Solutions to Problems for 2D & 3D Heat and Wave Equations
نویسنده
چکیده
(i) Solve for u (x, y, t) subject to an initial condition u (x, y, 0) = 100. (ii) Find the smallest eigenvalue λ and the first term approximation (i.e. the term with e−λκt). (iii) For fixed t = t0 ≫ 0, sketch the level curves u = constant as solid lines and the heat flow lines as dotted lines, in the xy-plane. (iv) Of all rectangular plates of equal area, which will cool the slowest? Hint: for each type of plate, the smallest eigenvalue gives the rate of cooling. (v) Does a square plate, side length L, subject to the BCs (1) cool more or less rapidly than a rod of length L, with insulated sides, and with ends maintained at 0 C? You may use the results we derived in class for the rod, without derivation. Solution: (i) We separate variables as
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