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hw12Apr2008

# hw12Apr2008 - CAAM 336 DIFFERENTIAL EQUATIONS Problem Set...

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CAAM 336 · DIFFERENTIAL EQUATIONS Problem Set 12 Due Wednesday 9 April 2008 in class. 1. [60 points] We wish to approximate the solution to the heat equation ∂u ∂t - 2 u ∂x 2 = 100 tx, 0 x 1 , t 0 with homogeneous Dirichlet boundary conditions u (0 , t ) = u (1 , t ) = 0 and initial condition u ( x, 0) = 0 using the finite element method (method of lines). Let N 1, h = 1 / ( N + 1), and x k = kh for k = 0 , . . . , N + 1. We shall construct approximations using the hat functions 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 ); 0 , otherwise . The approximate solution shall have the form u N ( x, t ) = N summationdisplay k =1 a k ( t ) φ k ( x ) . (a) Write down the system of ordinary differential equations that determines the coefficients a k ( t ), k = 1 , . . . , N . Specify the entries in the mass and stiffness matrices and the load vector. (You may use results from previous homeworks and class as convenient.) (b) Write a MATLAB code that uses the backward Euler method to solve for the coefficients a k ( t ).

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hw12Apr2008 - CAAM 336 DIFFERENTIAL EQUATIONS Problem Set...

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