Questions tagged [heat-equation]
For questions related to the solution and analysis of the heat equation.
1,276
questions
1
vote
0answers
13 views
Derivative of integral representation of Greens Function
I am interested in taking the differentiation of an integral representation containing the fundamental solution of the heat equation, hence the Greens function.
The equation of interest I want to ...
0
votes
0answers
12 views
Laplace equation with certain boundary conditions
I'm trying to solve the equation $u_{xx}+u_{yy} = 0$ on $[0,1] \times [0,\pi]$ with conditions $u(x,0)=u(x,\pi)=u(0,t) = 0$ and $u(1,t) = f(t).$
Is this the right method?
I split $u = X(x)Y(y)$ then I ...
-1
votes
0answers
18 views
Which methods are applicable to solve problem of heat conduction on the surface of ball? [closed]
The problem is of heat conduction on the surface of ball. The surface of the ball is given as $r^2=x^2+y^2+z^2=R^2$ at zero temperature and initial temperature is $f(x)$? I also want to model the ...
1
vote
0answers
35 views
Derivative of integral representation on boundary using the fundamental solution of the heat equation
I have a problem on how to obtain the derivative of an integral representation which includes the fundamental solution of the heat equation.
The equation of interest is given as:
$ f(x,t)=\sqrt{\...
0
votes
1answer
61 views
Solve the following equation $\frac{\partial ^2u}{\partial x^2}=\frac{\partial u}{\partial t}.\quad 0<x<1, 0<t,$
Solve the following equation $$\frac{\partial ^2u}{\partial x^2}=\frac{\partial u}{\partial t}.\quad 0<x<1, 0<t,$$ with
$$u(0,t)=-1\\ -\frac{\partial u(1,t)}{ \partial x}=(u(1,t)-1) \\ u(x,0)=...
-1
votes
0answers
16 views
How to deduce the constant in the following heat equation?
If $u(x, t)=Ae^{-t}\sin x$ solves the following initial boundary value problem $$u_t=u_{xx},\ \ \ 0<x<\pi, t>0,\ \ u(0, t)=u(\pi, t)=0\ \ \text{for}\ \ t>0,$$ $$u(x, 0)=\begin{cases}60, \...
1
vote
0answers
28 views
Heat kernel bounds on graph
I'm studying the heat kernel of the continuous time simple random walk $X_t$ on $\mathbb{Z}^d$. I know of the carne varopoulos bound for the heat kernel. But I'm lookong for a similiar bound for the ...
5
votes
1answer
90 views
+50
Energy estimate for heat equation
Consider the following heat equation in a bounded smooth domain $D \subset \mathbb{R}^d$:
\begin{align*}
u_t -\triangle u &= f, \qquad (x,t) \in D \times (0,T)\\
\partial_n u &=0, \qquad (x,t) ...
2
votes
0answers
28 views
Solving advection-diffusion equation $ P_t (x,t) = DP_{xx} -V P_x$ for $P_t(x,t=0)=-V\delta'(x)$.
I would like to solve the advection diffusion equation
$$ \partial_t P(x,t) = -V \partial_x P(x,t) + D \partial_x^2P(x,t) $$
for the conditions $P(x,0) = \delta(x)$, $P(x\rightarrow\infty,t) = 0$, ...
0
votes
0answers
16 views
heat equation dif solve [closed]
$\begin{array}{c}u_{t}=u_{x x}, \quad x \in \mathbb{R}, t>0 \\ u(x, 0)=x e^{-x^{2}}, & x \in \mathbb{R}\end{array}$
can you solve the heat equation
0
votes
1answer
23 views
A question on heat transfer and soil temperature profile
This measured soil temperature profile seem strange to me. We know from heat equation that heat transfers at infinite speed in media so if there's a boundary change in temperature, any interior point ...
0
votes
0answers
38 views
Estimative and uniqueness for cubic heat equation
Consider the following problem
$$
\begin{cases}
u_t - \Delta u= -u^3, \quad x \in \Omega,\ 0 < t < T\\
u(x,0)=f(x), \quad x \in \Omega \\
u(x,t)=0, \quad x \in \partial\Omega.
\end{cases}
$$
...
1
vote
1answer
46 views
If $u(ax,a^2t)=u(x,t)\Rightarrow \exists v$ such that $u(x,t)=v\left(\frac{x}{\sqrt{t}}\right)$.
