PS3 - ( y, t ) solves the 2D heat equation w t = k ( w xx +...

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MATH 412: Problem Set 3 due Thursday, February 21, 2008 1. Consider the following differential equation for u ( x, t ): ± u xx = βu, x Ω , u = 0 , x Ω . Although this is strictly speaking an ODE, use an energy argument to show without solving that for “interesting” solutions ( u 6 = 0) to exist, β must be negative. 2. Strauss 2.4 #16 - diffusion equation with decay 3. Strauss 2.4 #17 - diffusion equation with time-dependent decay 4. (rewrite of Strauss 2.4 #19) If u ( x, t ) and v ( x, t ) are any two solutions to the 1D heat equation, then show that w ( x, y, t ) = u ( x, t ) v
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Unformatted text preview: ( y, t ) solves the 2D heat equation w t = k ( w xx + w yy ) 5. Show explicitly that each of the following is a solution to the 1D heat equation: (a) the heat kernel S ( x, t ) = 1 4 kt exp(-x 2 / 4 kt ) (ignore the singularity at t = 0) (b) the separation of variables solution u n ( x, t ) = sin( nx ) exp(-n 2 kt ) Note that t appears in two dierent places in the two exponentials! 6. Strauss 3.1 #1 -heat equation on the half line 7. Strauss 3.2 #5 -wave equation on the half line...
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