HOMEWORK 2_W13(1) - DREXEL UNIVERSITY Department of...

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DREXEL UNIVERSITY Department of Mechanical Engineering & Mechanics Applied Engineering Analytical & Numerical Methods II MEM 592 - Winter 2013 HOMEWORK #2: Due Thursday, January 24 @ 6 30 pm 1. [35 points] A. [10] Consider the initial value problem (IVP): ݕ ሺݐሻ ൌ ݂ሺݐ, ݕሻ, ݕሺݐ ሻ ൌ ݕ (1) Knowing that the truncation error for the implicit (or backward) Euler method is given by ܶ ௬ሺ௧ ೖశభ ሻି௬ሺ௧ െ ݂ሺݐ ௞ାଵ , ݕሺݐ ௞ାଵ ሻሻ (2) and assuming sufficient differentiability of y, calculate the truncation error and state the order of accuracy. B. [10] Check the stability of the Backward Euler Method. C. [15] Consider the initial value problem ݔ ሺݐሻ ൌ െݔሺݐሻ, ݔሺ0ሻ ൌ 1 . Find its analytical solution. Use the 2nd-order accurate numerical method ݔ ௞ାଵ ൌ ݔ ሺ݂ሺݐ , ݔ ሻ ൅ ݂ሺݐ ௞ାଵ , ݔ ௞ାଵ ሻ ሻ (3) and find a condition that will guarantee convergence. Using this condition pick a value for ݄ and show whether or not your method yields qualitatively meaningful results. 2. [25 points] Consider the IVP ݕ ሺݔሻ ൌ ߣ ݕሺݔሻ ൅ ሺ1 െ ߣሻ cosሺݔሻ െ ሺ1 ൅ ߣሻ sin ሺݔሻ, ݕሺ0ሻ ൌ 1 (4) with true solution ݕሺݔሻ ൌ sinሺݔሻ ൅ cos ሺݔሻ . A. [20] Implement the Euler and Heun’s methods in MATLAB. Plot the analytical, Euler’s and Heun’s solutions for ߣ ൌ െ1 and h =0.5, h =0.1, h =0.01 in the interval ሾ0,5ሿ . Comment your results.
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  • Spring '14
  • Numerical Analysis, IVP, Numerical ordinary differential equations, Applied Engineering Analytical & Numerical Methods II

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