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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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 ...
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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 ...
celebi's user avatar
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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}\\ ...
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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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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)} ...
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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 ...
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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 ...
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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 ...
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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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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 (\...
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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} ...
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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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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\}) \...
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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 ...
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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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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 ...
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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)...
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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 (...
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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 $f : \mathbb{R}^n \times [0,\infty) \to \mathbb{R}$. ...
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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 \...
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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 $\Delta$ ...
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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 ...
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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 ...
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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 ...
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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. ...
makadamij's user avatar
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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_{\...
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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 $$ u(t, R^-) = u(t, R^+) - ku_r(t, R^+) $$ ...
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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), &\...
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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|_{\...
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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}{\...
jack gatz's user avatar
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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 ...
Li boang's user avatar
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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 ...
Orkiel's user avatar
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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 ...
eraldcoil's user avatar
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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 ...
Orkiel's user avatar
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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 ...
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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(...
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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 ...
900edges's user avatar
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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. ...
Daniel Muñoz's user avatar
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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 ...
Kenneth Ng's user avatar
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 ...
STL's user avatar
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2 votes
1 answer
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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}{\...
jack gatz's user avatar
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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 $$ u_t(t,x)=\kappa u_{xx}(t,x) +q(t,x) \,,$$ ...
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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 \...
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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 ...
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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=...
Lilla's user avatar
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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 ...
Dean W's user avatar
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1 vote
1 answer
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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 &...
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