Initial Temperature - Partial Differential Equations - Solved Exam, Exams of Differential Equations

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MATH848 - April 11, 2011 NAME: Exam JI - 50 Points In order to receive full credit, SHOW ALL YOUR WORK. Full credit will be given only if all reasoning and work is provided. When applicable, please enclose your final answers in boxes. 1. (10 Points) Conceptual Questions. For the following questions assume thet we arc considering the physical prohlem on a bounded domain, x € [0,1]. (a) Write down the heat aud wave equations and also any initial conditions needed for unique solutions. AL ane ur om 2D 4 a ‘ oon a i homogeneous heat problem with bound- ary conditions, u,(0,t) — 0, u,(1, 4) = 0. Describe the physical meaning of these boundary conditions and graph the temperature profile for t +> oe. Lo =Ta v f ay ‘ 1 Lek s kee Los 4 Vad ae \ > (cy Assume that the following graph is the initial displaccment of an elastic string modeled by the fiomoe- . Describe the pleysical geneous wave equation subject to boundary conditions u(0,!) — 0, u(1 boundary conditions and describe time-dynamics of the points, 2, and Py, on this ineaning of the: uming that the string has no initial velocity. v clastic string