Q5 (By Hand & MATLAB, 20 pts) Consider the initial value problem for 1

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Q5 By Hand Matlab 20 Pts Consider The Initial Value Problem For 1 T 6 Ty 1001t Y 951ty 50000y 49049t 1 (121.39 KiB) Viewed 27 times

Q5 (By Hand & MATLAB, 20 pts) Consider the initial value problem for 1 <t< 6 ty" + 1001t²y" – 951ty' + 50000y = -49049t, with y(1) = 1, y'(1) = y" (1) = 0). 1 2004001 t cos(7 lnt) - 128993 99 The true solution is y(t) = -t+ 1002050 1002056 t sin(7 lnt) + 100205071000 · (a) Formulate this third order linear ODE as a system of first order ODEs. Explore the eigenvalues of the Jacobian Jf of f(t, u) at to = 1 and comment on the stiffness. (b) Solve this problem by RK4 with appropriate number of steps N = 2P (p is an integer). What is the smallest N needed to avoid instabilities? (c) Solve this problem by BDF4 (three input parameters) with N = 16, 32, ..., 4096 (multiplied by 2) steps. Evaluate ||y(tk) – yk]lo for each N and record the run time for N = 4096. Then find the Jacobian J af and pass it to BDF4 to invoke the New- ду ton's method solving for Yn+4. Compare ||y(tk)-yk ||oo and the run time for N = 4096. =