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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-...
focalors's user avatar
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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 ...
l'étudiant's user avatar
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2 answers
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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....
Nerey's user avatar
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2 answers
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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:=(\...
Konstruktor's user avatar
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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 ...
cute dunkey's user avatar
2 votes
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61 views

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(...
mathematico's user avatar
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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{...
J.J.Zou's user avatar
2 votes
0 answers
91 views

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? ...
Alex's user avatar
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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 $u(0,x)$ ...
mathematico's user avatar
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1 answer
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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,...
mathematico's user avatar
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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 ...
900edges's user avatar
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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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Solving a heat equation, with puzzling passages.

I have the heat equation $$ u_t=\frac{1}{2}u_{xx},\quad x\in \mathbb{R}\\ u(0,x)=x^2 $$ I tried to solve it with these passages: $$u(t,x)=\int_{-\infty}^{+\infty} G(t,x-y) \phi(y) dy$$ $$u(t,x)=\frac{...
Clyde A. Jansen's user avatar
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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-...
Nils's user avatar
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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. ...
Krum Kutsarov's user avatar
4 votes
0 answers
120 views

Heat Equation : Can a singularity develop away from origin?

Consider the following mixed problem for a radial, nonlinear, 2D-heat equation $$ \begin{cases} u_t = u_{rr} + \dfrac{1}{r}u_r + F(t,r,u,u_r), \quad (t,r) \in [0,+\infty) \times [0,1] \\ u(0,r) = f(r) ...
Desura's user avatar
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Heat equation with Dirichlet boundary condition on a sole point

so I am working with the heat equation: $\frac{\partial G}{\partial t}(r,t) = D \Delta G(r,t)$ subject to the initial condition $G(r>0,t=0) = 0$ and the boundary condition $G(0,t>0) = 1$. I have ...
Nerey's user avatar
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Decay estimate for a heat-like kernel.

Consider a integral kernel $G(x,t)$, which satisfies $$\hat G(\xi ,t)\le Ce^{-c|\xi|^2t},$$ for some positive constant $c,C>0$. I want to show that it satisfies the following decay estimate for ...
Varnothing S's user avatar
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1 answer
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solution to a moving boundary heat equation

I was reading Rubinstein's book 'The Stefan Problem'(https://www.scirp.org/reference/referencespapers?referenceid=1620640) where in chapter2, page 98 he gave an analytical solution to the 1-d moving ...
Maskoff's user avatar
  • 617
5 votes
1 answer
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Bound for solution heat equation with Neumann condition

Assume that $H$ solves $$ H_t = aH_{xx},\\ H(0,x) = 0,\\ H_x(t,0) = h(t). $$ A solution of this equation can be given in integral form as $$ H(t, x)=-\int_0^t h(s) \bar{K}(t-s, z,x) d s, $$ where $\...
Alek Murt's user avatar
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1 answer
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pde with special boundary condition

https://www.phys.ens.fr/~ebrunet/Papers/BrunetDerrida.15.pdf I found in this aricle of Éric Brunet and Bernard Derrida "AnExactly Solvable Travelling Wave Equation", the following : the ...
IHEM's user avatar
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How to show that a convolution of function with the heat kernel can only improve its Holder regularity

Let $D$ be a closed connected set in $\mathbb{R}^d$ and let $f\colon D\to\mathbb{R}$ be an $\alpha$-Holder continuous function. I want to show that if I convolve this function with a heat kernel $p_t$,...
Evgeny Egorov's user avatar
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Finite difference method: Difference in calculations

I'm solving the basic heat equation. $$ \phi_t = c \phi_{xx}$$ which can be written in implicit FDM as follows: $$ \frac{\phi^{n+1}_i - \phi^n_i}{\Delta t} = c \left(\frac{\phi^{n+1}_{i+1} - 2\phi^{n+...
Syed Ali Mohsin Bukhari's user avatar
3 votes
1 answer
117 views

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 ...
Falcon's user avatar
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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 ...
Mango Warrior's user avatar
1 vote
1 answer
158 views

Confusion about one initial/boundary value problem for heat equation

This is a follow-up question to this. The referenced question arose while I was trying to solve $$ \begin{cases} u_t = \frac{1}{2} \Delta u, & x \in X, \\ u ( 0, x ) = 1, & x \in X, \\ u ( t, ...
tsnao's user avatar
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1 answer
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A naive question regarding the Navier Stokes equation

Assume that a function satisfies the heat equation. Does it also satisy the NS equations when you define the pressure such that its gradient is equal to the negative of the nonlinear term and the ...
user11937's user avatar
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Difference of boundary conditions between elliptic and parabolic PDEs

For elliptic PDEs we know that the boundary condition no matter Dirichlet, Neumann or mixed, needs to be defined over the whole boundary of the given volume $\partial V$. Only then we will get a ...
Krum Kutsarov's user avatar
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1 answer
47 views

Do we have to set boundary at $0$?

