1. Consider the function u(x,t) defined for 0 <= x <= pi and t >= 0 which satisf
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1. Consider the function u(x,t) defined for 0 <= x <= pi and t >= 0 which satisfies different initial conditions: u_tt = c^2u_xx; u(0, t) = u(pi, t) = 0; u(x0) = 0; u_t(x,0) = x( pi-x): Describe the difference in the physical situation that this models to that of the previous problem, and then solve it. Note that many of the steps will be identical--you do not need to re-derive the solutions of anything you have previously solved! 2.Consider the following boundary value problem for the 2D heat equation: u_t = c^2(Nabla)^2u; u(x,0, t) = u(0, y, t) = u(pi, y, t) = 0; u(x,pi , t) = x(pi - x): (a) What is the steady-state solution, u_ss(x, y)? [Hint: Look at a previous problem on Laplace's equation]. Sketch it. (b) Dene v(x, y, t) = u(x, y, t) - u_ss(x, y), where u_ss(x, y) is your solution to Part (a). Rewrite the PDE including the boundary conditions in terms of v instead of u. The resulting PDE is homogeneous because v(x, y, t) = 0 is a solution. (c) Write down the general solution to this PDE by adding the steady-state solution to the solution of the related homogeneous problem (which you've already solved!).Explanation / Answer
Hey everything is covered in this pdf http://www.math.uchicago.edu/~may/VIGRE/VIGRE2010/REUPapers/Alevy.pdf
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