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# 14.2 - 14.2 A Discretization Method BVP Revisited = = = We...

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10/12/2011 1 14.2 A Discretization Method 10/11/2011 BVP Revisited ? ′′ = 𝑓 ?, ?, ? , ? ∈ [?, ?] ? ? = ?, ? ? = ? We have looked at how to use IVP to help us solve BVP using the “shooting method” Now, we take a different approach: starting from the boundary values and try to fill in the intermediate values How? What if we know (or pretend that we know) the function values at ? 𝑖 = ? + 𝑖ℎ, 𝑖 = 1,2, … , 𝑛?

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10/12/2011 2 Writing ? 𝑖 = ? ? 𝑖 , ? 𝑖 = ? ? 𝑖 , … The DE at ? 𝑖 are: ? 𝑖 ′′ = 𝑓 ? 𝑖 , ? 𝑖 , ? 𝑖 , 𝑖 = 1,2, … , 𝑛 − 1 ? 0 = ?, ? 𝑛 = ? Using ? 𝑖 to estimate ? 𝑖 and ? 𝑖 ′′ : ? 𝑖 = 𝑥 𝑖+1 −𝑥 𝑖−1 2ℎ + 𝑂(ℎ 2 ) (Why?) ? 𝑖 ′′ = 𝑥 𝑖+1 −2𝑥 𝑖 +𝑥 𝑖−1 2 + 𝑂(ℎ 2 ) (Why?) So, the DE becomes: 1 2 ? 𝑖+1 − 2? 𝑖 + ? 𝑖−1 = 𝑓 ? 𝑖 , ? 𝑖 , ? 𝑖+1 − ? 𝑖−1 2ℎ + 𝑂 2 ? 0 = ?, ? 𝑛 = ? The Linear Case Assume 𝑓 ?, ?, ? = ? ? + ? ? ? ? + ? ? ?′(?) Then the discretized system reduces to ? 0 = ? 1 2 ? 𝑖+1 − 2? 𝑖 + ? 𝑖−1 = ? 𝑖 + ? 𝑖 ? 𝑖 + ? 𝑖 1 2ℎ (? 𝑖+1

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14.2 - 14.2 A Discretization Method BVP Revisited = = = We...

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