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Transcript
EGR 511 NUMERICAL METHODS _______________________ LAST NAME, FIRST Problem set #9 1. Use the Galerkin Finite-Element Method to approximate the solution to the boundary-value problem 2 2 x, for 0 x 1 with y(0) = y(1) = 0 4 4 16 using x0 = 0, x1 = 0.3, x2 = 0.7, x3 = 1 and compare the results to the actual solution 2 1 1 y(x) = - cos x sin x + cos x. 2 2 4 3 3 6 y” + y= cos 2. Show that the finite difference discretization of (x + 1)uxx + (y2 + 1)uyy - u = 1 0 x 1, 0 y 1, x = y = 1/3 with u(0,y) = y, u(1,y) = y2, u(x,0) = 0, u(x,1) = 1 is given by 10 4 1 0 5 9 3 u11 3 13 17 20 4 0 u 3 3 12 = 9 9 2 5 17 10 0 u21 27 3 3 9 u 5 13 19 22 56 0 27 3 9 3 3. Solve the one-dimensional heat conduction equation d 2T = f(x) dx 2 for a 10-cm rod with boundary conditions of T(0, t) = 50 and T(10, t) = 100 and a uniform heat source of f(x) = 20. Use the trial function T = ax2 + bx + c 4. a) Use the Rayleigh-Ritz method to approximate the solution of y” = 3x + 1, y(0) = 0, y(1) = 0, using a quadratic in x as the approximating function. b) Solve the problem by collocation, setting the residual to zero at x = 0.5. c) Solve the problem by Galerkin’s method. 5. Develop the elements equations for a 10-cm rod with boundary conditions of T(0, t) = 40 and T(10, t) = 100 and a uniform heat source of f(x) = 20. Employ four equal-size elements of length = 2.5 cm. Compute the temperature distribution for the entire rod. dc d 2c 6 Use Galerkin’s method to develop an element equation for D 2 U - kc = 0. dx dx