Questions tagged [heat-equation]
For questions related to the solution and analysis of the heat equation.
1,566
questions
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24
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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 ...
- 1
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1
answer
40
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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 ...
- 327
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0
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23
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Proving $u(x,t) \leq \alpha x(1-x) e^{-\beta t}$ for Heat Diffusion using the Maximum Principle
I have $u(x,t)$ to be defined as the solution to the following partial differential equation for heat diffusion over the domain $S = (0, 1) \times (0, \infty)$.
$$
\begin{cases}
u_t - u_{xx} &= 0 ...
- 1,124
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1
answer
21
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Smoothness of heat kernel on Lipschitz and polygon (cornered) domain
I'm wondering about the spatial smoothness of the heat kernel $K(t,x,x_0)$ on Lipschitz and polygon domains (or cornered domains). It's well known that $K(t,x,x_0)$ is smooth in $t$ for very general ...
- 81
-3
votes
1
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Can someone help me solve this 1D heat equation question? [closed]
pg1
pg2
pg3
Need help with this one D heat equation with mixed boundary condition:
$$\begin{array}{}
\dfrac{\partial ^2 T}{\partial x^2}\end{array} = \dfrac{1}{\alpha}\dfrac{\partial T}{\partial t}\\
...
0
votes
0
answers
16
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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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50
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System involving parabolic heat equation
Consider the following system of partial differential equations:
$$
\begin{cases}
2\sqrt{s}\dfrac{\partial}{\partial s} \sqrt{\mp \Omega_s(x)}=\sqrt{x}\sqrt{\pm\dfrac{\partial}{\partial x}\Omega_s(x)} ...
- 840
1
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0
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34
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PDEs: Boundary term in the weak variational form of the 2D heat equation
I am looking at a tutorial on the Fenics software library for solving PDEs with finite elements. I have a question about the handling of the boundary term when we find the weak form for the 2D heat ...
- 2,109
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0
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42
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How to show that the value of the heat kernel decreases as we move away from the heat source for a bounded domain?
The free-space heat kernel is given by
$K(t,x,y) = \frac{1}{(4\pi t)^{d/2}} e^{-|x-y|^2/4t}$, with $x,y \in \mathbb{R}^d$ and $t>0$.
This expression shows that the heat kernel decreases as the ...
- 49
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29
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Definition of spatial and temporal function spaces in Brezis
There are two points which are not clear for me in Brezis's book Functional Analysis: The space $L^2 (0, \infty; H^1_0(\Omega))$, what is its definition?
I found in Evans book the definition of ...
- 31
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1
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79
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How to solve a heat equation with homogeneous boundary conditions of the form $u_t = u_{xx} +W(x,t)u$ using the method of variation of parameters?
I am trying to solve this problem:
$$u_t(x,t)=u_{xx}(x,t)+e^{-t}u(x,t),x\in (0,\pi),t>0\\u(0,t)=u(\pi,t)=0,t>0\\u(x,0)=\sin(2x),x\in(0,\pi)$$
but I don't know how to deal with the $e^{-t}$ term. ...
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31
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Semigroup of heat equation: $\|P^\kappa_t \|_{L^p \to L^{p'}} \le c t^{-\frac{d(p'-p)}{2pp'}}$
For any $\kappa>0$, we consider the Gaussian heat kernel
$$
p^\kappa_t (x) := (\kappa \pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{\kappa t}},
\quad t>0, x \in {\mathbb R}^d.
$$
Let $L^0 := L^0 (\...
- 16.1k
2
votes
1
answer
53
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How to get generator of this Gaussian contraction semigroup?
$X=C_0(\mathbb R^n)$ is the closure of Schwartz function space $\mathcal S(\mathbb R^n)$ under the $L^{\infty}(\mathbb R^n)$ norm. Define
$$
T_tu=\left\{
\begin{aligned}
...
- 31
2
votes
0
answers
94
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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 ...
- 2,165
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41
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The unique solution of non-homogeneous heat equation and its boundedness (maximum principle) / Is there a mistake in the textbook?.
