Answer Happy

all the steps: +V0 a) (I) = 1+inx QUESTION 1 Solve the following initial and boundary problems. Check your answer by substituting and explain { (10 Marks) b) (20) and (0) Is this equation elliptic, parabolic or hyperbolic? (15 Marks) [25 Marks) QUESTION 2 Assume that for a € R./ is 2-periodic, 8(t) = 11-r) and (x) = f(-a) Denote by A and B! the Fourier coefficients of For all > 1. show that (Explain all the steps!): (a) -- and !-- (10 Marks) (b) = A4 - A.Co - nina and B A vit pa - BCN (15 Marks) [25 Marks] QUESTION 3 17 Consider the temperature distribution in a 2 x 1 rectangular plate as described in the figure below. Show that the steady-state temperature distribution (x,y) in the plate is given by 19 1 (1+(-1))in il 2 (1) 2 Calculate the steady-sato temperature at the centre of the plate. [20 Marks] Ice at loe at Ice at Boiling water at 100 QUESTION 4 Consider the initial boundary value problem below Du + f(xt),0<<< Land > 0 M (0.1) - (...) (t) t > 0 (0) - 1.0 ECL Use the maximum principle for the heat equation, to prove that the above initial value problem has at most one solution (give all details) [30 Marks)