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

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2
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0answers
16 views

Heat equation with mixed boundary conditions

I am trying to solve the following problem $$\left\{\begin{matrix}\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2},& 0<x<1,t>0,\\ u(0,t)=\frac{\partial u}{\partial ...
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1answer
14 views

uniqueness of heat equations and the squared integrable assumption

I am looking at the classical proof of uniqueness for the heat equation in Evans. Clearly, we differentiate under the integral sign of the square of $w$. A very basic question is, why are the ...
0
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0answers
23 views

long time behavior of heat equation

Given the heat equation \begin{align} {{u}_{t}}-{{u}_{xx}}&=0,\quad x\in \mathbb R,\,t>0 \\ u\left( x,0 \right)&=f\left( x \right),\quad x\in \mathbb R. \end{align} If ...
1
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1answer
18 views

checking that an initial condition holds for the heat equation

I'm trying to follow a video lecture on solving the heat equation. $I) \space u_t = ku_{xx}, x \in \mathbb{R}, t > 0$ $II) \space u(x,0)=\phi (x), $ $k$ is const, $\phi (x) $ is a ...
2
votes
0answers
37 views
+50

Polynomial in the components of the curvature tensor

Consider a closed Riemannian manifold $(M,g)$ of dimension n and let $K(t,x,y)$ be its heat kernel. Then it is known that the heat kernel has an asymptotic expansion as $t\downarrow 0$: ...
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0answers
10 views

PDE Von Neumann Problem- Physical Interpretation

The Von Neumann Problem is as such: $\Delta u = f(x,y,z)$ in $\ D$ $\frac {\partial u} {\partial n} = 0$ on bdy $\ D$. Using this you can prove that for there to be a solution to this Von Neumann ...
0
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1answer
12 views

Solving the heat equation with piecewise IC

I have the solution to the heat equation, with the BC's and everything but the IC applied. So I am just trying to solve for the coefficients, the solution without the coefficients is $$u(x,t) = ...
0
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1answer
17 views

What does $u_0(x)$ represent?

I am looking at the heat equation and in my notes it says the initial temperature distribution $u(0,x)=u_0(x)$. what does this mean? What does $u_0(x)$ represent?
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1answer
9 views

Where does $u(t,x) \to u(t,x)-a-(b-a)x$ come from?

I know the heat equation is $$\frac{\partial}{\partial t} u(t,x)=\frac{\partial^2}{\partial x^2} u(t,x)$$ I know that $u(t,x)$ is the temperature distribution at time $t$ at the point $x$. We assume ...
0
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0answers
18 views

Solutions to Heat equation $ \int_{\mathbb{-\infty}}^{\infty} \frac{\partial^2 T(x, t)}{dt^2} dx = 0 $

I was wondering about the motion of heat and came across this differential equation. $$ \int_{\mathbb{-\infty}}^{\infty} \frac{\partial^2 T(x, t)}{dt^2} dx = 0 $$ $T(x,t)$ represents temperature ...
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0answers
21 views

Bounding a Subsolution of the Heat Equation

As the title suggests, I'd like to bound a subsolution of the heat equation. I have \begin{align*} u_t - \Delta u &\le 0 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text{ in } \mathbb R^n \times (0,\infty) \\ ...
0
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0answers
42 views

Barenblatt solution for diffusion on the whole line

Question It is my second course on PDEs and the teacher asked us to find solutions like: $$u(x,t) = t^\alpha · f\left(\frac{|x|^2}{t^\beta}\right)$$ for the diffusion (with $k=1$): $$u_t - u_{xx} = ...
2
votes
1answer
131 views

Regularity of heat kernel

I'm trying to find some references dealing with regularity and properties of the heat/Gaussian kernel $$ G_t\left(x,y\right) = \frac{1}{\sqrt{2\pi t}}\, e^{-\left.\left(x-y\right)^2\right/2t}, ...
1
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1answer
33 views

