Questions tagged [heat-equation]
For questions related to the solution and analysis of the heat equation.
1,120
questions
0
votes
0answers
23 views
Find solution of heat equation on right plane with source
Find a formula for the solution to
$$
\begin{cases}
w_t−kw_{xx}=f(x,t) &\text{for } x>0,t>0\\
w(x,0)=g(x) &\text{for } x>0\\
w_x(0,t)=h(t) &\text{for } t>0\\
\end{cases}$$
Hint:...
1
vote
0answers
27 views
Interpreting the solution of the heat equation on the whole line
Can someone provided some insight on what is meant by "weighted average"?
Allow me to give some exposition.
So given the heat equation on the whole line
$\begin{cases}
u_{t}-ku_{xx}=0\quad\...
2
votes
1answer
24 views
Bar problem and heat conduction equation
A thin bar, defined through $ x\in [0,l] $ has a temparature distribution $ \theta (x,t) $
and has at the point $ x=0 $ a temperature of $0$ .
At the other end there is a heat emission to another ...
2
votes
0answers
36 views
Evaluate $\frac{1}{2\sqrt{\pi b}}\int_{-\infty}^\infty\exp\left(-e^{x+a} - \frac{x^2}{4b}\right)dx$
So, I'm trying to evaluate the integral
$$
I(a,b) = \frac{1}{2\sqrt{\pi b}}\int_{-\infty}^\infty\exp\left(-e^{x+a} - \frac{x^2}{4b}\right)dx,
$$
where $a$ and $b$ are real numbers with $b>0$. This ...
0
votes
1answer
18 views
Find an explicit solution of the following system
I need to find an explicit solution $u(x,t)$ of the system
$$\begin{cases}
\partial_x^2 u = \frac{1}{k} \partial_t u \\
u(x,0) = f(x)
\end{cases}$$
where $f(x) = x^2e^{-x^2}$, $\,x \in \mathbb{R}, \,t&...
-1
votes
0answers
58 views
Heat Equation with Neumann boundary condition interval domain
Consider the heat equation on the interval $a \leq x \leq b$ with Newmman boundary conditions:
$$u_{t}(x,t)-u_{xx}(x,t)=0 \quad \textit{ for } a \leq x \leq b, t>0$$
$$u(x,0)=g(x) \quad \textit{ ...
0
votes
0answers
31 views
2D Heat Equation with Inhomogeneous Neumann Boundary Conditions
I would like to solve the 2D heat equation $u_t = \kappa \Delta u$ on a box ($[0,1]^2$ or $[0,2\pi]^2$, whichever is simpler), given Neumann boundary conditions - zero (no flux) on the bottom, left, ...
0
votes
0answers
10 views
Boundary conditions of the heat equation in a 1D rod
Suppose I have 2 rods (A and B) of 0.5m each. Rod A is at 0 degrees and rod B is at 100 degrees.
They are put into contact at the midpoint
I understand how the heat equation is created.
I read that ...
0
votes
0answers
11 views
Discretization of inhomogeneous Dirichlet boundary conditions for 2D heat equation
We have the following PDE on $\Omega = [0,1] \times [0,1]$: \begin{equation} \begin{split} \partial_t u &= \Delta u \; \; \text{for} \; \; (x,y) \in \Omega \\ u &= -1 \; \; \text{on} \; \; \...
0
votes
0answers
24 views
Showing that $\int_{0}^{\infty}k(x,z)f(x)dx$ is analytic with $k$ heat kernel.
Let $k(x,z)=\frac{1}{(4\pi z)^{n/2}}e^{|x|/{4z}}$ heat kernel. Show that $F(z):=\int_{0}^{\infty}k(x,z)f(x)dx$ is analytic, where $f\in L^1(\mathbb{R})\cap L^2(\mathbb{R}.)$
Will there be some theorem ...
0
votes
0answers
23 views
2D heat equation: boundary conditions that make physical sense?
I want to solve the heat equation $$u_t = c(u_{xx} + u_{yy})$$ on a rectangle with uniform initial temperature on the whole rectangle and constant, nonzero temperature along each side of the rectangle ...
2
votes
0answers
53 views
Rate of decay of Fourier coefficients on heat equation solution
Let $u(x,t) = \sum_{j=1}^\infty u_j^0 e^{-j^2t}\sin(jx)$ be the solution for the heat equation, where $u_j^0 = \sqrt{\frac{2}{\pi}}\int_0^\pi u^0(x)\sin(jx)\;dx$ are the Fourier coefficients of the ...
