Questions tagged [heat-equation]

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

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1
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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 ...
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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 ...
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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 ...
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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{\...
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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)=...
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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, \...
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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 ...
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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) ...
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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 $P(x,0) = \delta(x)$, $P(x\rightarrow\infty,t) = 0$, ...
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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
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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 ...
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38 views

Estimative and uniqueness for cubic heat equation

Consider the following problem $$ \begin{cases} u_t - \Delta u= -u^3, \quad x \in \Omega,\ 0 < t < T\\ u(x,0)=f(x), \quad x \in \Omega \\ u(x,t)=0, \quad x \in \partial\Omega. \end{cases} $$ ...
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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-\...
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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. ...
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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 ...
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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 ...
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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 ...
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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{...
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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 $x =$ ...
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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]...
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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 ...
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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 ...
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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<\...
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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 ...
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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 ...
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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 ...
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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 ...
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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\|}}. \...
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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 ...
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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,...
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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{...
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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 \...
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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
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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,...
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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+...
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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{...
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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
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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
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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 ...
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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
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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)&...
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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
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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
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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 ...
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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}...
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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 \\ ...
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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. ...
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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}...
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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
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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,\ ...

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