1. (10 points) Use Laplace transforms to estimate drawdown s(r, t) caused by pumping from a zero- radius well. The well operates at the constant flow rate Q during the time interval [0, 7] and then shuts off, resulting in time-varying flow rate Qpump (t) for 0 T Single-phase flow in a reservoir with hydraulic properties D and K and thickness b is described by the single-phase flow equatio ds Var (1) subject to the initial and boundary conditions S(7, 0) = 0, $(00, t) = 0, @pump (t) (2) Compare (graphically) your solution with the solution for constant pumping rate. 2. (10 points) Forced vibrations with a damper are described by coupled ODEs (10) du (11) subject to initial conditions du "(0)(0) - 70)-0) =0. (12) Use the Laplace transformation to find u(t) and v(t). Plot these two functions on the same graph. 3. (10 points) Solve analytically a Fredholm integral equation of the second kind, u(t) at + u(T ) sin(t T)dr. (20) 4. (20 points) Use the Laplace transformation to solve the wave equation, 720, 120 (24) subject to the initial conditions "(z, 0) - $(1) 0321 at ( 2,0) = 0 (25) and the boundary conditions "(0, t) = 0, "( I -+ 00, t) = 0. (26) This problem describes vertical displacements in a long string that has been initially displaced