Question

3. (20 points) Denote u(ar, y) the steady-state temperature in a rectangle area 0 z 10, 0yS 1. Find the temperature in the rectangle if the temperature on the up side is kept at 0°, the lower side at 10° while the temperature on the left side is S0)= sin(y) and the right side is insulated. Answer the following questions. (a) (10 points) Write the Dirichlet problem including the Laplace's equation in two dimensions and the boundary conditions. (b) (10 points) The above Dirichlet problem can be solved by superposition. Let u(r, y)(, y) uCru) Write the required equations (namely, the boundary value problem) to solve r(2,t) and w(z. y), respectively

Answer 1

-3 u(x.)=o (10, Temperature lve llaing The So To u(n,0) =100(io,0X Fametion We hve Disnichlet = o u(x.) = 0 Nao Pt() Bneck tlem Na Se Trs Balem I 7 VIN,0)=10 V(,)= 0

nablem2 with Sin Ty w(o.) 11 rite, Cen + slution ofBraem Where " md ne ladinn un.)e and Then The Required Saludien wr Th Guertinn Thy In ldiun As There n king The Se Not ntndard Lalce Elin. ludion Alrs IYou Need There You A Turt Annuer Iill Broride Comment sldion Alus Sahe It We Can Adthod departin tWritle nd Jainy uper lasition binipe Tren