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Unformatted text preview: 18.03 Problem Set 2 Spring 2009 Due in boxes at Room 2-106, 12:55 pm, Friday, Feb 13 Part A (16 points) (at due date) Write the names of any person, web site, or materials you consulted. Write “No C” (no consultation) on your paper if you consulted no outside materials/people. Lec 2, Fri, Feb 5 Read: EP 6.1, 6.2 Work 1C-4. Lec 3, Mon, Feb 8 Read: EP 1.5 Work: EP 1.5/1, 2, 38; 1D-4 (use λ 1 = 1, λ 2 = 2 in both parts) Lec 4, Wed, Feb 10 Read: EP 1.6 Work: EP 1.5: 5, 13; 1B-5a, 9a Part B (32 points) 1. (Lec 2, Fri, Feb 5) [8pts: 2 + 1 + 3 + 2] a) The solution to y ′ = y with y (0) = 1 is y = e x . What is the Euler approximation to y (1) = e with n equal steps? First fix n , and find y k , k = 0 , 1 , 2 , . . . to see the pattern. b) Explain, using y ′′ , whether the approximation is larger/smaller than the exact answer. (You can also see it at D’AIMP, if you wish.) c) Use the math.mit.edu/daimp Euler’s Method to study y ′ = y 2- x . Find the solution to y ′ = y 2- x with initial condition y (0) =- . 40 using Euler’s method with step sizes h . Make a table with five column headings, namely, h , the estimated value of y est (1) (including actual value, which correponds to h = 0), error E , E/h , and E/h 2 . Does your table support the claim that the error is approximately proportional to h ? d) Turn in two separate printouts showing two ways the numerical approximation can go wrong. The first way is that the long term behavior is nothing like the true limit. The second way is that two numerical solutions...
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- Spring '09
- Numerical Analysis, Numerical ordinary differential equations