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Unformatted text preview: 10/7/2011 1 14.1 BVP, 1 Shooting Method Use our knowledge of IVP to solve BVP • Now, we can solve IVP. • Consider the following IVP of order 2: ? ′′ = ?, ?, ? ′ , ? ∈ [¡, ?] ? ¡ = ? , ? ′ ¡ = ? ′ • We can solve this IVP on interval [¡, ?] with a step length ℎ = ¢−£ by RK4, say. • Now, consider a BVP: ? ′′ = ?, ?, ? ′ , ? ∈ [¡, ?] ? ¡ = ? , ? ? = 10/7/2011 2 How to solve the BVP? • At least two possibilities: (i) How can we use our knowledge of solving IVP to help us to solve BVP? (ii) Use the boundary value to estimate intermediate values as controlled by the ODE • Other ideas? • Let try the first idea in (i) Shooting Method for Nonlinear ODE-BVPs • Nonlinear ODE • • Consider with guessed slope z • • Use the difference between x ( b ) and x b to adjust x ’( a ) • ? = ?(?) is a function of the guessed value z • Use secant method or Newton method to find the correct z value with ? = ?...
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- Fall '11
- Method, Boundary value problem, Secant method, Root-finding algorithm, nonlinear shooting