Question

1. Consider Poisson's equation Au=Vu=-1, with A=Ví = & +i domain shown in Figure 1 in the Lx M rectangular Figure 1: Damain with the boundary conditions in Q1 with boundary canditians u(0, y) = u(L, y)= u(z,0) 0 and u(r, M) 1 (a) Asume M L and determine the values af uat points A, B, C and D (b) Generalize your answer in part (a) for M#L 2. Consider Laplace's equation Au = V2u = 0, with A = v2 Figure 2. The values of the function is u 10 alang the inner walls af the domain Along the vertical outer wall u u 0 and akng the horizontal outer wall uy values of u at paints 1, 2, 3 and 4 in the domain, shown in =0. Determine the Figure 2: Damain with the boundary conditions in 02 3. (Statianary temperature field u(x, y) in the domain in Figure 2 satisfies the Laplace equation Au Vu 0. Assurne that the value of u is fixed to u 0along all inner and outer boundaries Can you predict the value of the temperature in all points af the domain without carrying out calculations? Justify your answer

Answer 1

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is soluction of Lba Lablace To qet Poinen's Mk nn JASinh This Substitute n M Sin hnm Snhnny A L mn wa's Sim drcy o Sinnny L MA Sinhn ucx,y) 4 tird omy value inside the rect By this you Vu=o appli cakla Some seperation of variahle matod
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