The problem that I am trying to solve is similar to this exerciese in Evans' book.
We are given a solution $u\in C^\infty(\mathbb{R}^n\times(0,\infty))$ of the equation
\begin{equation}
\partial_tu-\...
1
vote
1answer
51 views
Understanding the heat equation
I've been reading about the heat equation recently and while I do grasp the mathematical principles involved, I'm struggling a bit with understanding the more in-depth details in a more general way.
...
0
votes
0answers
31 views
Heat equation of a rod which has constant heat of $\theta$ at on one end and $\alpha$ at the other end
I’m studying PDE and am currently on heat equation subject .
$$\frac{\partial u}{\partial t}=c^2 \frac{\partial ^2 u}{\partial x^2}$$
This book I’m reading worked on heat equation of a rod for ...
0
votes
0answers
18 views
How to have the bidomain heat equation as two coupled equations
If we consider a bidomain the heat equation will be:
$$\frac{\partial\Phi_0}{\partial t} -D_0 \frac{\partial^2\Phi_0}{\partial^2x}=q_0(x,t)$$
for $0<x<s$
$$\frac{\partial\Phi_1}{\partial t} -D_1 ...
0
votes
1answer
20 views
Transform heat equation to add drift/transport term
Let $f \in C^{1,2}((0,\infty)\times \mathbb R)$ be a solution to the heat equation:
$$ \partial_t f(t,x) =\partial_x^2f(t,x). $$
Given a constant $c\in \mathbb R$, is there a reasonable transformation ...
2
votes
0answers
54 views
How to find the boundary condition in method of Separation of variables?
I am trying to find the solution of the following Initial boundary- value problem.
\begin{equation*}
u_t = k \Delta u ~~~~~~0<x<1,~~~~ 0<y<1 ~~~~,0<t<1
\end{equation*}
\begin{...
1
vote
1answer
85 views
Given boundary conditions and initial condition, solve the PDE
Question, solve the given PDE :
$$ \frac{\partial C}{\partial t} = a \frac{\partial^2 C}{\partial x^2} -kC $$
where a, k are constants.
boundary conditions : $C= C_0$ at $x=0$, $C=0 $ at $x =$ ...
0
votes
1answer
38 views
Proof for equality of energy for heat equation.
Given:
Heat equation: $(1)\begin{cases}\frac{\partial u}{\partial t }-\frac{\partial^2 u}{\partial x^2}=f\quad pp. \text{in}\quad \Omega\times]0,T[\\
u=0\quad pp. \text{in}\quad \partial\Omega\times]...
0
votes
0answers
56 views
I don't understand the definition of the heat kernel in Getzler's book
Consider the following (special case) of Definition $2.15$ on page $75$ of Getzler's book:
Let $E\to M$ be a bundle and
$$
H\colon\Gamma(M,E)\to\Gamma(M,E)
$$
a generalized Laplacian.$^1$ The heat ...
0
votes
0answers
13 views
Possible analytical solution of 3D heat equation in finite cylindrical domain with radiative boundaries?
Currently, I am trying to find an analytical solution for the following situation. I have a cylinder of radius $R$ and length $l$. On the upper circular surface (say at $z = 0$), I have a Gaussian ...
1
vote
0answers
35 views
A question about particular solution of Heat equation
I have a question on particular solution of 1-D Heat equation
$$\frac{\partial^{2} T(x, t)}{\partial x^{2}}=\frac{1}{\alpha} \frac{\partial T(x, t)}{\partial t} \quad \text { in } \quad 0<x<\...
-1
votes
1answer
45 views
Uniqueness of heat equation solution
Let $T(\mathbf{x},t)$ denote the temperature at location $\bf x$ and time $t$ in a closed and bounded region $R$ with thermal conductivity $k(\mathbf{x})>0$, density $\rho$ and specific heat ...
1
vote
0answers
24 views
Free Electron Galerkin FEM
SCHRODINGER’S EQUATION
$$-ih u_{t}(x,y,z,t) = \frac{h^2}{2m} u_{xx}(x,y,z,t)+ \frac{e^2}{r}u(x,y,z,t)$$
The potential $\frac{e^2}{r}$ is a variable coefficient.
So, let’s take the free Schrodinger ...
0
votes
0answers
19 views
On Evans' proof of smoothness of solution to the heat equation in parabolic cylinders
I am slightly confused by Evans' proof of this theorem (Theorem 8 in Section 2.4 in the second edition of the book "Partial Differential Equation". By heat equation it is meant the ...
