The instantaneous source solution. Consider the function u(x, t) = 1/squareroot

Question

The instantaneous source solution. Consider the function u(x, t) = 1/squareroot t exp(-x^2/4t) (a) Show that u(x, t) is a solution to the heat equation partial differential u/partial differential t = partial differential^2 u/partial differential x^2 (b) Show that the area under the curve for the solution is conserved. That is, defining m = integral^infinity_-infinity u(x, t) dx show that dm/dt = 0 so that m = constant. Hence the area under the curve for the solution does not change in time. (c) Make rough sketches of the solution for various times t > 0.

Explanation / Answer

du(x,t)/dt

= t(-1/2) [e-x2/4t (x2/4)(-1/t2)] + t(-3/2)(-1/2) e(-x2/4t)

= e(-x2/4t) [ x2/4t5/2 - 1/2t3/2 ]

= - e(-x2/4t) (1/2t3/2) (1-x2/2t)

d2u/dx2

initially

du/dx = 1/sqrt(t) (e-x2/4t) (-2x/4t)

d2u/dx2 = -1/2t3/2 [ e-x2/4t + x (-2x/4t) e-x2/4t ]

= -1/2t3/2 ( e-x2/4t ) ( 1 - x2/2t)

hence satisfy heat equation

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