 14, Problem 20

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Solve Problem 9 with boundary conditions u(c, t) = 200, u(r, 0) = 0. With these imposed conditions, one would expect intuitively that at any interior point of the plate, \(u(r, t) \rightarrow 200\) as \(t \rightarrow \infty\). Assume that c = 10 and that the plate is cast iron so that k = 0.1 (approximately). Use a CAS as an aid in finding the numerical values of the first five eigenvalues \(\lambda_{1}, \lambda_{2}, \lambda_{3}, \lambda_{4}, \lambda_{5}\) of the boundary-value problem and the five coefficients \(A_{1}, A_{2}, A_{3}, A_{4}, A_{5}\) in the solution u(r, t). Let the corresponding approximate solution be denoted by \(S_{5}(r, t)\). Plot \(S_{5}(5, t)\) and \(S_{5}(0, t)\) on a sufficiently large time interval [0, T ]. Use the plots of \(S_{5}(5, t)\) and \(S_{5}(0, t)\)  to estimate the times (in seconds) for which \(u(5, t) \approx 100\) and \(u(0, t) \approx 100\). Repeat for \(u(5, t) \approx 200\) and \(u(0, t) \approx 200\).