The solution to du/dt = Au = 0 1 1 0 u (eigenvalues i and i) goes around in a circle: u
Chapter 5, Problem 5.3.16(choose chapter or problem)
The solution to du/dt = Au = 0 1 1 0 u (eigenvalues i and i) goes around in a circle: u = (cost,sint). Suppose we approximate du/dt by forward, backward, and centered differences F, B, C: (F) un+1 un = Aun or un+1 = (I +A)un (this is Eulers method). (B) un+1 un = Aun+1 or un+1 = (I A) 1un (backward Euler). (C) un+1 un = 1 2 A(un+1 +un) or un+1 = (I 1 2 A) 1 (I + 1 2 A)un. Find the eigenvalues of I +A, (IA) 1 , and (I 1 2 A) 1 (I + 1 2 A). For which difference equation does the solution un stay on a circle?
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