Differential Equations Lecture Work Solutions 323

# Differential Equations Lecture Work Solutions 323 - Î Î = 1...

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3. Determine the stability requirement to solve the 1-D heat equation with a source term ∂u ∂t = α 2 u ∂x 2 + ku Use the central-space, forward-time difference method. Does the von Neumann necessary condition make physical sense for this type of computational problem? The method is u n +1 j = (1 + k t 2 r ) u n j + r u n j +1 + u n j 1 Substitute a Fourier mode and we get the following equation for
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Unformatted text preview: Î» Î» = 1 + k âˆ† t âˆ’ 2 r + 2 r cos Î² = 1 + 2 r (cos Î² âˆ’ 1 | {z } = âˆ’ 2 sin 2 Î² 2 ) + k âˆ† t Î» = 1 âˆ’ 4 r sin 2 Î² 2 + k âˆ† t If r â‰¤ 1 2 then Î» â‰¤ 1 + O (âˆ† t ) The âˆ† t term makes sense since ku term allows the solution to grow in time and thus Î» (and the numerical solution) must be allowed to grow. 323...
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