What if we change one of Fourier's law of heat conduction

I'm studying PDE heat diffusion on 1-D rod using the textbook. It states four intuitions leading to Fourier's law of heat conduction $\phi=-K_0\frac{\partial u}{\partial x}$, where $\phi$ is the heat ...
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0answers
25 views

Propagation speed for the wave equation

Let $g\in C_c(\mathbb{R}^n,[0,\infty))$ with $\int g(x)dx=1$. Denote $\Phi$ the Heat kernel given by $\Phi(x,t)=\frac{1}{(4\pi t)^{n/2}}e^{-\frac{|x-y|^2}{4t}}$. Let ...
0
votes
0answers
14 views

Pde Problem with Delta initiative condition.

The problem is to solve the heat equation $\frac{\partial T}{\partial t} = ~\frac{\partial^2 T}{\partial x^2}$ $T=T(x,t)$ inside a disc of radius $R$ with initial condition $T(x,0)=\delta^{(2)}(x)$ ...
0
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0answers
23 views

Does the average value vary proportionally with the sum of values?

This seems to me a fairly fundamental problem but I'd just like some clarification on the answer from someone else. My problem is related to heat transfer through a sphere, so here goes: If I have ...
0
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0answers
22 views

constrained heat equation

Consider the following constrained minimum energy problem for 1-D heat equation for $x\in[0,1],t\in[0,\infty)$: $$u(x,t)=\underset{{0\leq u(x,t)\leq1}}{argmin}~\frac{\partial}{\partial ...
1
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0answers
39 views

Finite difference discretisation of the heat equation

Here is the equation to be discretised: $$ k\left(\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2}\right) = \dot{q} $$ Using the following discretisation scheme: $$ ...
1
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2answers
31 views

Properties of the heat kernel

The function $$g_t(x) = \frac{1}{(4\pi t)^{n/2}}\exp\left(-\frac{|x|^2}{4t}\right)~~~\text{for}~~~t>0~~~\text{and}~~~x\in\mathbb R^n$$ denotes the heat kernel. I want to show that $g_{s+t}=g_s*g_t$ ...
0
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0answers
10 views

Integrate Heat Kernel Along a Spatial Variable

Hi I am reading Jost's PDEs 3rd edition. The books wants to show $$\int_{y\in\Omega}\:q(x,y,t)\:dy\leq1\: for\:t\geq0\:\:(5.3.37)$$,where $q(x,y,t)$ is the heat kernel for a bounded domain $\Omega$ in ...
0
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0answers
11 views

Maximum principle for heat equation with Neumann boundary condition

please help me with this problem. Let $u\in C^2_1([0,l)\times (0,T))\cap C^0([0,l]\times [0,T])$ be a solution of the heat equation $u_t-u_{xx}=0$ such that $u_x(0,t)=0$. Prove that the maximum of $u$ ...
0
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1answer
20 views

Balancing a PDE for Integration

I'm stuck trying to understand a the mathematical step in a heat transfer texbook's example (found on Step 3 of .pdf Page 77 here). Re: Radial Heat Conduction in a Tube The PDE is simplified to: $$ ...
1
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2answers
38 views

Solutions to the heat equation, spatial or time decay?

In reading standard texts there seems to be two standard solution to the heat equation: $$\frac{\partial T}{\partial t}=D\frac{\partial^2T}{\partial x^2}$$ The first is obtained when assuming a real ...
0
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0answers
25 views

A question arised from Jost's text book: PDEs 3rd edition.

This is a question which I encountered in Jost's Partial Differential Equations 3rd edition. The original issue is to utilize iteration to find a solution of $$\Delta u-\frac{\partial u}{\partial ...
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0answers
10 views

Are there any practical mean of zeroth-order square term?

