2

Question 6. Consider the Dirichlet Problem Au = 0 on D = (0, 1) x (0,2) u(0,y) = u(x, 0) = u(x, 2) = 0, u(1,y) = g(y). where function g has a Riemann integrable derivative and g(0) = g(2) = 0. a) Derive a formal series solution in terms of the Fourier series of g: g(y) = ) an sin inTy 2 "b) Prove that the formal series solution derived in a) converges uniformly to a classical solution of the given Poisson problem. c) If g(v) = v(2 - y) calculate an approximation of the solution given by the sum of grittithe first three nonzero coefficients of the series solution