Questions tagged [heat-equation]

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

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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{\...
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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 \...
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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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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}{...
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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 ...
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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 $c$, ...
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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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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} ...
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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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$u_{t}=$ $au_{xx}$ $+cu$ $+F(x,t)$ 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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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):=...
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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 ...
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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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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 ...
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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 ...
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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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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 ...
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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 ...
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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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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 ...
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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&...
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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,...
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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 ...
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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 ...
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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\...
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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<...
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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 $\...
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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 ...
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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 ...
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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, \...
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1 answer
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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 \...
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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 ...
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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 \...
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2 answers
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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 ...
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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 ...
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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 ...
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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 ...
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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 (...
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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 ...
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4 votes
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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, \\ &...
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3 votes
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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 $\...
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2 votes
1 answer
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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+\...
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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 $w$ ...
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2 answers
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CN FEM Stability [closed]

For Crank-Nicolson FEM for solving $u_{t}-\Delta u=f$, how can I show that it is stable?
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3 votes
1 answer
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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}(\...
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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 ...
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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}{\...
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2 votes
1 answer
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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(...
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3 votes
1 answer
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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 ...
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2 votes
0 answers
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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 \, ...
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