Consider the temperature u.x; t / in a bare slender wire
Chapter , Problem 22(choose chapter or problem)
Consider the temperature u.x; t / in a bare slender wire with u.0; t / D u.L; t / D 0 and u.x; 0/ D f .x/. Instead of being laterally insulated, the wire loses heat to a surrounding medium (at fixed temperature zero) at a rate proportional to u.x; t /. (a) Conclude from that @u @t D k @2u @x2 hu; where h is a positive constant. (b) Then substitute u.x; t / D eht v.x; t / to show that v.x; t / satisfies the boundary value problem having the solution given in (30) and (31). Hence conclude that u.x; t / D eht X1 nD1 cn exp n22kt=L2 sin nx L ; where cn D 2 L Z L 0 f .x/sin nx L dx: 23. Con
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