0
votes
0answers
9 views
If $\hat{g}(\omega)=e^{-4 a\pi^2 \omega^2 t}$ then how I show $g=\dfrac{1}{4\pi \sqrt{a\pi t}}\exp\Big(\frac{-x^2}{16a\pi^2 t}\Big)$? [duplicate]
In this paper ,( page 3),Author solved heat equation using Fourier transformation method ,such that he come up to $u(x,t)=f(x)*g(t)$ with $$\hat{g}(\omega)=e^{-4 a\pi^2 \omega^2 t}$$ and $f(x)$ is ...
1
vote
0answers
50 views
Are the solutions of this two PDEs equal at the final time?
Consider the Cauchy problem given by
\begin{align}(1)
\begin{cases}
\partial_t u= -\partial_x [f(x)u],\;\; t\in [0,\|\Delta\|]\\
u(0,x)=\frac{1}{\sqrt{2\pi\|\Delta\|}}e^{-\frac{x^2}{2\|\Delta\|}}.
\...
0
votes
0answers
17 views
Continuity of Heat Semigroup in $L^3(\mathbb{R}^3)$
I want to prove the strong continuity of $S(t)x$ in $L^3(\mathbb{R}^3)$, this is $S(t)x \in C\left([0 , \infty) , L^3(\mathbb{R}^3)\right)$, where $x \in L^3(\mathbb{R}^3)$ and $S(t)$ is the Heat ...
1
vote
0answers
58 views
Follow the steps to solve the problem: $u_t = \alpha^2 (u_{xx} +u_{yy}), \hspace{0.5cm} (x,y) \in D=\{ (x,y) \in \mathbb{R}^2 \colon x^2 + y^2 < 1\}$
Let $D =\{ (x,y) \in \mathbb{R}^2 \colon x^2 + y^2 < 1\}$ and consider the problem
\begin{cases}
u_t = \alpha^2 (u_{xx} + u_{yy}), \hspace{0.5cm} (x,y) \in D, \hspace{0.4cm} t>0\\
u(x,y,0) = f(x,...
0
votes
2answers
54 views
Solve the problem $u_t = c^2 u_{xx} + g(x,t)$, $(x,t) \in (0,L) \times (0,\infty)$
Studying some notes on partial differential equations about the heat equation, I came across the following problem and found it interesting because of its general form:
\begin{equation}
(*)\begin{...
0
votes
0answers
34 views
The heat equation with affine initial data.
Suppose that $u \in C(\mathbb{R} \times [0, \infty)) \cap C^2(\mathbb{R} \times (0,\infty))$
$$u_{t}=u_{xx}, \quad \text{in} \quad \mathbb{R} \times (0, \infty),$$
$$ u(x,0)=Ax+B, \quad x \in \...
1
vote
0answers
16 views
How to prove the existence of weak solutions of parabolic PDEs using Rothe's method?
I am reading a textbook on PDE (Introduction to Elliptic and Parabolic Partial Differential Equations by Z. Wu, J. Yin and C. Wang, written in Chinese) where a proof of the existence of weak solutions ...
3
votes
1answer
84 views
Circularly Symmetric Heat Equation
The circularly symmetric heat equation is
$\frac{\partial{u}}{\partial{t}} = k\frac{1}{r}\frac{\partial}{\partial{r}}(r\frac{\partial{u}}{\partial{r}})$
When we have the boundary conditions being
$u(a,...
0
votes
1answer
37 views
multidimensional fourier series for heat equation
In the additional notes (luckily not examinable) for my PDE modelling module I stumbled across the following statement and was wondering how it was derived
$$\partial_t\theta=\alpha\left(\partial^2_x+...
1
vote
1answer
45 views
Initial values in Duhamels Principle
I read the following statement in an article about photoacoustic tomography and don't understand a detail about it.
According to Duhamel's Principle, the inhomogeneous equation
\begin{equation}
\label{...
2
votes
0answers
21 views
Application of the mean-value property for the heat equation
I am wondering if there exists some application of the following classical result (I write the version that appears in Evans' book):
Theorem: Let $u \in C_{1}^{2}(U_{T})$ solve the heat equation. Then
...
