MIT OpenCourseWare http://ocw.mit.edu 18.085 Computational Science and Engineering IFall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.18.085 Prof. Strang Quiz 2 November 7, 2003 1 2 3 Your name is: Grading Total 1) (36 pts.) (a) For − d2u = δ(x − a) with u(0) = u(1) = 0, the solution is linear on dx2 both sides of x = a (graph = triangle from two straight lines). What is wrong with the next sentence? Integrating both sides of the equation from x = 0 to x = 1, the area unde r the triangle graph is 1. The base is 1 so the height must be umax = 2. (b) If u(x, a) is the true solution to that standard problem in part (a), and the load point x = a approaches x = 1, what function does the solution u approach? Give a physical reason for your answer or a math reason or both.xxx 22) (40 pts.) Which of these 5 equations can be solved?? If the equation has a solution, please find one. If not why not ?? (a) ⎡ ⎤ ⎣ ∂u/∂x div ⎦ = 1 ∂u/∂y (b) ⎡ ⎤ ∂s/∂y div ⎣ ⎦ = 1 −∂s/∂x (c) uxx+uyy = 0 in the unit circle and u(1, θ) = sin 4θ around the boundary (d) Find a family of curves u(x, y) = C that is everywhere perpendicular 2to the family of curves x + x2 − y = C. (e) d4u/dx4 = δ(x) [point load at x = 0, not requiring boundary con-ditions, any solution u(x) is OK]. 3xxx 43) (24 pts.) I want to solve Laplace’s equation (really Poisson’s equation) in 3D with no boundaries (the whole space). The right side is a point load δ at the origin (0, 0, 0). So −div(grad u) = δ. (a) Integrate both sides over a sphere of radius R around the origin: � � � −div(grad u) dxdydz = 1 . Use the divergence theorem (or Gauss-Green) to transform that triple integral into an integral on the surface of the sphere of radius R. (b) What is the normal vector n in that integral? Knowing that u must be radially symmetric, ∂u is a constant on the sphere of radius R. What ∂r is that constant? (c) So what is u(r)? 5xxx
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