# Questions tagged [heat-equation]

For questions related to the solution and analysis of the heat equation.

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### Confusion about Separation of Variable with Periodic Boundary Conditions

I'm terribly confused about the separation of variable method for PDE (heat eq particularly). The given conditions are: $u_t(x,t)=u_{xx}(x,t), u(0,t)=u(L,t), u_x(0,t)=u_x(L,t), u(x,0)=f(x)$ Since I ...
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### Is the heat equation unique under general boundary conditions?

Energy methods can be used to show that the heat equation has a unique solution, but this requires specific boundary conditions (from what I know, that $u=0$ on the boundary, though I assume one can ...
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### Equivalence with the Weierstrass transform

I have the following expression $$\frac{1}{\sqrt{4\pi t}}\int_{-\infty}^{+\infty}dx~ f(x-y) e^{-x^2/4 t} \tag{1},~~\forall ~y \in \mathbb{R}.$$. I am trying to relate it with the generalized ...
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### Heat from a geothermal well: your take?

Imagine digging a cylinder-shaped (vertical) bore-well of depth $L$ and diameter $r$ ($L\gg r$). The (infinitely thin) cylinder-wall is made watertight and we split the well in half using a kind of ...
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### Insulated boundary heat equation

I don't fully understand why the boundary insulated rod heat problem is mathematically described by the following boundary heat equation on $[0,1]$: \begin{align*} u_t &= u_{xx}\\ u_x(0,t) &= ...
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### Gradient Estimate of the heat equation on Riemannian manifolds

From the book Geometric Analysis by Peter Li, we have the gradient estimate of heat equation as follows: Theorem Let $M^m$ be a complete manifold with boundary. Assume that $p \in M$ and $\rho>0$ ...
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