PARTIAL DIFFERENTIAL EQUATION QUESTION

ANSWER ALL PARTS PLEASE

(10 points) For each pair of a flux function, o(z,t), and a source/sink function, f(r.). give the conservation law associated with density (I. t). (a) (1,t) = utvuz,t), f(r.t) = cos(I)(2. t). (b) 0(,t) = } (u(x, t))": f(, t) = Vuzz(x, t). (10 points) Consider the initial value problem U + cos(t)uz = ute u(2.0) = r* cos(I). What is the solution, u(1.1)? (10 points) Let u(x, t) satisfy the boundary value problem Ug = c uzz for 0 0, u(0,t) = u((,t) = 0 for t >0. Show that r(") (x, t) = cos(nad) sin (972) is a solution of the above equation for any n What initial conditions does n (%, t) satisfy? (10 points) For each pair of a flux function, o(z,t), and a source/sink function, f(r.). give the conservation law associated with density (I. t). (a) (1,t) = utvuz,t), f(r.t) = cos(I)(2. t). (b) 0(,t) = } (u(x, t))": f(, t) = Vuzz(x, t). (10 points) Consider the initial value problem U + cos(t)uz = ute u(2.0) = r* cos(I). What is the solution, u(1.1)? (10 points) Let u(x, t) satisfy the boundary value problem Ug = c uzz for 0 0, u(0,t) = u((,t) = 0 for t >0. Show that r(") (x, t) = cos(nad) sin (972) is a solution of the above equation for any n What initial conditions does n (%, t) satisfy