Consider the heat equation
delta(u)/delta(t) = (delta^2)(u)/delta(x^2)
Show that if u(c, t) = (t^alpha)psi(E) where E = x/sqrt(t) and alpha is a constant, then psi(E) satisfies the ordinary differential equation
alpha(psi) = 1/2 E(psi) = psi, where ' = d/dE
is independent of t only if alpha = - 1/2. Further, show that if alpha - 1/2 then
C - 1/2 E(psi) = psi
where C is an arbitrary constant. From this last ordinary differential equation, and assuming C = 0, deduce that
u(x, t) = A/sqrt(t) e^-x^3/At
is a solution of the heat equation (here A is an arbitrary constant).
Show that as t tends to zero from above,
lim(1/sqrt(t) * e^(-x^2/4t)) = 0 for x =/ 0
and that for all t > 0
The integral of 1/sqrt(t) * e^-x^3/4t dx = B
where B is a finite constant
Given that the integral e^-x^2 dx = sqrt(pi), find B. What physical and/or probabilistic interpretation might one give to u(x,t)?
This solution investigates the heat equation in an attached Word document.