0
votes
0answers
18 views
Linear PDE with inhomogenous bounndary conditions.
Basically I am trying to solve the following problem:
Having the heat equation $$\partial u(x,t)/\partial t=-\epsilon \partial^2u(x,t)/\partial x^2$$
$$ \partial u(0,t)/\partial x=f(t)$$ $$ \partial u(...
0
votes
1answer
39 views
Heat Equation - Reducing the boundary condition
I should solve the heat equation:
$$ \frac{\partial }{\partial t}u(t,x) = \frac{\partial^2 }{\partial x^2} u(t,x) $$
with boundary conditions $u(t,0)=u_0, \: u(t,1)=u_1$ and initial condition $u(0,x)=...
0
votes
0answers
26 views
What is the Stationary Diffusion Equation Intuitively?
I'm having a little trouble understanding what the stationary diffusion equation does. As I understand it, the standard heat equation is $u_t = -k\Delta u$, where $k$ is for conductivity.
My intuitive ...
1
vote
0answers
29 views
1D Heat Equation with Inhomogeneous Neumann Boundary Conditions
I'm trying to solve the 1D heat equation with inhomogeneous boundary conditions.
$u_t=Ku_{xx}$
$u_x(0,t)=F_0 \hspace{0.2cm} and\hspace{0.2cm} u_x(L,t)=F_1$
$u(x,0)=f(x)$
My first step was to write the ...
4
votes
1answer
57 views
Heat equation solution using Fourier transform
I want to solve the equation
$$x^2\frac{\partial^2 u(x,t)}{\partial x^2}+ax\frac{\partial u(x,t)}{\partial x}=\frac{\partial u(x,t)}{\partial t}$$
with $u(x,0)=f(x)$ for $0<x<\infty$ and $t>0$...
2
votes
0answers
39 views
Construct solution of $u_t + (|u|)_x = \epsilon u_{xx}$ from solution of heat equation
I am trying to find the vanishing viscosity limit of
$\begin{alignat}{3}
\begin{cases}
u^{\epsilon}_t + (|u^{\epsilon}|)_x &= \epsilon u^{\epsilon}_{xx} &\mathrm{in} &\mathbb{R} \times (0,...
0
votes
1answer
27 views
Does the heat equation have a unique solution with these mixed boundary conditions
Does the heat equation $u_t - u_{xx} = 0$ on the unit square with $\forall 0 \leq x \leq 1: u(x,0)=0$,
$\forall 0 \leq t \leq 1: u(0,t)=0$, $\forall 0 \leq t \leq 1: u_x(1,t)=0$ have a unique solution?...
0
votes
0answers
16 views
Heat equation and its gradient flow
Let $u$ be the solution of the heat equation in $\Omega\subset \mathbb R$. So $u(x, t)$ is the solution of
$$\frac{\partial}{\partial t}u(x,t)=\frac{1}{2}\frac{\partial^2}{\partial x^2}u(x, t), \...
0
votes
1answer
61 views
Change of Variables in Heat Equation
I am looking how to apply a change the variables to the heat equation. It may be quite simple but I want to make sure that I did it correctly (or find my error if I made one). The equation is given as
...
0
votes
0answers
28 views
Shouldn't heat kernel on a weighted graph be inverse proportional to the shortest path length?
The heat kernel $k: \mathbb{R_{\geq 0}} \times \Omega \times \Omega \to \Bbb{R}_{\geq 0}$ on some domain $\Omega$ calculates the amount of energy (or information) transferred in time $t$ from $x \in \...
0
votes
0answers
21 views
Heat equation in a rectangle
I'm trying to solve the following problem.
Consider the heat equation in the rectangle $[0,\pi] \times [0,\pi]$ with Dirichlet boundary conditions. Calling $\Omega:=(0,\pi) \times (0,\pi)$ we have ...
0
votes
0answers
24 views
Heat equation in spherical coordinates: heat flow at boundary
At steady-state we can write the one-dimensional heat equation as
$$\frac{\partial}{\partial r}\left(r^2k\frac{\partial T}{\partial r}\right)=-r^2Q$$
which has the solution
$$T(r)=-\frac{r^2Q}{6k}+\...
1
vote
0answers
31 views
Is my interpretation of this proof of a maximum principle for the discrete heat equation correct?
