Questions tagged [heat-equation]
For questions related to the solution and analysis of the heat equation.
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Method of manufactured solution for discontinues material properties
I am trying to verify a 1D unsteady heat equation. I used Finite volume method. My domain has an interface and conductivity changes after the interface. I am trying to verify my code using Method of ...
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Formulating a solution ansatz for the 1D heat equation in polar coordinates to learn the PDE in a PINN setting
Hello Math Stack Exchange Community,
I am working on solving a partial differential equation (PDE) with a neural network in a PINN-like fashion, and I am seeking advice on identifying an appropriate ...
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Basic Solution to the Heat Equation
As a learning example, I am trying to derive the solution to the basic Heat Equation (https://en.wikipedia.org/wiki/Heat_equation) using Fourier Transforms.
As I understand, the Heat Equation can ...
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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:
\begin{equation} \label{chap2:GDiffusionSystem}
\begin{aligned}
& \frac{\partial G}{\partial t}(r,t) = C \, \Delta G(r,t) \hspace{0....
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Heat semigroup on $C_b(\mathbb R)$
Let $(X,\|\cdot\|)\in \{(L^2(\mathbb R),\|\cdot\|_{L^2}),(L^\infty(\mathbb R),\|\cdot\|_{L^\infty}),(C_b(\mathbb R),\|\cdot\|_{\infty})\}$
I have a question regarding the heat semigroup $$T_tf:=(\...
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How to prove this non-increasing property of heat equation
Consider the heat equation system:
\begin{cases}
u_t = u_{xx} + f(x), & \\
u|_{x=0} = u|_{x=l} = 0, & \\
u|_{t=0} = 0, &
\end{cases}
where $f(x) \leq 0 $ for $ 0 \leq x \leq l $. The goal ...
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Longtime behaviour of the heat kernel on the real line for bounded initial conditions
Let $u(t,x)$ be the fundamental solution to the heat equation $u_t = \frac{1}{2}u_{xx}$ with initial condition $u(0,\cdot)$. That is, $u(t,x) = \int_{\mathbb{R}}p_{t}(x-y)u(0,y)\mathrm{d}y$ where $p_t(...
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Convergence in comparison principles of parabolic pdes
I assume that the following equations all have sufficiently smooth strong solutions.
I have demonstrated that the solution to the (imaginary-time) Schrödinger equation:
\begin{equation*}
\begin{...
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Brownian Motion / heat flow generated by Hodge Laplacian
Let $\square_M = - (dd^* + d^*d)$ be the Hodge Laplacian on the differential forms $\Omega(M)$ (or if you wish, on a fixed $\Omega^k(M)$). What is the stochastic process generated by this operator? ...
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Long time behaviour of the integral of the solution to the heat equation
I am interested at longtime behaviour of the solution to the one space dimension heat equation. That is, the solution to the equation $$u_{t} = \frac{1}{2}u_{xx},$$
with initial condition $u(0,x)$ ...
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How to show conservation of mass for the heat equation?
I have a question about a property of the solutions to the heat equation. Let $u(t,x)$ be a solution to the (one-space dimension) heat equation $$u_t = u_{xx}.$$ Is it true that $\int_{\mathbb{R}}u(t,...
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Maximum principle for reaction diffusion equation?
Consider the heat equation $$u_t = u_{xx} \;\; \text{ for } \;\; x\in \Omega, t \in [0,+\infty) \,. $$
The strong maximum principle states that if the solution $u$ attains its maximum in the interior ...
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Discretization of the heat equation: is the bilinear form $a(u,v) = (u,v)_{L^2} - \tau (u',v')_{L^2}$ coercive for every $\tau > 0$?
I'm working on the heat equation in 1D on the domain $\Omega = (0,1)$
$$ \partial_t u(t,x) - \partial_x^2 u(t,x) = 0 $$
with boundary conditions $u(t,0) = u(t,1) = 0$ and initial temperature ...
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