The boundary condition of heat transfer (19) -ux (1I, t) =
Chapter 12, Problem 12.5(choose chapter or problem)
The boundary condition of heat transfer (19) -ux (1I, t) = k[II('IT. t) - uo] applies when a bar of length 'IT with c = I is laterally insulated, the left end x = 0 is kept at OC, and at the right end heat is flowing into air of constant temperature uo. Let k = I for simplicity, and 110 = O. Show that a solution is lI(x, t) = sin px e-p2t where P is a solution of tan p1I = - p. Show graphically that this equation has infinitely many positive solutions PI> P2, P3, ... , where Pn > 11 - i and lim (Pn - 11 + ~) = O. (Formula (19) is also known n_cc as radiation boundary condition, but this is misleading; see Ref. [C3], p. 19.)
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