I am looking for help on this proof of a maximum principle for the discrete heat equation. The following is from Introduction to Partial Differential Equations (Tveito, Winther).
Consider the Heat ...
1
vote
0answers
70 views
How to determine stability of nonlinear diffusion equation using explicit finite difference scheme?
I'm trying to determine stability criteria for a particular case of nonlinear diffusion
$$
\frac{\partial u}{\partial t} = \frac{\partial }{\partial x}\left(g(u)\frac{\partial u}{\partial x}\right),
$$...
0
votes
1answer
25 views
How to prove $v(x,t) = x\cdot Du(x,t)+2tu_t(x,t)$ is also a solution of the heat equation
How to prove that
$$
v(x,t) = x \cdot Du(x,t)+2tu_t(x,t)$
$$
is also a solution of the heat equation. $u_t(x,t)-\Delta u(x,t)=0$
Where $ u:R^{d \times1}\rightarrow R$ and $"\cdot" $ is dot ...
0
votes
0answers
17 views
What is the physical interpretation of $u_x(0,t)$, $u_x(1,t)$ and a negative diffusivity constant in 1D Heat Equation on $(0,1)$?
I'm looking for help with the following problem. We have the heat equation $u_t=Du_{xx}$ modelling the temperature across a conducting material, defined for $0<x<1, t>0$ with boundary ...
0
votes
1answer
28 views
Solution to a special differential equation
I am wondering whether the following differential equation can be solved.
$$\frac{\partial^{2}f}{\partial x^2}+ \frac{\partial^{2}f}{\partial y^2}+ \frac{\partial^{2}f}{\partial z^2}+ \alpha \frac{\...
0
votes
0answers
70 views
Is the general solution to the Heat Equation $u_t=\gamma u_{xx}$ a homogeneous function?
Is this solution correct?
The heat equation is
$$\frac{\partial u(x,t)}{\partial t} = \frac{x_0^2}{t_0} \frac{\partial^2 u(x,t)}{\partial x^2}, \mbox{where } \frac{x_0^2}{t_0}=\mbox{"gamma"}$$
...
1
vote
1answer
64 views
Heat equation with Gaussian boundary condition
Let $$ S(x,t)= \frac{1}{2\sqrt{\pi t}}e^\frac{-x^2}{4t},\quad
-\infty < x < \infty,\quad t>0$$
a. Find the solution to the equation
$$u_t = u_{xx},\quad -\infty < x < \infty,\...
0
votes
0answers
13 views
Heat equation in a three dimensional ball (for a Fourier analysis class)
I need help with the heat eqn in a three dimensional ball - I'm completely lost:
given equation, boundary conditions, and radius
1
vote
2answers
31 views
Interpretation of the heat equation
Let $u=u(x,t)$ a solution of $$\begin{cases}\partial _tu=\partial _{xx}u\\ u(x,0)=f(x)\end{cases}$$
I can compute the solution, but I can't interpret this sort of equation. For an ODE $v'(t)=f(v(t))$,...
0
votes
0answers
33 views
I do not understand what this PDE problem is looking for?!
This is an exercise in Qing book. the given hint is not helping me with what the problem is looking for and how to begin. I would appreciate any help.
ps. I am studying uniqueness of solutions of ...
0
votes
0answers
15 views
One-dimensional heat equation with nonlinear Robin-type boundary condition
I am trying to find an analytical solution to the following heat equation with nonlinear Robin-type boundary condition:
$$
\frac{\partial}{\partial t} u(t, x) = D \frac{\partial^2}{\partial x^2} u \...
0
votes
0answers
11 views
Prove the existence of cut off function
I want to prove the existence of cut off function:
"a smooth "cut-off" function $\Gamma$ such that $0\leq \Gamma(x) \leq 1$, $\Gamma(x)=1$ for $x \in B_{R}(0)$ and $\Gamma(x)=0$ for $x \notin B_{2R}(0)...
0
votes
0answers
14 views
Modified 3D heat equation with a quadratic term
Consider a generic 2D manifold $\mathcal{M}$ with Lorentzian metric $g_{\mu\nu}$ and a $\xi$-dependent scalar field $\phi:\mathcal{M}\times\mathbb{R}\longrightarrow\mathbb{R}$, where $\xi$ is a real ...
