Questions tagged [heat-equation]
For questions related to the solution and analysis of the heat equation.
1,413
questions
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2-D radial-axial Laplace equation in cylindrical coordinates (anisotropic diffusion & Dirichlet BCs)
I am trying to solve the 2-D Laplace equation in the cylindrical coordinates, as below.
$\lambda _{r}[\frac{\partial^2 T}{\partial r^2}+\frac{1}{r}\frac{\partial T}{\partial r}]+\lambda _{z}\frac{\...
0
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0
answers
19
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3-D Heat Transfer equation with internal loss generation (cylindrical coordinates)
I want to further ask about the general solution to the 3-D heat transfer equation with constant internal loss generation in the cylindrical coordinates, as follows.
$\frac{1}{r}\frac{\partial \...
1
vote
0
answers
19
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Heat equation with function squared.
Can I seek some help on transforming the following heat equation
$$0 = \frac{\partial X(x,y)}{\partial x} + \frac{X^2(x,y)}{2k} + \frac{\sigma^2}{4 x^2} \frac{\partial^2 X(x,y)}{\partial y^2}$$
into a ...
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0
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13
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stuck on boundary conditions heat equation
Given
$u_t=au_{xx}+cu_{x}$
$u(x,0)=f(x)$
$u_x(0,t)=u(1,t)=0$
Solve with separation of variables
my attempt:
setting $u=e^{bx}v$ and and trying to eliminate the $v_x$ factor we get that $b=\frac{-c}{...
1
vote
1
answer
29
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$3D$ Heat transfer equation with internal loss generation (zero boundary temperatures )
I am sincerely asking the analytical solution to the $3D$ Heat transfer equation with constant internal loss generation.
I don't know how to find the particular solution for the Poisson equation. The ...
0
votes
0
answers
8
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Heat equation on a cylinder with Newton cooling - coordinate change
I have the heat equation on a cylinder shell $\Omega$ of infinite length with inner radius $R_1$ and outer radius $R_2$. The material of the shell has density $\rho$, specific thermal capacity $c$, ...
0
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0
answers
35
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Uniqueness of the solution of Heat equation. (Partial Differential Equation.)
Here is the heat equation ;
$u$ : continuous on $[0,\infty)\times \mathbb R$, $C^2$ on $(0,\infty)\times \mathbb R$
$u(t,x+1)=u(t,x)$ for all $(t,x)$.
$u_t(t,x)=u_{xx} (t,x) \ (t>0, x\in \mathbb R)$...
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0
answers
16
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Solve the non-homogeneous heat equation
inGiven $u_t=au_{xx}+cu+F(x,t)$
$u(x,0)=f(x)$
$u_x(0,t)=(1,t)=0$
Find the solution for the non-homogenous problem
My attempt:
I solved for the homogenous first and got that $u(x,t)=\sum_{0}^{\infty} ...
-4
votes
0
answers
23
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How to slove this Partial differential equation? [closed]
Q. No.1 A string of length L has its ends x = 0 and x = L fixed. It is released from rest in the
position
2
4
.
) – (Lxx
u
L
= Find an expression for the displacement of the string at any subsequent
...
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17
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$u_{t}=$ $au_{xx}$ $+cu$ $+F(x,t)$ find the best constant for c which the solution does not increase.
$u_{t}=$ $au_{xx}$ $+cu$ $+F(x,t)$
$u(x,0)=$ $f(x)\geq 0$
$u(0,t)=0$ $u(1,t)=0$
$1.$For the homogeneous equation $F(x,t)=0$ find the best apriori bound for c ,
such that for any $f(x)$ the ...
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0
answers
17
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values of $\lambda$for $v(x,t):=u(\lambda x,6t)$ to be another solution of the heat equation?
Let $Q=\mathbb{R}^n\times (0,\,+\infty)$ and $u:Q\to\mathbb{R}$ a solution (classical) of the heat equation $\partial_tu-D\Delta{u}=0 $in $Q$, where $D>0$ is an assigned coefficient. Let $v(x,t):=...
1
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0
answers
47
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find a bound for c in the heat equation
Given the heat equation
$u_t=au_{xx}+cu$
$u(x,0)=f(x)\geq0$
$u(0,t)=u(1,t)=0$
find apriori bound on c that doesn't grow for every $f$ without solving.
my attempt:
multiplying by $u$ and integrating on ...