2
votes
0answers
35 views
A heat equation with the conditions $hu(0,t)- u_{x}(0,t) = 0$ and $u(L,t)+ u_{x}(L,t) = 0$
I must solve the following problem with initial boundary value:
\begin{cases}
u_t = c^2 u_{xx} \hspace{3cm} 0<x<L, \hspace{0.3cm} t>0, \hspace{0.3cm}c>0\\
u(x,0) = f(x) \hspace{2.3cm} 0 \...
2
votes
0answers
35 views
Heat equation with non-zero boundary condition
I have the following PDE with initial+boundary conditions:
$U_t = - U_{xx}$
$U(x,0) = 0$ for $x>0$ , $U(0,t) = 1$ , $U(1,t) = 0$.
The solution is for $0<x<1$ and $t>0$.
Is there an ...
0
votes
0answers
73 views
Find the solution to this initial boundary value problem: $u_t = c^2 u_{xx}$, $-L < x < L$, $t>0$, $c>0$ …
Heat conduction in a thin circular ring (consider it as a rod, bent in the shape of a circular ring joining the two ends) of length $2L$, labeled $-L$ to $L$ leads to the equation $u_t = c^2 u_{xx}$, $...
3
votes
1answer
54 views
On Duhamel's principle for the inhomogeneous heat equation with irregular data and regularity of solution
Consider the following inhomogenous initial value problem for the heat equation, that is
$\begin{align}
\dot{u}-u^{\prime \prime}&=f(u) &&\quad x \in \mathbb R, \;t>0, \\
u(\cdot,0)&...
1
vote
0answers
23 views
Physical Intuition for Mean Value Property of Heat Equation.
I am trying to think of an intuitive explanation for the mean value property for the heat equation:
$$u(x,t) = \frac{1}{4r^2} \int_{E(x,t,r)} u(y,s) \frac{|y|^2}{|s|^2} dy ds$$
where $E(x,t,r)$ is the ...
1
vote
0answers
59 views
Steady State Solution of a 1D Heat Equation
I was given the 1D heat equation
$\frac{\partial u}{\partial t}=u+\frac{\partial^2 u}{\partial x^2}$
with the boundary conditions of
$0 < x < \pi$ , $u(0,t)=0$ , $\frac{\partial u}{\partial x}(\...
1
vote
1answer
77 views
Passing derivative through integral of a convolution
I'm trying to solve an exercise on Folland's book of real analysis on the part of fourier transforms and its applications on PDE's, when working with the heat equation one may show that given the heat ...
0
votes
1answer
80 views
Solution to the heat equation [closed]
Let $X \sim \mathcal{N} (0, I_n)$ be a standard gaussian in $\mathbb{R}^n$ and $Y$ be a another random variable with a smooth density $f$. I want to show that for $s>0$, the density of $Y + \sqrt{s}...
0
votes
0answers
15 views
What's the physical interpretation of that Non-Homogeneous Heat Equation with Neumann boundary conditions?
For all $u = u\left(x,t\right)$
That's the equation with all the other conditions:
\begin{equation}
\begin{cases}
u_t=u_{xx}+4u+x^2-2t-4x^2t+2\cdot \cos ^2\left(x\right), 0 < x < \pi \\
...
0
votes
1answer
19 views
What's the meaning of Neumann BCs in heat conduction problems?
What's the meaning of Neumann BCs in heat conduction problems?
Such as given here:
https://web1.eng.famu.fsu.edu/~dommelen/pdes/style_a/svbex.html
Why does one specify $u_x=g_0$ at the end of the rod. ...
0
votes
0answers
15 views
Solve a linear heat equation and regularity
Page 37 of the following note I am reading
A short course on Mean field games -
Ceremadehttps://www.ceremade.dauphine.fr › ~cardaliaguet
has the following: If $F(x,t)$ and $G(x,t)$ belong to $C^{1/2}...
0
votes
0answers
39 views
Is the heat equation (e.g. 2D) solvable without boundary conditions? Or are the boundary conditions always present?
Is the heat equation (e.g. 2D) solvable without boundary conditions? Or are the boundary conditions always present?
I've been puzzled a bit, since it seems that w/o boundary conditions there's no ...
0
votes
0answers
25 views
Separation of Variables Hiccup
Find the temperature $u(x,\ t)$ in a unit length rod modeled by
$u_t = 4u_{xx}$
$u(0,\ t) = 0$
$u(1,\ t) = 0$
$u(x,\ 0) = x - x^2$
Breaking out the steady-state temperature, $u(x,\ t) = s(x) + v(x,\ ...
