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HW_1 - MAE 290B Homework 1 Numerical Methods in Science and...

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MAE 290B - Homework # 1 Numerical Methods in Science and Engineering Prof. Alison Marsden Due date: Thurs Jan 20, 2011 Problem 1 - Euler Method. A physical phenomena is governed by the linear differential equation dv dt = - 0 . 2 v - 2 cos(2 t ) v 2 subject to the initial value v (0) = 1. (a) Solve this equation analytically. (b) Write a program to solve the equation for 0 < t 7 using the explicit Euler scheme with the following time steps: h = 0 . 2 , 0 . 05 , 0 . 025 , 0 . 006. Plot the four numerical solutions along with the exact solution on one graph with axes 0 < x < 7 and 0 < y < 1 . 4. Discuss your results. (c) In practical problems, the exact solution is not always available. To obtain an accurate solution, we keep reducing the time step (usually by a factor of 2) until two consecutive numerical solutions are nearly the same. Assuming that you do not know the exact solution for the present equation, do you think that the solution corresponding to h = 0 . 006 is accurate (to plotting accuracy)? Justify your answer. In case you find it not accurate enough, obtain a better one. Problem 2 - Apollo Orbit. The following differential equations describe the motion of a body in orbit about two much heavier bodies. An example would be an Apollo capsule in an earth-moon orbit. The three bodies determine a plane in space and a two-dimensional Cartesian coordinate in this plane. The origin is at the center of mass of the two heavy bodies, the x -axis is the line through these two bodies, and the distance between them is taken as the unit. Thus, if μ is the ratio of the mass of the moon to that of the earth, then
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