A solution of c(x, t) t = D 2c(x, t) x2 is the function c(x, t) = 1 _ 4 Dt exp _ x2 4Dt

Chapter 10, Problem 66

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A solution of c(x, t) t = D 2c(x, t) x2 is the function c(x, t) = 1 _ 4 Dt exp _ x2 4Dt _ for x R and t > 0. (a) Show that a local maximum of c(x, t) occurs at x = 0 for fixed t . (b) Show that c(0, t), t > 0, is a decreasing function of t. (c) Find lim t0+ c(x, t) when x = 0 and when x _= 0. (d) Use the fact that % eu2/2 du = _ 2 to show that, for t > 0, % c(x, t) dx = 1 (e) The function c(x, t) can be interpreted as the concentration of a substance diffusing in space. Explain the meaning of % c(x, t) dx = 1 and use your results in (c) and (d) to explain why this means that initially (i.e., at t = 0) the entire amount of the substance was released at the origin. Mathematically, we can specify such an initial condition (in which the substance is concentrated at the origin at time 0) by the -function (x), with the property that (x) = 0, for x _= 0 and % (x) dx = 1