0
votes
1
answer
33
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Heat equation as gradient flow of Dirichlet energy
I am looking for a reference which rigorously explores the heat equation as a gradient flow of the Dirichlet energy (say in $L^2$ ? or some other inner-product space). I don't know this literature ...
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0
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31
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Isometries and Solutions to Heat Equation
Let $M$, $N$ be two Riemannian manifolds, $\Delta_{M}$ and $\Delta_{N}$ be the Laplace-Beltrami operators on $M$ and $N$ respectively and let $\phi: M \to N$ be an isometry. Suppose $u_{N}(x,t)$ is a ...
0
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0
answers
12
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Convolution with Laplace kernel and differential equation.
Convolution with the Heat kernel, you will get the solution of the heat equation. The heat kernel is a Gaussian kernel with the parameter t appropriately introduced. Then, does the Laplace kernel with ...
0
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1
answer
17
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Question in Convergence of a integral in the Heat Kernel and Dirac delta function.
In Convergence of a integral - heat Kernel and dirac delta function
Why $$\lim_{t\to 0+}\int_{|x|>\delta}K_t(x)|\varphi(x)-\varphi(0)|\,dx=0?$$
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19
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Solving an Inhomogeneous heat equation subjected to some conditions by using the solution of its homogeneous version subjected to the same conditions.
Suppose the PDE is $\frac{\partial u}{\partial t}=k\frac{\partial ^2 u}{\partial x^2}$ subjected to the conditions u(0,t)=u($\pi$,t)=0 for $t\geq0$ and u(x,0)=$4\sin(3x)$. The solution in this can be ...
0
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0
answers
25
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Solving the heat equation using diffusion kernel
Given the heat equation is: $\frac{\partial U}{\partial t} - \frac{1}{2}\sigma\frac{\partial^2U}{\partial z^2} = 0$
I need to solve for U through integration, but using the fact that the Normal ...
1
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1
answer
41
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Fourier transform and heat equation on quarter plane
Question:
Consider $u_t = k(u_{xx} + u_{yy})$, x, y > 0
subject to the boundary conditions
u(0, y, t) = 0 and $u_y(x,0,t) = 0$ and initial condtion
u(x, y, 0) = f(x, y).
If we would have $-\infty &...
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0
answers
8
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two-sided Robin bounary conditions in heat transfer problem
If I have a wall and I know its thermal conductivity $h_1(\tau), h_2(\tau)$ (time-dependent) of two sides, could I get the analytic solution of the temperature field $t$? It seems that the materials ...
1
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0
answers
29
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Forward linear heat equation - how does solution operator looks like?
at the moment I am dealing with the forward linear heat equation
$$\begin{align} u_t(x,t) &= u_{xx}(x,t), \quad 0<x<\pi, 0<t<T, \\
u(0,t)& = u(\pi, t) = 0, \quad 0<t&...
1
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0
answers
74
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Estimates on Derivates for One Dimensional Heat Equation
$\textbf{The problem}$:
Let $[-r_0,r_0]$ be a segment in $\mathbb{R}$, let $T>0$ and $u$ be a smooth function satisfying:
\begin{align}
\begin{cases}
u_t-\Delta u = 0 & \qquad \text{on $[-r_0,...
1
vote
1
answer
65
views
Solving an IVBP for a metal rod dipped from cold to warm water
I have to solve the diffusion equation for a solid rod, which is in a bath of 100 degrees. At $t=0$ it is moved to a second bath, where the water temperature is 0 degrees.
I prepare the diffusion ...
0
votes
1
answer
24
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does $u(1/2,1/2)>v(1/2,1/2)$
given
$u_t=u_{xx}+e^t\sin(x)$
$u(x,0)=x^2$
$u(0,t)=u(1,t)=t^2$
and
$v_t=v_{xx}+e^t\sin(x)$
$v(x,0)=x$
$v(0,t)=v(1,t)=t$
Does $u(1/2,1/2)>v(1/2,1/2)$?
My attempt:
Define $w=u-v$ and the system of ...
1
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0
answers
35
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How to solve this partial differential equation (heat-diffusion equation)
I'm having trouble in solving a specific partial differential equation. It writes:
$$
\dfrac{\partial p}{\partial t} = c \left( \dfrac{\partial^{2} p}{\partial x_{1}^{2}} + \cos^2\left(\theta\...