Consider the following Cauchy problem for heat equation: $u_t - \Delta_xu=f(x,t), x \in \mathbb{R}_n, t>0; u|_{t=0}=\phi(x), x \in \mathbb{R}_n$ where $u \in C^2(\{x \in \mathbb{R}_n; t>0\}) \...
- 641
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18
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Is the kernel of the Laplacian fractional operator positive and us a Schwartz function?
Let $p_t(x):=\int_{\mathbb{R}^n} \mathrm{e}^{ix\cdot \xi}\mathrm{e}^{-t|\xi|^2}\,d\xi$ be the heat's kernel. within the properties of the kernel $p_t$, are fulfilled that
$p_t(x)>0$ for all $t\geq ...
- 3,220
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1
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38
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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) &= ...
- 1,149
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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 $\rho>0$ ...
- 477
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50
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A reference for solution of non homogeneous heat equation on bounded domain
In PDE's book from Evans, is said that
$$
u(x,t) = \int_0^t \int_{\mathbb{R}^N} \frac{1}{(4\pi (t-s))^{N/2}} e^{-\frac{|x-y|^2}{4(t-s)}} f(y,s) dy ds
$$
is a solution of
$$
\begin{cases}
u_t - \Delta ...
- 833
1
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0
answers
24
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Mild/Semigroup solution to parabolic conservation law is weak solution
Let $f \in L^2(\mathbb{R}) \cap L^\infty(\mathbb{R})$, $u_0 \in C_0^\infty(\mathbb{R})$ and consider the PDE
$$
\partial_t u (t, x) -\partial_x^2 u(t, x) = \partial_xf(u(t, x)), \quad (t, x) \in (0, T)...
- 6,054
0
votes
1
answer
51
views
Physical meaning of a heat equation with a term $\alpha u$ [closed]
I would like to know if there is any physical meaning for the equation
$$
\begin{cases}
u_t - u_{xx} + \alpha u = f(x), (a,b)\times(0, +\infty) \\
u(x,0)=u_0(x), x \in (a,b)\\
u(a,t)=u(b,t)=0, t \in (...
- 833
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votes
0
answers
20
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Space-time convolution with the heat kernel - still continuous?
Let $K(x,t):=\frac{1}{(4\pi t)^{n/2}}e^{-\lvert x \rvert^2/(4t)}$ be the $n$-dimensional heat kernel.
Also, consider a locally-integrabl functon $f : \mathbb{R}^n \times [0,\infty) \to \mathbb{R}$. ...
- 6,796
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0
answers
32
views
If $f(x,t) : \mathbb{T}^3 \times (0,\infty) \to \mathbb{R}$ solves the periodic heat equation and $f(x,0)=0$, is it true that $f$ is identically zero?
Let $\mathbb{T}^3 : = (\mathbb{R}/\mathbb{Z})^3$ be the $3$-dimensional torus and $f(x,t) : \mathbb{T}^3 \times [0,\infty) \to \mathbb{R}$ be a function such that
$f$ is smooth on $\mathbb{T}^3 \...
- 6,796
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0
answers
39
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Maximal $L^p$-regularity of Laplace-Beltrami operator $\Delta$ on closed manifold
It's well-known that the Dirichlet Laplacian $\Delta$ on flat domain is R-sectorial on $\Sigma_{\pi}$ in $L^p$ space for all $p\in (1,\infty)$.
I'm wondering if the Laplace-Beltrami operator $\Delta$ ...
- 81
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0
answers
43
views
When a kernel is positive?
I am trying to verify that the heat's kernel is positive using the inverse Fourier transform. For this, I calculate the heat's kernel by means of a contour integral. After verifying this fact, my ...
- 3,220
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0
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26
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Estimation of heat flux at the boundary
So for heat equation $$T_t - k \Delta T = f(t)1_{r <= R}(r)$$ with initial condition $T(r,0) = 0$. where $r = ||(x,y,z)||$. $f(t)$ could be any positive function that it's integral over time is ...
- 651
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1
answer
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The meaning of the "mass matrix" on a PDE
Caveat
Note, I have reviewed the question below with a similar name, and this is not a duplicate. I am asking about the Mass matrix on a PDE while the reference below is asking about the Mass matrix ...
