Scientific Computing

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CS205 Homework #8 Review Session Notes Properties of First Order ODEs Most properties of first order scalar ODEs ( y = λy ) extend naturally to their vector-valued counterparts ( y = Ay ). 1. Existence and Uniqueness . Consider the following scalar and vector-valued initial value problems: y = λy, y ( t 0 ) = y 0 y = Ay , y ( t 0 ) = y 0 In both cases, a unique solution exists. 2. Well-Posedness . We say that an initial value problem is strictly well-posed if its analytic solution decays to zero as t → ∞ , regardless of the initial value condition itself. For the scalar ODE y = λy the necessary and sufficient condition for well- posedness is λ < 0. For the vector-valued case, the corresponding condition is that Re { λ } < 0 for any eigenvalue λ of A . There is also a “relaxed” definition for well-posedness, which is much less common. The relaxed definition requires only that the analytic solution to an initial value problem remains bounded , regardless of the initial conditions. The necessary and sufficient condition for the scalar case is λ
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