According to Evans' PDE (351 page), for a linear parabolic equation,the second order term describes diffusion, first order term describes transport , and the zeroth order term describes creation or ...
0
votes
1answer
37 views

Effective Boundary Condition for a Heat Equation with Variable Conductivity

Consider a heat equation in one space dimension $$\frac{\partial u(t,x)}{\partial t} = \frac12\Theta(x)\frac{\partial^2u(t,x)}{\partial x^2} \tag{1}$$ where Heavyside function $$ \Theta(x) = ...
0
votes
1answer
24 views

Heat equation with nonhomogeneous boundary conditions

Let ${{Q}_{L}}=\left( 0,L \right)\times \left( 0,T \right]\subset {{R}^{2}}$ and ${{u}_{L}}\in C\left( {{{\bar{Q}}}_{L}} \right)\cap {{C}^{2}}\left( {{Q}_{L}} \right)$ be a solution of the ...
1
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0answers
31 views

How define the entropy of heat equation?

Today, I report a paper about Ricci flow, I saw entropy. As I know, entropy is a physical term.And I know it is used to describe how far the system from heat death.But I don't know the equation of ...
0
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1answer
24 views

Mixed Neumann/Dirichlet eigenvalue problem where one boundary condition is a nonzero constant

$$ u_t(x,t) = u_{xx}(x,t), 0<x<l, t>0$$ $$ u(0,t) = 20,\quad u_x(l,t) = 0$$ $$ u(x,0) = \psi(x)$$ Here, $u(0,t) = 20$ is causing a problem with the resultant eigenvalue problem. I ...
1
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1answer
42 views

Estimates of the derivatives of the fundamental solution of heat equations in $\mathbb{R}^n$

Let $$n_t(z)=\frac{1}{(4\pi t)^{n/2}}e^{-\frac{1}{4t}\|z\|^2}.$$ And the differentiation w.r.t. $x$, denotes as $D_x^{r}$ is defined as $$\frac{\partial^{|r|}}{\partial x_1^{r_1}\cdots\partial ...
1
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1answer
34 views

explicit solution to heat equation without an integral sign

Consider the 1-dimensional heat equation: $$\left\{ \begin{align} & {{u}_{t}}\left( x,t \right)={{u}_{xx}}\left( x,t \right),\text{ }x\in R,\text{ }t>0 \\ & u\left( x,0 ...
0
votes
1answer
34 views

Substitution in PDE (heat equation)

Members, i have a little question concerning the following subsitution for the heat equation $u_t=u_{xx}$. The substitution is the following: $$t = \zeta x^2$$ ...
0
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0answers
14 views

Invariants of a PDE by Lie Symmetries

I have a little Problem in understanding how to derive invariants form Lie Symmetries (or theire infinitesimals). As one can show the heat equation $u_t=u_{xx}$ has the following symmetries ...
0
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0answers
9 views

Reference for a Heat Process in a Wedge

I would like to ask about an explicit suggestion/reference for the following type of heat processes: Roughly, assume we have a "wedge" $W$ of the following form - a domain in $\mathbb{R}^n$ with a ...
4
votes
1answer
34 views

Suppose $X_t$ is a brownian motion with $X_0 \sim u_0$. What is the probability density of $X_t$? (heat equation)

Suppose $u_0(x) = 2x$ for $0 \leq x \leq 1$ and $u_0(x)=0$ otherwise. Suppose $X_t$ is a brownian motion with $X_0 \sim u_0$. What is the probability density of $X_t$? Since $X_t$ is a brownian ...
4
votes
1answer
28 views

How do I find an $A_0$ and $A_n$ which satisfy the initial conditions of this heat equation?

