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Differential Equations: Solution to Heat Equation

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)?

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This solution investigates the heat equation in an attached Word document.

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