- 2,109
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0
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38
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trouble understanding the heat equation solution derived in this article
I'm an undergraduat EE student and I'm analysing a scientific paper regarding the theory of the photoacoustic effect. Although the paper is pure physics, my problem is more of a mathematical nature. ...
6
votes
1
answer
139
views
dominated convergence theorem on the heat kernel
Let $f \in C_c^{\infty}(\mathbb{R}^d)$ and $K_t = (4\pi t)^{-\frac{d}{2}}e^{-\frac{x^2}{4t}} $ for every $t>0$.
I want to prove this :
\begin{align*}
\frac{d}{dt}(K_t*f)(x)&=\frac{d}{dt}\int_{\...
- 145
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votes
0
answers
85
views
How do I use this interface condition?
I am solving heat equation that is having a jump condition. Precisely,
$$
u_t -\alpha\nabla^2 u = f(t)1_{r \leq R}(r)
$$ for $r \geq 0$, and the condition is
$$
u(t, R^-) = u(t, R^+) - ku_r(t, R^+)
$$
...
- 651
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votes
0
answers
20
views
Space-variant diffusion with infinite speeds: eigendecomposition and matrix exponential
The heat diffusion equation on some domain $\Omega$ with Neumann boundary conditions on $\partial\Omega$ and normal $n$ is given as:
\begin{alignat}{3}
\partial_t u(t,x) &= \Delta u(t,x), &\...
- 1,843
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0
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49
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Heat equation: proving that smaller diffusion leads to bigger solution via energy methods
Let $\Omega$ be a bounded Lipschitz domain and denote by $u_\alpha$ the solution of the heat equation
$$u_t -\alpha \Delta u = f$$
with $f \in L^2(0,T;L^2(\Omega))$, $u(0) = u_0$ given and $u|_{\...
- 409
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votes
1
answer
31
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Non homogeneous Heat equation in polar coordinates with non homogeneous BC's
I'm trying to work around my way of this problem
$$a\frac{1}{r}\frac{\partial }{\partial r}(r\frac{\partial u}{\partial r})+be^{-y(r-a)}=\frac{\partial u}{\partial t}$$
$$\left.\frac{\partial u}{\...
- 63
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39
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Heat equation with separation of variables
Solve the following Heat equation:
$u_t=u_{xx}, \quad 0<x<1, t>0$
$u(t,0)=1=u(t,1), \quad t\geq 0$
$u(0,x)=1-\sin(\pi x) \quad 0\leq x\leq 1$
What I have tried:
$u(t,x)=T(t)X(x)$
$\Rightarrow ...
- 13
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votes
0
answers
25
views
Plotting the 3D Heat Equation in 2D Slices
I am trying to plot the temperature distribution of a cake using the heat equation in 3D in MATLAB. The boundary conditions are
.
I am trying to plot the temperature distribution of a cake shaped as a ...
- 11
1
vote
0
answers
74
views
Verifying that the Laplacian is the infinitesimal generator of a semigroup.
Let
\begin{align}
\frac{\partial}{\partial_t}u(t,x)&=\Delta u(t,x),\quad t>0,\, x\in\mathbb{R}^n\\
u(0,x)&=f(x)
\end{align}
where $f\in L^2(\mathbb{R}^n)$ (Heat's equation)
The heat kernel ...
- 3,220
1
vote
2
answers
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Nonhomogeneous Neumann Boundary Conditions for the 3D Heat Equation [closed]
I have recently came accross an interesting boundary problem regarding the temperature content inside a cake in an oven. The solution assumes the boundary conditions to be fixed at the oven ...
- 11
4
votes
1
answer
78
views
Doubt on the definition of heat kernel
The heat kernel of a compact Riemannian manifold is the only smooth function $k=k(t,x,y):\mathbb{R}_{>0}\times M \times M\to \mathbb{R} $ such that $k(\cdot,x,\cdot)$ is a solution to the heat ...
