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hwk9 - Physics 604 Homework#9 Fall 11 Dr Drake 1 Arfken...

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Physics 604 Homework #9 Fall ‘11 Dr. Drake 1. Arfken 15.3.16, 15.3.17, 15.4.3. For 15.4.3 also calculate the solution by inte- grating across x = 0 to obtain the jump in the first derivative and then match to the solutions in the outer region. 2. The temperature in a one-dimensional medium satisfies a diffusion equation with diffusion coefficient D . At t = 0 a very localized source of heat is turned on. The equation satisfied by T is ∂T ∂t - D 2 ∂x 2 T = H ( t ) δ ( x ) where H = 0 for t < 0 and H = 1 for t > 0. Assume that T = 0 for t < 0. (a) Solve for the space/time dependence of T ( x, t ) by first completing a Laplace transform and then a Fourier transform of the equation and then complet- ing the subsequent inverse Laplace transform. The inversion of the Fourier transform can not be done exactly so you will have an integral representa- tion for the solution given by a k space integral. (b) Consider the time dependence of T at x = 0. The k space integral can again not be carried out but by defining an appropriate dimensionless
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