Questions tagged [heat-equation]

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

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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:...
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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\...
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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 ...
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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 ...
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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&...
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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{ ...
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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, ...
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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 ...
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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} \; \; \...
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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 ...
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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 ...
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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 ...
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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(...
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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)=...
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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 ...
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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 ...
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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 $t>0$...
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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,...
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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?...
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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), \...
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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 ...
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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 \...
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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 ...
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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}+\...
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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 ...
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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 $$ \frac{\partial u}{\partial t} = \frac{\partial }{\partial x}\left(g(u)\frac{\partial u}{\partial x}\right), $$...
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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 ...
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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 ...
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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{\...
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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 $$\frac{\partial u(x,t)}{\partial t} = \frac{x_0^2}{t_0} \frac{\partial^2 u(x,t)}{\partial x^2}, \mbox{where } \frac{x_0^2}{t_0}=\mbox{"gamma"}$$ ...
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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,\...
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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
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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 $v'(t)=f(v(t))$,...
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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 ...
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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 \...
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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)...
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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 ...
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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},\ ...
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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 ...
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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&...
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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, ...
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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 ...
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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: ...
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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)...
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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 ...
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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.
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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,...
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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
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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\...
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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,\...

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