Let's say I have the heat equation $\frac {\partial u}{\partial t} = k\frac {\partial^2 u}{\partial x^2}$, $0 \lt x \lt L$, $t \gt 0$, subject to the boundary conditions $$\begin{cases} \frac ...
0
votes
0answers
15 views

Solve 1D Inhomogeneous Heat Equation with Convection

Here's a problem that I came across in Partial Differential Equations. I was asked to find the solution for an inhomogeneous heat equation with convection: \begin{align} u_t = hu_{xx} + \mu u_x + ...
0
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1answer
42 views

Exercise on heat equation

I have the following exercise: Consider the heat equation $$u_t = k \Delta u \quad (*)$$ on a domain $\Omega$ with boundary conditions $u _{|\partial \Omega} =0$ and initial data $\phi(x)$. ...
0
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1answer
10 views

Heat-Equation with different initial values

I look for the solution for the heat-equation with the initial value problem $(\mathbb{I})$: $u(0,x)=f(x)=\begin{cases} B, & \text{if }x\ge 0 \\ A, & \text{if }x<0 \end{cases}$ Now i ...
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votes
0answers
13 views

Method of lines for elliptical PDEs (Laplace Heat Equation)

Consider the following steady state problem $$\Delta T = 0,\,\,\,\, (x,y) \in \Omega, \space \space 0 \leq x \leq 4 ,\space \space \space\space 0 \leq y \leq 2 $$ $$ T(0,y) = 300, \space \space ...
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0answers
25 views

Continuity Requirements for PDE Strong Solution

Consider a stationary heat transfer problem as an example, the governing equation is the following second order pde: $$-k\Delta T(x)=\dot{q}(x) $$ In the all text books it said that the strong ...
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1answer
36 views

well defined solution of heat equations

Let $\mathcal{S}(\mathbb{R})$ be the space of rapidly decreasing function on $\mathbb{R}$. Let $\Phi_t$ be defined as $$\Phi_t(x)=\frac{1}{\sqrt{4\pi t}}e^{-\frac{|x|^2}{4t}},\quad ...
0
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0answers
13 views

scalar valued semigroup vs vector valued semigroup

I am reading a chapter on heat equation, semigroup in the PDE book written by Jost. Some confusions arises when a 1D problem generalised to multidimensional. In the bottom right of the following, i.e. ...
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0answers
45 views

How does this integration work?

In dealing with a question about equilibrium values for the heat equation $$\frac{\partial u}{\partial t}=\frac k r \frac{\partial}{\partial r}\left(\frac{\partial u}{\partial r}\right)$$ after a ...
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2answers
57 views

Laplace $2$-D Heat Conduction

Consider the following steady state problem $$\Delta T = 0,\,\,\,\, (x,y) \in \Omega, \space \space 0 \leq x \leq 4 ,\space \space \space\space 0 \leq y \leq 2 $$ $$ T(0,y) = 300, \space \space ...
0
votes
0answers
21 views

How to solve a system of diffusion equations (in $u,v$) in two regions with $u(x,t=0) = u_0$ and $v(x,t=0) = 0$

Let $u=u(x,t)$ be defined on $x\in[0,L/2]$ and $v=v(x,t)$ on $x\in[L/2,L]$. For $t>0$ we have the diffusion system $$\frac{\partial u}{\partial t}=D\frac{\partial^2 u}{\partial^2 x} \qquad ...
5
votes
0answers
105 views

Theoretical link between the graph diffusion/heat kernel and spectral clustering

The graph diffusion kernel of a graph is the exponential of its Laplacian $\exp(-\beta L)$ (or a similar expression depending on how you define the kernel). If you have labels on some vertices, you ...
0
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0answers
17 views

Solution of heat equation with cross terms on rectangle.

I would like to find the fundamental solution of the following PDE $$ u_t = \frac12 u_{xx} + \rho u_{xy} + \frac12 u_{yy} $$ on the rectangle $[-a,a]\times[-b,b]$, with $a,b>0$ and with homogeneous ...
1
vote
1answer
32 views

Is the Flux equal to gradient in Vector analysis?

I am trying to get some appreciation of the concepts of flux and continuity equation in vector analysis. Let's keep ourselves to three spatial dimensions, $x, y$ and $z$ Assume the density is ...