a.) Give a solution to the two-dimensional Laplace equation () on a rectangular domain, with sides a and b, if = 0 is identical on one side, and increases linearly between a given initial and final value, but for the other two sides, n = 0 (i.e. the derivative in the normal direction is zero; this problem is therefore a mixed boundary value problem). Calculate the coefficients of the solution only.

b.) Here the boundary should be a trough of width a, infinitely long in the y direction, with Neumanntype boundary conditions on all sides: the derivative in the normal direction is zero on the sides, and at the bottom of the trough a step function: B0 on one side and +B0 on the other side. Here we also give the sum of series form of the solution, calculate the coefficients only.