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exam2 - M ATH 1 52 FALL 2004 MIDTERM 2 20 Problem#1 a Using...

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MATH 152, FALL 2004: MIDTERM #2 Problem #1 2 0 a) Using Fourier Transform solve the initial value roblem with diffusion equation with variable dissipation for K > 0, -cc < x < cc and t > 0. b) Write the solution u above more explicitly when 4 ( x ) = e-"' For this second part you may need to remember that if F denotes the Fourier transform, then ~ ( e ~ " ~ / ~ ) ( ( ) = (2~)~/~61-5'/' and F ( f (ax))(c) = a - ' f ( ~ / a ) for any constant a. Problem #2 $0 Solve the initial and boundary value problem f o r 0 < x < c a , V , a 1 c > 0 a n d a > c . Hint: Solve first the problem with h(x, t ) = V = 0. Problem #3 @ 30 Consider the equation with a > 0. This equation models a one dimensional road with heat loss through the lateral sides with zero outside temperature. Suppose the road has length L and the boundary conditions a) The equilibrium temperatures are the functions u constant with respect to time, hence they are solutions of Find all the solutions u ( x ) of (5). b) Solve the boundary problem given by (3) and (4) with initial data u(x,
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