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I am attempting to solve the 1 d heat equation using separation of variables.

1d heat equation:

##\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2}##

I used the standard separation of variables to get a solution. Without including boundary conditions right now, the solutions are:

##T(x,t) = f(x) \times g(t)##

where

##f(x) = A_1(sin(\sqrt{\lambda} x) + cos(\sqrt{\lambda} x))## and

##g(t) = A_1 e^{-\lambda \alpha t}##

other terms:

##\alpha =## thermal conductivity

##\lambda =## some constant (solved for in boundary conditions)

##A_1 =## some constant

I've worked through and have a decent grasp on many of the fixed boundary conditions solutions for this problem. I'm trying to model ground temperature with increasing depth, so I know need to have a periodic boundary condition at the surface to represent seasons and then constant temperature deep underground.

Boundary conditions:

##T(0,t) = T_0 + Asin(\frac{2 \pi t}{365} )## (ground surface)

##T(\infty, t) = T_0 ##

##\lim_{x \to \infty} \frac{d T}{d x} = 0 ##

This paper gave the magnitude of thermal oscillations as decreasing in proportion to ##e^\frac{-x}{d_p}## where ##d_p = \sqrt{\alpha t_p/\pi}## is the period depth, t_p is the period time (365 days). I'm not sure how he got that decaying exponential, some kind of coefficient, but I haven't dealt with periodic boundary conditions before and it looks like he didn't use numerical methods to get this one?

Any idea how he was able to calculate the dampening of oscillations??

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# A Damped Thermal Oscillations

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