Hi all I have a fundamental question about heat equation, so far all the heat equation set up I have experienced asasume boundary at a boundary at $x=0$, (assuming it's one-dimensional). I'm wondering ...
Maskoff's user avatar
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0 answers
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Regularity of difference of fundamental solutions to heat equation

Let $K_t^{(1)}$ and $K_t^{(2)}$ be two fundamental solutions to the heat equation $\partial_t-\Delta$ on a non-compact Riemannian manifold $(M,g)$, i.e. $$(\partial_t -\Delta_x)K_t^{(i)}(x,y)=0\,,\...
crimsonmist's user avatar
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Similarity Solutions for the 2D Heat Equation in Polar Coordinates (with radial symmetry)

I'm attempting to solve the 2D heat equation expressed in polar coordinates, where $ \frac{\partial u}{\partial \theta} = 0 $ due to radial symmetry. This simplifies the equation to $ D (u_{rr} + \...
Kamal Ahmadov's user avatar
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0 answers
37 views

About kernel, Green functions.

Consider the heat equation \begin{align} \partial_t u(t,x)&=\Delta u(t,x),\quad t>0,\, x\in\mathbb{R}^n\\ u(0,x)&=u_0(x) \end{align} The Green function (or kernel) is $G_t(x)=\int_{\mathbb{...
eraldcoil's user avatar
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2 votes
0 answers
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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 $$\frac{\partial T}{\partial t} = \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} $$ ...
user89699's user avatar
  • 181
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0 answers
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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 ...
leon buck's user avatar
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0 answers
72 views

Solution to nonhomogeneous heat equation via Green function

Consider the heat equation $$ u_t = \epsilon u_{xx} \quad x\in (0,1)$$ where $\epsilon >0$ is constant. Suppose that this equation is subject to the boundary conditions $$ u(0,t) = \alpha (t), \...
Galois's user avatar
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0 answers
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Heat equation with periodic BC using complex Fourier series.

$k>0,$ $\begin{cases} {u_t = ku_{xx}}, -\pi \le x \le \pi, t>0\\ {u(x,0)=f(x)}\\ u(-\pi,t)=u(\pi,t), u_x(-\pi,t)=u_x(\pi,t), t>0 \end{cases}$ I want to solve this heat equation using complex ...
JAEMTO's user avatar
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0 answers
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solution in backwards heat equation acting well posed forward in time

I have a question about the equation $$ \tau\frac{\partial u^2}{\partial\tau^2} = -x \frac{\partial u}{\partial x}$$ What boundary/initial value data do I need to make the following solution unique ...
zeta space's user avatar
1 vote
0 answers
47 views

Confusion on definition of fundamental solution for the heat equation

As mentioned on Wikipedia, a fundamental solution for a linear differential operator $L$ is a function (or distribution) $G$ such that $$LG = \delta$$ which by linearity of $L$ gives the following ...
CBBAM's user avatar
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54 views

Matlab code for solving inhomogeneous 1D heat equation using Crank-Nicolson

I'm trying to solve the 1D heat equation using Crank-Nicolson in Matlab. The problem is, my numerical solution is not the same as the exact solution. (I already checked the plots and their shapes are ...
Kim Juhee's user avatar
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0 answers
33 views

gluing time-like separated solutions of pde together

Take $$ s\frac{\partial^2}{\partial s^2}\phi_s(x)=-x\frac{\partial}{\partial x}\phi_s(x) \tag{1}$$ which is well posed forward in time under the right conditions, $\forall s >0$ and ill posed ...
zeta space's user avatar
1 vote
0 answers
58 views

Heat Equation with Singular Initial Condition

I'm trying to solve the 2D heat equation in cartesian coordinates with initial condition of the form $1/r$. The problem is as follows: $$ \begin{equation} \begin{cases} \frac{\partial u}{\partial t} = ...
justsome1's user avatar
7 votes
4 answers
306 views

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 ...
dtttruc's user avatar
  • 75
2 votes
1 answer
196 views

How to actually find the stream function for a simple Laplace problem?

Let's assume we're solving a $2D$ Laplace problem, in a domain (if necessary simply connected) $\Omega \subset \mathbb{R}^2$, with a Dirichlet boundary $\Gamma_D$ and a Neumann boundary $\Gamma_N$: $$\...
Collapse's user avatar
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0 answers
56 views

does the regularized heat kernel have the semigroup property?

Consider the heat kernel on the 2D-torus $$ K(t,z-z_1):= \sum_{m \in \mathbb{Z}^2} \exp\Big(-(1-4\pi^2|m|^2)t +2\pi i m \cdot (z-z_1)\Big) $$ It satisfies the semigroup property $$ K(t_1+t_2,z_1-z_2)=\...
Marco's user avatar
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0 answers
27 views

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 ...
Villhaze's user avatar
1 vote
0 answers
20 views

Pointwise convergence of heat kernels on perturbed Riemannian manifolds

Thank you in advance for your comments! Let $(M,g)$ be a Riemannian manifold (in general non-compact, connected). Then the associated Dirichlet Laplacian $\Delta_g$ generates the heat semigroup $(e^{s\...
crimsonmist's user avatar
2 votes
1 answer
125 views

Variational formulation for the heat equation

Let $J = (0,T)$, $T > 0$, $G = (a,b) \subset \mathbb{R}$ (finite interval), and $f \in C(J;L^2(G))$. I consider the heat equation with zero Dirichlet boundary conditions and with initial value $u_0 ...
julian2000P's user avatar
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0 answers
38 views

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 ...
Furkan's user avatar
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4 votes
1 answer
150 views

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 ...
0xbadf00d's user avatar
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Solution to a 1D Fokker-Planck equation with 2 absorbing boundaries and 1 continuity boundary

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 ...
Stig's user avatar
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