Question

(b) . Write the k-th step of the trapezoidal method as a root-finding problem Ğ = is Y+1 where the unknown (e)Find the Jacobian matrix of the vector function from the previous part. (dWrite a function in its own file with definition [Y] dampedPendulum(L, T) function alpha, beta, d, h, that approximates the solution to the equivalent system you derived in part (a) with L: the length of the pendulum string alpha: the initial displacement beta: the initial velocity d: the damping coefficient h: the timestep between nodes T: the final time Y: the approximation of the exact solution using the trapezoidal method, stored in two rows MATH353, Spring 2019 Assignment 9 You will need to approximately solve a nonlinear equation at every step. Use the Newton's method for systems with initial guesses, maximum iterations, tolerance and stopping condition of your choice. You may use your own NevtonMethod2D.m from Assignment 6 or the one posted under Files/Code on Canvas. In your prob10: Call your dampedPendulum function to solve (approximately) the damped pendulum equation with parameters L 0.3, T 10,g- 9.8,h- 0.01 and the rest as in the following table Part | α | β | d (B) |π/2 | 5 | 0 (C) | π-1/3 | 2 (D) I π/6 I 0 I 10 from intermediate steps and function Output: Suppress all output and comment all fprintf messages calls, and i. Plot the displacement and velocity for times in [0, T] using your approximations on the nodes. Gen- erate a separate figure for each part ii. Make a comment on your results from each part. Do your approximations behave as expected? Justify the results according to the values of the parameters for each part Please answer all parts, thank you Problem 1 : Damped pendulum & Consider the following second order IVP for the damped pendulum y(0)a y'(0) -β where d >0 is the damping coefficient. The larger d is, the faster the system loses energy and stabilizes to its equilibrium position y -0,y'0 (a) Derive an equivalent system of two ODEs and write it in the standard form init,1 init,2 f2(t,Y) (b) Write the k-th step of the trapezoidal method as a root-finding problem G = 0, where the unknown (c) Find the Jacobian matrix of the vector function G from the previous part.

Answer 1

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