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hw7 - CAAM 336 DIFFERENTIAL EQUATIONS Problem Set 7 Posted...

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CAAM 336 · DIFFERENTIAL EQUATIONS Problem Set 7 Posted Wednesday 6 October 2010. Due Wednesday 13 October 2010, 5pm. General advice: You may compute any integrals you encounter using symbolic mathematics tools such as WolframAlpha, Mathematica, or the Symbolic Math Toolbox in MATLAB. This problem set counts for 50 points, i.e., half the value of the earlier problem sets. The late policy will function as usual on this problem set. 1. [50 points] Use the finite element method to solve the differential equation - ( u 0 ( x ) κ ( x )) 0 = 2 x, 0 < x < 1 for κ ( x ) = 1 + x 2 , subject to homogeneous Dirichlet boundary conditions, u (0) = u (1) = 0 , with the approximation space V N given by the piecewise linear hat functions that featured on the last problem set: For n 1, h = 1 / ( N + 1), and x k = kh for k = 0 , . . . , N + 1, we have 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 . (a) Write MATLAB code that constructs the stiffness matrix K for a given value of N , with κ ( x ) = 1 + x 2 . [You may edit the fem_demo1.m code from the class website. You should compute all necessary integrals (by hand or using a symbolic package) so as to obtain clean formulas that depend on
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