0
votes
0answers
26 views
Show that boundary value problem of the heat equation has no solution
Given the equation $$u_t(x,t)=a^2u_{xx}(x,t) + f(x,t),\ f(x,t)=cos(\frac{\pi x}{l})e^{-\omega t},\ x \in (0,l),\ t>0$$
and boundary values $$u(x,0)=cos(\frac{3\pi}{2l}x),\ u(0,t)=e^{-\alpha t},\ ...
0
votes
0answers
30 views
Application of PDE for solving steady state heat problem.
A thin rectangular homogeneous thermally conducting plate occupies the region $0 \leq x \leq a$,
$0 \leq y \leq b$. The edge $y = 0$ is held at temperature $Tx(x − a)$, where T is a constant and the
...
0
votes
0answers
5 views
Limit of solution to heat equation with mixed boundaries
This question concerns the heat equation $\frac{\partial T(x,t)}{\partial t}=\frac{\partial ^2T(x,t)}{\partial x^2}$ with mixed boundary conditions $\frac{\partial T(0,t)}{\partial x}=T(1,t)=0$ for $t&...
0
votes
0answers
25 views
Non-homogeneous heat equation using the Fourier transform with a particular term
I´m trying to solve $u_t(x,t) = u_{xx}(x,t) - xu_x(x,t), x \in R, t > 0,$ and $u(x,0) = f(x)$ using the Fourier transform. I have seen an example where instead of $xu_x$ they put a known function, ...
0
votes
1answer
36 views
Heat equation with zero Neumann boundary condition
I have a question ask to find Fourier cosine series, so my approach as follow:
Given $\phi(x) = x^2$ and $x \in [0,1]$. For $n \neq 0$, directly applying the formulas, we have
$$A_n = 2 \int_0^1 x^2 ...
0
votes
0answers
38 views
Approach to solve a Coupled system of PDE [Heat transfer in cylindrical coordinates]
I have the following two PDEs, which describe steady-state coupled heat transport between a externally heated axi-symmetric solid body (Eq. 1, $T(r,z)$) and a fluid (Eq. 2, $t(z)$) flowing inside it:
...
0
votes
0answers
37 views
Periodic extension for Heat Equation with time-dependent source
I have the PDE
$$\frac{\partial u}{\partial t} = \frac{\partial^2u}{\partial x^2} - tu, \quad 0<x<1$$
subject to the boundary conditions (BCs) $u(0,t)=0, u_x(1,t)=0$ and initial condition (IC)...
1
vote
1answer
40 views
Reducing $u_t = u_{xx} + u$ to the heat equation, and finding a solution
Been given the following problem:
Consider the PDE
$$u_t = u_{xx} + u, 0 < x < 1$$
$$u(0, t) = u(1, t) = 0, t > 0$$
$$u(x, 0) = f(x), 0 < x < 1$$
(a) Use the method of ...
0
votes
0answers
14 views
Show that the heat kernel satisfies the identity- Semi group property of the solution process for the heat equation
I was thinking of using Greens identity and few more theorems to solve this but I couldnt get it. Can someone suggest me or help me here? Appreciate your support & help.
0
votes
0answers
8 views
Discretization error of implicit finite difference scheme with robin boundary by maximum principle
I am handling the following heat equation with Robin condition by implicit finite difference scheme:
\begin{cases}
u_t = \frac{1}{2}u_{xx}, (t,x) \in [0,T]\times[0,1], \\
u(t,0) = 0 = u(0,...
0
votes
0answers
46 views
Cauchy problem for the heat equation
I need to find the solution of the Cauchy problem for the heat equation :
$$
U_t = U_{xx} + 3t²
$$
where $U(x,0) = sin(x)$
I would appreciate any help
Thanks in advance
2
votes
0answers
26 views
Heat equation in Cauchy-Dirichlet problem
I have this Cauchy-Dirichlet problem and i have to prove that the solution is non-negative in $\mathbb{R}^{+}\times\left[0,1\right]$. How can i do?
\begin{cases}
u_t(t,x) - u_{xx} (t,x)= tx \quad\...
0
votes
0answers
13 views
1D Heat Equation with Non-Standard Mixed Conditons on a Single Boundary Point
I'd like to solve a specific 1D heat equation with non-standard mixed conditions.
Ultimately, I'd like to solve something like
$$u_t = u_{xx} +q(x)$$
$$u(0,t)=0,\quad u_x(0,t) = 0,\quad u(x,0) = 0,\...