1
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0
answers
54
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A horribly difficult diffusion problem
We have a semi-finite glasstube, defined by the interval $0<x<\infty$, which contains water. At time $t=0$, the water is clean on $0<x<L$, but polluted with concentration $Q$ on $L<x<...
0
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0
answers
21
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Maximum principle for heat equation with modified RHS term
Let $\Omega$ be a bounded domain in $\mathbb{R^n}$ and $u_0\in C(\bar{\Omega})$ and if $u\in C^{2,1}(\Omega\times (0,\infty))\cap C((\bar{\Omega})×[0,\infty))$ is a solution of $ u_t-\Delta u=0$ in $\...
0
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0
answers
67
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Heat Equation with Mixed Inhomogeneous Dirichlet and Homogeneous Neumann Boundary Conditions
I'm going through a mock exam for my finals and have the following heat equation problem:
$\frac{\partial y}{\partial t} = \frac{\partial^2 y}{\partial x^2}, 0<x<1, t>0$
with initial ...
0
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0
answers
9
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On coefficients of hyperbolic and harmonic solutions to the heat equation
In this post I show an example of finding the coefficients of the heat equation. The same principles applies for the Laplace equation and for the wave equation. However one question that turns to ...
1
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0
answers
60
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Solving Backward Heat Equation with a Backward Heat Kernel?
Let $D>0$ be a constant. Imagine we have the following forward heat conduction problem:
\begin{align*}
\begin{cases}
\partial_t u = D \partial_x^2u &, \quad (x,t) \in \mathbb{R} \times (0, \...
1
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1
answer
40
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1D Transient Heat Equation with an Inhomogeneous Boundary Condition
I am trying to solve the one-dimensional transient heat equation with a specified flux in one end ($x=0$) and perfect insulation on the other ($x=L$):
$$\frac{\partial T(x,t)}{\partial x} = \alpha \...
1
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0
answers
32
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How the green function for the relativistic heat equation converges to the green function of the heat equation?
The relativistic heat equation or telegraphers equation is:
$$
(\alpha\partial_t^2 + \beta\partial_t - \omega\,\nabla^2_{\text{3D}})G_R = \delta
$$
if $\alpha \rightarrow 0$ the solution must ...
0
votes
1
answer
31
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How to use maximum principle with an equation similar to the heat equation?
Suppose that $u(t,x)\in C_t^1C_x^2(\Omega_T)\cap C(\overline{\Omega_T})$ satisfying
$$
\begin{cases}
\partial _tu-\Delta u+c\left( x \right) u\le 0,\left( t,x \right) \in \Omega _T,\\
u\left( t,x \...
5
votes
2
answers
84
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Interchanging spatial Fourier transform and time derivative for heat kernel
Let $K_t := (4\pi t)^{-n / 2}e^{|x|^2 / 4t}$ for $x \in \mathbb{R}^n$ and $t \in (0, \infty)$. I would like to show that
$$
\tag{1}
\partial_t \widehat{K_t} = \widehat{\partial_t K_t},
$$
(which makes ...
0
votes
0
answers
40
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Inhomogeneous Boundary Value Problem. How would I solve?
The heat equation:
$$\begin{align} \frac{\partial u}{\partial t} &= {9} \frac{\partial^2 u}{\partial x^2}\,, \qquad 0<x<{3}, \quad t \gt 0\, \\ \end{align}$$
Has boundary and initial ...
2
votes
0
answers
44
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How are anisotropic heat-equation and Fokker-Planck equation related?
Consider first a rescaled brownian motion $X_t$, in $\mathbb{R}^n$ which fulfills the SDE
$$dX_t = \sqrt{2} dB_t,$$
where $B_t$ is brownian motion. Then the density $p(t,x,x_0)$ of the process ...
0
votes
1
answer
17
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Finding the coefficient of the Heat equation
I have used a rapid way to solve the heat equation, with von Neumann conditions:
\begin{equation}
u_t-\alpha u_{xx}=0 \ \ \ 0<x<L, t>0 \\
u_x(0,t)=u_x(L,t)=0, \ \ \ t>0
\end{equation}
and
...
