Answer Happy

5. Consider a rectangular plate with width w and hight h. If the temperature at all edges are kept to be zero, except at the bottom edge, where it is specified by an arbitrary function f(x) as illustrated in the given figure. Then, the steady-state temperature distribution, u(x, y), inside the plate is modelled by the following boundary value problem: Au Urn + uyy = 0. 0 <r<w, 0<y<h, u(0, y) = u(w, y) = 0, 0<y<h, uz,0) = f(c), ur, h) = 0,0<r<w. (0,h) A un = 0 (w, h) 4 = 0 Au = 0 = 0 (0,0) 1 - 10 (1,0) Using separation of variable method, the solution is given by ππ1

(0,0) u = f(x) (w,0) Using separation of variable method, the solution is given by u(x, y) = { [an x, an cosh g)+b, sinh sin 75 y)] n=1 where, the coefficients, an and bn, to be determined using the boundary conditions at the bottom and top edges. Find the steady-state temperature for the following cases: (i) w=h=1 and f () = 3 sin(27x) + 2 sin(542). x 0. 0 <r <1 0< (ii) w = 2. h=1 and f(x) = 11. 1<:<2. 1 <r