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This was one of the last questions in my math tutorial so unfortunately I didn't get around to attempting it and our class doesn't post the tutorial answers online so I'm hoping you can help show me the working through this problem as I just keep hitting walls, your time is appreciated:

This is a tough one and I'm afraid I don't have a good answer. As I'm sure you're aware, this PDE is similar to the heat conduction equation but with an extra term,

∂u/∂x

which stymies me in being able to use a Fourier transform (or any orthogonal family of functions) to match the initial condition. For the record, separation of variables gives

u(x,t) = θ(t)χ(x)

θ'/θ = (-∂χ/∂x)/χ + (∂^2χ/∂x^2)/χ = k = separation constant

θ(t) = Cexp(-kt)

χ(x) = Aexp(-x/2)･exp(iβx)

where β = β(k) = real exponent .

The IC means χ(x) = f(x), -∞ < x < ∞, which would be OK if we could find an orthogonal family of functions upon which to project f(x). If we didn't have the exp(-x/2) term, then a Fourier Transform would work (sines and cosines). The ODE for χ(x) itself is not in the form of a Sturm-Liouville equation, which would provide some hope.

I'm very curious where this problem came from and what method is used to solve it.

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