# SampleQ_m1 - MATH 412: Sample Questions for Midterm 1 1....

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MATH 412: Sample Questions for Midterm 1 1. Say whether or not the following deﬁnes a vector space (and explain): the set of all solutions u ( x, t ) to the semi-inﬁnite diﬀusion equation u t = ku xx for x (0 , + ), with u ( x, 0) = f ( x ) and (a) u (0 , t ) = 0 (b) u x (0 , t ) = 0 2. Write the Conservation Law equation for the density function u ( x, t ) in R n . Specify what ﬂux function ~ F leads to the heat equation, and substitute it to obtain the heat equation in R n . 3. Consider the 1D heat equation between 0 and L , with Neumann boundary conditions speciﬁed: u t = ku xx , 0 < x < L, t > 0 u x (0 , t ) = Q 1 , t > 0 u x ( L, t ) = Q 2 , t > 0 u ( x, 0) = φ ( x ) , 0 < x < L, where Q 1 and Q 2 are given constants. Find the steady state solution(s), and discuss their dependence on Q 1 , Q 2 , and φ ( x ). 4. Solve the Cauchy problem for u ( x, t ): ± u t + cu x = - λu 3 , x R , t > 0 u ( x, 0) = exp( - x 3 ) , with λ > 0 and c > 0, and assuming

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## This note was uploaded on 07/23/2008 for the course MATH 412 taught by Professor Belmonte during the Spring '08 term at Pennsylvania State University, University Park.

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SampleQ_m1 - MATH 412: Sample Questions for Midterm 1 1....

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