# Questions tagged [heat-equation]

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

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### Difference scheme for time-reversed heat conduction equation

I am working on solving the time-reversed heat conduction equation(assuming a two-dimensional space with Dirichlet boundary conditions). I have implemented the finite difference method, using first-...
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### Evans - existence of parabolic PDE, why does $B[u_m,v;t]\to B[u,v;t]$?

In Evans book, chapter 7.1, he establishes existence of weak solutions of $$\partial_t u + Lu = f$$ where $Lu:= -\nabla\cdot (A\nabla u) + b\cdot\nabla u + cu$. He first shows that for any $m$, the ...
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### Heat equation in an unbounded domain

so, I'm considering the following problem: \label{chap2:GDiffusionSystem} \begin{aligned} & \frac{\partial G}{\partial t}(r,t) = C \, \Delta G(r,t) \hspace{0....
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### Energy estimate for the harmonic map heat flow

I am working through Struwes paper "On the evolution of harmonic maps in higher dimensions" (https://projecteuclid.org/journals/journal-of-differential-geometry/volume-28/issue-3/On-the-...
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### Solution of the parabolic PDE using Green's function

Green's function for the parabolic PDE is defined as: $$\Delta G(\vec{x},t,\vec{\xi},\theta)=\delta(\vec{x}-\vec{\xi},t-\theta)$$ Where $G$ satifies the homogeneous initial and boundary conditions. ...
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### Find self-similar solution for the heat equation

I would like to find solutions to the equation $$\partial_t \phi = \partial_{rr}\phi + \frac{1}{r}\partial_r \phi - \frac{1}{r^2}\phi$$ of the form $\phi(t, r) = t^{-\gamma} g(r/\sqrt{t})$. This ...
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### Asymptotic estimate of the inverse Fourier transform of $e^{-\xi^{2q}}$

In V. Yu. Krylov's paper [1], he estimates the inverse Fourier transform of $e^{-\xi^{2q}}$ using harmonic measure arguments (which I don't understand since the reference is written in Russian ...
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### Hottest point of a convex area of the plane

Consider 2d transient heat conduction inside a convex area of the plane A. So we have $$\frac{\partial T}{\partial t} = \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2}$$ ...
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### What are the meanings of these functions in the annular fin equations?

I'm an engineering student building an excel calculator for annular heatsinks based on the annular fin equations and a variety of experimentally derived functions for convective heat transfer ...
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### Uniqueness and continuous dependence on the data of Heat equation.

Let two smooth $v_1$ and $v_2$ both satisfy the system $$\partial_t{v}-\Delta v=f \quad \text{in} \quad U \times (0,\infty),$$ $$v = g \quad \text{on} \quad \partial U \times (0,\infty),$$ for some ...
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### Convert $\partial t$ into $\partial x$ in a heat equation.

We shall consider functions $h=h(t, x):[0, \infty) \times \mathbb{R} \rightarrow(0, \infty)$ which are $2 \pi$-periodic with respect to $x$, belong to $C^{\infty}([0, \infty) \times \mathbb{R})$ and ...
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### fractional heat equation and spectral method

I want to apply a spectral method for the weak formulation of the equation $(-\Delta)^su=f$ $s>0$ with zero Dirichlet boundary conditions, where $(-\Delta)^s$ shall be the fractional laplacian on ...
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### Estimate the norm of the (stochastic) heat equation with time-dependent diffusion coefficient

I'm considering the following (stochastic) PDE: $${\rm d}U_t=\kappa(t)\Delta U_t{\rm d}t+\sigma W_t\tag1$$ on $[0,1)^2$ with Neumann boundary conditions, where $\kappa:[0,T]\to(0,\infty)$ is linear ...
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Consider the following simple Fokker-Planck equation: $$\partial_t f(x,t) = a \partial_x^2 f(x,t)$$ which holds on the intervals $x\in(0,c)$ and $(c,b)$. with $0<c<b$. $0$ and $b$ are ...