- 3,502
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votes
1
answer
28
views
Exponential $L^2$ convergence of the solution of this PDE
I have the following PDE
$$u_t-au_{xx}=1,$$ with $a>0$ and boundary conditions $$u(t,0)=0=u(t,\pi),$$
$$u(0,x)=u_0(x),$$ for $t>0$ and some $u_0(x)\in L^2_{(0,\pi)}.$ I have to prove that $$||u(...
user1194733
0
votes
0
answers
27
views
SDE boundary conditions
So the probability density of Brownian motion on $\mathbb{R}$ is described by the heat equation on $\mathbb{R}$. So the Brownian motion is the SDE corresponding to the heat equation PDE.
How do we ...
- 1,812
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votes
2
answers
88
views
How to impose boundary conditions in heat equation when solving using Fourier transform?
Given the following PDE: $u_t(x,t)=u_{xx}(x,t)$, where the subindices are partial differentiation. Using the Fourier transform $(\mathcal{F})$ for the spatial frequency domain $(\omega)$, in the eq. ...
- 427
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votes
0
answers
35
views
Solution of backward heat equation
I am attempting to solve the following PDE: for $y >0$ and $t>0$,
$$V_t(y,t) + a V_{yy}(y,t) - b V_y(y,t) = f(y,t), \ V(y,0) = 0 . $$
By some changes of variables, I covert the above Euler-type ...
- 389
1
vote
1
answer
93
views
Solving the 1D Heat Equation on [a,b] rather than [0,L]
Solve the 1D Heat Equation on $x \in [a,b]$
$$ \frac{\partial ^2 T}{\partial x^2} = \frac{1}{\alpha} \frac{\partial T}{\partial t}$$
$$ T(a,t) = T(b,t) = 0, T(x,0) = T_0(x) $$
Now, I know that ...
- 86
2
votes
1
answer
89
views
Laplace equation (polar coordinates) with non-homogeneous boundary conditions
I've been trying to solve this problem with separation of variables where T is a function of r and z
$$\frac{1}{r}\frac{\partial }{\partial r}(r\frac{\partial T}{\partial r})+\frac{\partial^2T}{\...
- 63
0
votes
0
answers
13
views
Heat equation on finite interval with decaying heat source
I am trying to solve the heat equation on a finite interval with a localised decaying heat source, but I am stuck.
Specifically: Consider the equation
$$ u_t(t,x)=\kappa u_{xx}(t,x) +q(t,x) \,,$$
...
- 2,205
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votes
0
answers
28
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Fourier series of $\sin (n\rho) \,\frac{1-e^{(\lambda -\zeta n^2)t}}{\zeta n^2-\lambda}$
I'm looking for a closed form (if it exists) of the Fourier series
$$u(x,t)=u_0 e^{-\lambda t} \sum_{n=1}^\infty \sin (n\rho) \,\frac{1-e^{(\lambda -\zeta n^2)t}}{\zeta n^2-\lambda}\, \sin\!\left(n \...
- 2,205
1
vote
1
answer
40
views
Heat equation, u non-negative, bounded domain, mass conservation with $0$ Dirichlet condition?
I am given a non-negative solution $ u\geq 0$ to the heat equation on a bounded open subset $\mathbb{\Omega}$ of $\mathbb{R^n}$ and $t>0,$ so $(x,t) \in \Omega \times (0,\infty).$ I have ...
- 35
2
votes
0
answers
54
views
Reference for regularity of heat equation
Consider a Gelfand triple $V\subseteq H$, together $f=f_1+f_2 \in L^2(0,T,H) + H^1(0,T,V^{*}) $ and a symmetric and bounded, coercive operator $A: V\rightarrow V^*$.
Consider the heat equation
$$u'+Au=...
- 1,940
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votes
1
answer
47
views
Solving second order derivative with Neumann boundary conditions
I am trying to solve the equation:
$w''(z) = -\frac{q0-qm}{kc*hr}*exp(-\frac{z}{hr})$
with the boundary conditions:
$w'(0) = 0$ and $w'(L) = \frac{qm}{kc}$
I am not sure how to do this where the ...
- 1
1
vote
1
answer
62
views
LU factorization distribution over addition
I am applying the Crank-Nicolson-Method to solve the diffusion equation in 1D.
I need to solve the implicit equation
$$Au^{j+1} = Bu^{j} \equiv b$$
Note that
$$A= \begin{bmatrix}
\ddots & \vdots &...
- 1,380