Worksheet 13 Solutions - MA 2930, April 27, 2011 Worksheet...

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Unformatted text preview: MA 2930, April 27, 2011 Worksheet 13 Solutions 1. Find a PDE together with boundary/initial conditions that each of following sets of functions satisfies. (a) { cos( 10 x )sin(10 t ), cos( 3 10 x )sin(30 t ) } (b) { e- 5 t cos( x ), e- 20 t cos(2 x ) } (a) Since the functions involve (co)sines of both x and t , they must be solutions of the wave equation: a 2 u xx = u tt . To find a note that a times the argument of the x-function is the argument of the t- function, so a = 100. So the PDE is 10000 u xx = u tt . As to boundary conditions, the x part is a cosine, so u x (0 ,t ) = 0. We can take the other boundary condition to be u (5 ,t ) = 0 or u x (10 ,t ) = 0 - both are satisfied by both the solutions. Since the t part is a sine, u ( x, 0) = 0. (b) Since the functions involve cosine of x and exponential of t , they must be solutions of the heat equation: a 2 u xx = u t . To find a note that a 2 times the square of the argument of the x-function is the argument of the t- function, so a 2 = 5. So the PDE is 5 u xx = u t . As to boundary conditions, the x part is a cosine, so u x (0 ,t ) = 0. The other boundary condition must be u x ( ,t ) = 0. 2. Derive the polar form of the 2-dimensional wave equation from its cartesian form a 2 ( u xx + u yy ) = u tt . Use x = r cos ,y = r sin and use the chain rule for functions of two variables to express u xx and u yy in terms of partials with respect to r and ....
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Worksheet 13 Solutions - MA 2930, April 27, 2011 Worksheet...

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