In this assignment, the Newton-Raphson method is used to find the finite-difference solution to the two-point second-order nonlinear boundary value problem (BVP) u'' + exp(u)u' = µsin(2?x), u(0) = 1, u'(1) + (u(1))^3 = 0, (1) The assignment is to investigate the behaviour of the numerical scheme as a function of the grid spacing and also investigate the behaviour of the solution as a function of the parameter µ. Question 1 Write down an appropriate second-order accurate finite-difference scheme that corresponds to the discretised BVP (equation 1) using N + 1 equally spaced nodes. You should carefully define the grid spacing, grid points and introduce the appropriate notation to describe the numerical solution to BVP at each node. For the boundary conditions ensure second-order accuracy and use the backward-difference formula for u 0 where appropriate. Question 2 Write the difference equations in the form F(u) = 0 and calculate the entries of the Jacobian Jij = ?Fi/?uj . State the Newton-Raphson method used to solve F(u) = 0. What is the size of the matrix J, justify your answer. State a suitable initial guess for the solution to the BVP to be used to initialise the Newton-Raphson scheme, justify your answer.