hw4 - , x > 0 and w (0 , t ) = h ( t ) , t...

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Homework assignment 4 Due date: Monday April 21, 2008. 1. (# 9.5.5) Inside a circle of radius a , consider Poisson’s equation: 2 u = f ( x ) , u ( a, θ ) = h 1 ( θ ) for θ (0 , π ) , and u r ( a, θ ) = h 2 ( θ ) for θ ( π, 0) . Represent the solution u ( r, θ ) in terms of the Green’s function (assumed to be known). Write down the problem to which this Green’s function is a solution, but do not solve that problem. 2. (# 9.5.14) Using the method of images, solve 2 G = δ ( x x 0 ) , in the Frst quadrant x > 0 and y > 0 with G = 0 on the boundary. 3. (# 9.5.19) Determine the Green’s function inside a semi-disk (0 < r < a and 0 < θ < π ) 2 G = δ ( x x 0 ) , with G = 0 on the boundary. 4. (# 12.2.4) Solve using the method of characteristics: ∂w ∂t + c ∂w ∂x = 0 ( c > 0) , on the Frst quadrant x > 0 and t > 0 if w ( x, 0) = f ( x )
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Unformatted text preview: , x &gt; 0 and w (0 , t ) = h ( t ) , t &gt; . 5. (# 12.2.5 (a) and (d)) Solve using the method of characteristics: w t + c w x = e 2 x , w ( x, 0) = f ( x ) , and w t + 3 t w x = w, w ( x, 0) = f ( x ) , 6. (# 12.6.6 (a)) Consider the following trac ow problem: t + c ( ) x = 0 . Assume that u ( ) = u max (1 / max ) and c ( ) = d/d ( u ( )). Solve for ( x, t ) if ( x, 0) = b max , x &lt; , x &gt; . This initial condition corresponds to an inFnite line of trac stopped at a red light at x = 0 which is started by the light turning green. * MAP 4341; Instructor: Patrick De Leenheer. 1...
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