hw13Apr2006

hw13Apr2006 - CAAM 336 DIFFERENTIAL EQUATIONS Problem Set...

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Unformatted text preview: CAAM 336 · DIFFERENTIAL EQUATIONS Problem Set 13 Posted Friday 11 April, 2008. Due Friday 18 April, 2006 in class. 1. [50 points] We wish to approximate the solution to the homogenous wave equation ∂ 2 u ∂t 2- c 2 ∂ 2 u ∂x 2 = 0 , ≤ x ≤ 1 , t ≥ with homogeneous Dirichlet boundary conditions u (0 , t ) = u (1 , t ) = 0 and initial conditions u ( x, 0) = ψ ( x ) ∂u ∂t ( x, 0) = γ ( x ) using the finite element method (method of lines). a. Write down the weak formulation of the initial, boundary value vave problem by taking the L 2 (0 , 1) inner product of both sides with a test function v ( x ) ∈ C 2 D (0 , 1) . Integrate by parts to write the weak formulation in terms of the familiar energy inner product : a ( u, v ) = integraldisplay 1 c 2 ∂u ∂x dv dx dx b. Let N ≥ 1, h = 1 / ( N +1), and x k = kh for k = 0 , . . ., N +1. We define our basis of hat functions in the usual way φ k ( x ) = ( x- x k- 1 ) /h, x ∈ [ x k- 1 , x k ); ( x k +1- x ) /h, x ∈ [ x k , x k +1 ); , otherwise ....
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  • Fall '09
  • Tompson
  • Boundary value problem, Partial differential equation, Boundary conditions, homogeneous Dirichlet boundary, Dirichlet boundary conditions

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hw13Apr2006 - CAAM 336 DIFFERENTIAL EQUATIONS Problem Set...

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