= x, 0 < x ...

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= x, 0 < x <1 (1) Consider the following boundary value problem d du -((1+x)) = dx dx u(0) = u(1) = 0. (a) Determine the constant a of the exact solution u(x) = = 2 1 + a ln(1 + x). After finding the constant a, demonstrate that u(x) is the exact solution to the boundary value problem. (Note: You do not need to solve the DE.) (b) Write down the variational/weak form of the problem. (c) Apply the Galerkin method to construct approximate solutions in the subspaces Vy for N = 1, 2, and 3 using the following trigonometric basis functions $ = sin(xx), 02 = sin(2x), 3 = sin(3x) } Plot the exact and approximate solutions together, and calculate the percentage error in the approximations at x = 0.25, 0.50, and 0.75. (d) Repeat part (c) using the following polynomial basis functions - - { 0 = x(1 2), 62 = 2 ( 2 2) (1 2), 03 = = () ( 2) (1 2) } - x x -x 3 = x, 0 < x <1 (1) Consider the following boundary value problem d du -((1+x)) = dx dx u(0) = u(1) = 0. (a) Determine the constant a of the exact solution u(x) = = 2 1 + a ln(1 + x). After finding the constant a, demonstrate that u(x) is the exact solution to the boundary value problem. (Note: You do not need to solve the DE.) (b) Write down the variational/weak form of the problem. (c) Apply the Galerkin method to construct approximate solutions in the subspaces Vy for N = 1, 2, and 3 using the following trigonometric basis functions $ = sin(xx), 02 = sin(2x), 3 = sin(3x) } Plot the exact and approximate solutions together, and calculate the percentage error in the approximations at x = 0.25, 0.50, and 0.75. (d) Repeat part (c) using the following polynomial basis functions - - { 0 = x(1 2), 62 = 2 ( 2 2) (1 2), 03 = = () ( 2) (1 2) } - x x -x 3