temperature is To and initially the fluid is at temperature T1. If V is the speed of the fluid in the x direction, a problem describing the temperature u(x, y,t) is: ди + V Ət x > 0, 0 < y < b, t> 0, k- u(x,0, t) = 0, u(x, b, t) = 0, u(0, у, t) — То, и (х, у, 0) 3 Ti- Make a separation of variables as above. State and solve the eigenvalue problem for Y . Show that Un(X, Y, t) = ¢n(x – Vt)e-"ikle+Vt) 1,2, 3, ... are the eigen values for the Y problem, satisfies the sin µnY %3D where An = µ, for n = PDE and boundary conditions at y = 0 and y = b, without restrictions of øn (except differentiability). %3D

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