Question

Mark which statements below are true, using the following:

Consider the diffusion problem au Ou u(0, t) = 0, u(L, t) = 50 u(x,0-fx where FER is a constant, forcing term. Any attempt to solve this using separation of variables fails. This is because the PDE is not homogeneous. A more fruitful approach arises from splitting the solution into the sum of two parts, taking into account that all change eventually dies out. That is there is a transient part and a steady state part. From this we can proceed to separate, u(x, t-X(x)T(t) us(x), where the subscript designates the function as the steady limit and does not represent a derivative BEWARE: MARKING A STATEMENT TRUE THAT IS ACTUALLY FALSE RECEIVES A NEGATIVE MARKI A. Upon separation of variables, the time problem is given by , T (t) + αΤ (t) 0 The functions of the set (sin()n 1,2,3,...) are orthogonal on [0, L] The steady state solution is 50 ug(x) -x+Fx(x-L) im uf(x, t) 0 For the functions of the set {sin(X )n- 1,2,3,.. zero and negative values of are not considered because they are not orthogonal The time part of the solution is obtained by solving the following BVP Substituting the alternative split form of a solution with transient and steady parts gives " ' u, (x) + X (x)T (t) = X(x)T (t) cannot satisfy u(0, t) = 0, u(L, t) = 50 u(x, t)-usx) cannot be represented by a Fourier series because normal separability does not work The full solution is where 50T し2

Answer 1

  A is true, B is true, C is false, D is true, E is true, F is false, G is false, H is false, I is false, J is false.

let u (x,1)-v(x,17%, (x) hen where u, (x) is a solution for since t, (0)=0 ,-0 50 FL 50 FL now v(x,t) be the solutionof v(Z1)-u (L,1)-4(L) = 50-50-0 the suitable solution is since v(0,1)--(a)e-n -α = 0 then v(x,1)-Σ, in 8-1 (50 FL (1,9-y(x)-塔-(찔쓸) x (50 FL x2(50 FL hence (50 pL 2 L 2 hence the inal solutionis (50 FL