# Questions tagged [heat-equation]

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

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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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### 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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### 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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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 $L<x<... 0 votes 0 answers 21 views ### Maximum principle for heat equation with modified RHS term Let$\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$in$\... 67 views

### 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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### 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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### 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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### 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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### 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}(\...
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 ...