I have the following questions. My due date is on 3rd march.

1. What functionu(x) withu(0) = 0 andu(1) = 0 minimizesP(u) = ∫[½ (du/dx)2 +xu(x)]dx?

(The integral is from 0 to 1)

2. What functionw(x) withdw/dx=x(with unknown integration constant) minimizes

Q(w) = ∫(w2/2)dx? (The integral is from 0 to 1)

3. What functionsuandwminimizePandQwithdw/dx=xandu(0) =w(1) = 0? Verify the

strong duality –P=Q.

4. From the differential equation –d/dx(c du/dx) =f, derive the weak form (4) by multiplying by

test functionsvand integrating.

5. Show thatP(u) +Q(w)>0 for any admissibleuw: and

∫[(c/2) (du/dx)2 –fu+ (1/(2c))w2]dx>0 whenf= –dw/dxandu(0) =w(1) = 0.

(Integral is from 0 to 1)

6. Show thatP(u) +Q(w) = 0 for the optimaluandw, which satisfyw=cu’

Any help is welcome

Thanks