0
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0
answers
21
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Solve 2D heat equation with a sinusoidal source with the Euler scheme and the finite difference method (FDM)
I am trying to solve the heat equation of the following form:
$$
\frac{\partial u(x,y,t)}{\partial t} - \Delta u(x,y,t) = q \cdot sin(6.28\cdot x) \cdot cos(6.28\cdot y) \; \; \text{in }(0,1)^2\times (...
0
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0
answers
21
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Inhomogenous heat equation with initial conditions (1D)
Problem
Solve the heat equation
$$u_t - u_{xx} = e^{-t}cos(x)$$
with the initial conditions
$$u(t,x)|_{t=0} = cos(x)$$
Solution (attempt)
I know that the solution will be the sum of a particular ...
4
votes
0
answers
107
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Heat equation with time-varying Neumann condition
Suppose that $u$ is the solution to the heat equation with mixed Neumann and Robin boundary conditions
\begin{align}
&\partial_tu(t,x) = k \partial_{xx} u(t,x), &&t>0, 0<x<L, \\
&...
3
votes
0
answers
97
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How do you Solve a Mixed Inhomogeneous Dirichlet and Homogeneous Neumann Boundary Conditions of Standard Heat Equation?
I have been given a standard heat equation $$\frac{\partial u}{\partial t} - \frac{\partial^2 u}{\partial x^2} = 0$$ with $u(t,x)$ and the intial condition $u(0,x) = 0$ and the boundary condtions $\...
2
votes
1
answer
74
views
Diffusion equation with periodic boundary conditions
I've been looking for the solution of the following diffusion equation and I haven't been able to find it.
Can anyone help me, please?
$$U_t=kU_{xx} \\
U(x,0)=f(x)\\
U(0,t)=0 ;U(L,t)=A\sin(\omega t+\...
0
votes
1
answer
31
views
Change of Variables in PDEs
Problem.
Let $u_t^\varepsilon + a u_x^\varepsilon = \varepsilon u_{xx}^\varepsilon$ where $a \in \mathbb{R}$. Use change of variables $w^\varepsilon = u^\varepsilon(x + at, t)$ and show that $w$ ...
0
votes
2
answers
244
views
CN FEM Stability [closed]
For Crank-Nicolson FEM for solving $u_{t}-\Delta u=f$, how can I show that it is stable?
3
votes
1
answer
80
views
Why does the heat source become so hot?- Heat equation with heat source using finite difference method
I am trying to model the heat equation with heat source and Robin boundary conditions, i.e. the system:
\begin{align}
T_t\;=&\;\alpha\Delta T+\frac{1}{c_p\rho \text{Vol}(\Gamma)}1_{\Gamma}(\...
0
votes
0
answers
64
views
Crank Nicolson method error analysis
I am trying to solve the diffusion PDE:
$$ \frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$$
using the CN discretization. I have implemented the method in Matlab, quiet ...
0
votes
0
answers
34
views
Applying conservation of energy to dimensionless form of heat equation
This is part of a large exercise about dimensional analysis. Basically we have the 1-D heat equation in a rod with infinite length:
$$\frac{\partial u}{\partial t} = \kappa \frac{\partial^2 u}{\...
2
votes
1
answer
43
views
If $N_1(t) = \int_0^L(u(x,t)-u_0)^2\,dx$ is monotonically decreasing in time, what does that say about $u$?
Consider the heat equation in a rod of length $L$ with fixed temperatures at the endpoints:
\begin{equation}
\begin{cases}
\frac{\partial u}{\partial t} = \kappa \frac{\partial^2 u}{\partial x^2} \\
u(...
3
votes
1
answer
46
views
If $u$ satisfies the 1D heat equation, show that $u^2$ satisfies another PDE
Consider heat conduction in a rod described by
$$ \frac{\partial u}{\partial t} = \kappa \frac{\partial^2 u}{\partial x^2},$$
with constant thermal conductivity $\kappa$. Show that if $u$ satisfies ...
2
votes
0
answers
19
views
Solution to Heat-Like Equation with Diverging Initial Condition
I am trying to solve the equation
$$ \frac{\partial}{\partial\alpha}F(x; \alpha) = \lambda \frac{\partial^2}{\partial x^2}F(x; \alpha) \qquad (1)$$
with the condition $F(x; \alpha = 0) = \exp(\ln 2 \, ...