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You can use energy methods to show that solutions to the Heat Equation are unique. I was just wondering why it is then that separation of variables, the fundamental solution and the method of Fourier Transforms all give different solutions to the standard Heat Equation?

The problem I am considering is

$$u_t = Du_{xx}, \,\,\,\, 0 \leq x \leq L, \,\, t>0$$ $$u(x,0) = f(x)$$ $$u(0,t) = u(L,t) = 0.$$

Using separation of variables, I get the solution to be $$u(x,t) = \sum_{n=1}^{\infty} C_ne^{\frac{n^2\pi^2}{L^2}Dt}\sin\left(\frac{n\pi x}{L}\right) \text{ where } C_n = \frac{2}{L}\displaystyle \int_0^L f(x)\sin\left(\frac{n \pi x}{L}\right).$$

The Fundamental solution to the Heat Equation on the other hand is $$u(x,t) = \frac{1}{\sqrt{4\pi Dt}} \int_{-\infty}^{\infty} e^{-(x-y)^2/4Dt}f(y) \, dy$$ And then using Fourier Transforms gives the same answer as the Fundamental solution.

Is it just the case that the latter two are for solving the heat equation where $x$ is unbounded (and so there are no boundary conditions)?

And then is it the case that the solution is only unique when $x$ is bounded (and so the Fourier series separation of variables solutoin is unique for the problem it is solving, but the other two methods don't provide unique solutions for the problems they're solving?)

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    You need to give more information as to what specific problem you're referring to and what solutions you have that seem different. The solution is only unique if you're considering an actual boundary value problem. – 2017-01-04
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    @K.Power Edited – 2017-01-04
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    Your fundamental solution does not satisfy the boundary conditions of the given problem - it is the solution of a different problem for the heat equation on the whole real line, not a finite interval. The fourier transform technique also applies to the whole real line so gives the same solution as the fundamental one. Laplace transform can be used to find a solution appropriate to the half line. – 2017-01-04
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    @Paul Ah, okay. Just to make sure I understand though, just because Fourier transforms give the same solution as the fundamental one, that doesn't mean the solution is unique does it, as the solution is only unique for bounded intervals? – 2017-01-04
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    A partial differential equation consists of 2 parts, the PDE describing how the solution changes and the boundary conditions describing the solution behavour at the boundary. The boundary conditions have a profound effect on the solution and on the solution process. Once the boundary conditions are well posed the solution is unique (though not all methods will be appropriate in finding it) – 2017-01-04

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