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Unformatted text preview: CAAM 336 Â· 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 âˆ‚ 2 u âˆ‚t 2- c 2 âˆ‚ 2 u âˆ‚x 2 = 0 , â‰¤ x â‰¤ 1 , t â‰¥ with homogeneous Dirichlet boundary conditions u (0 , t ) = u (1 , t ) = 0 and initial conditions u ( x, 0) = Ïˆ ( x ) âˆ‚u âˆ‚t ( x, 0) = Î³ ( 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 ) âˆˆ 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 âˆ‚u âˆ‚x dv dx dx b. Let N â‰¥ 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 Ï† k ( x ) = ( x- x k- 1 ) /h, x âˆˆ [ x k- 1 , x k ); ( x k +1- x ) /h, x âˆˆ [ x k , x k +1 ); , otherwise ....
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- Fall '09
- Boundary value problem, Partial differential equation, Boundary conditions, homogeneous Dirichlet boundary, Dirichlet boundary conditions