Answer Happy

(a) Let h(x, y) = 3ry on the rectangle (0,4] x [0, ]. Write down the double Fourier sine series of h(z,y); that is, write 00 00 m2 h(x,y) = ΣΣ by sin sin(ny) m-1 n-1 where bmn= 3ry sin sin(ny) dy dr. bm,n = (b) Solve the initial value problem Du 1 3/2 Ot on the rectangle [0,4] x [0, ], subject to the boundary conditions u(z, 0, t) = u(z, n, t) = 0 and the initial condition u(z, y,0)=h(x, y) (from part (a)) 00 00 u(x,y,t) = Σ Σ m-In-1 6 u(0, y, t) = u(4, y, t) = 0 || and ma Au

(c) Solve the initial value problem on the rectangle [0, 4] x [0, π], subject to the boundary conditions and the initial conditions (where h(x, y) is from part (a)) 00 u(z,y,t) = m-1 n-1 8²u 01² u(0, y,t) = u(4, y, t) = 0 u(x, y,0) = 0 and and 24 Au Du Ət u(z, 0,t) = u(z, x, t) = 0 (x, y,0)=h(x, y).