MATH 257 Midterm: MATH 257+316 2010 Winter Test 1
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No calculators or electronic devices of any kind are permitted. There are 12 pages in this test including this cover page, blank pages, and the formula sheet. 3: consider the wave equation with initial conditions utt = uxx, u(x, 0) = f (x) = 1/2 < x < 1, otherwise. ut(x, 0) = 0: [10 marks] suppose the domain is in nite, < x < . Write down d"alembert"s solution, and carefully sketch u(x, 0), u(x, 1/2), u(x, 1): [10 marks] suppose instead the domain is nite, with re ecting (neumann) boundary conditions at x = 1: ux( 1, t) = 0, ux(1, t) = 0, Brie y describe how you would use the method of nite di erences to nd an approximate solution to this problem. Use the notation uk n u(xn, tk) to denote the values of u on the nite di erence mesh, and include how you propose to incorporate the boundary and initial conditions.
