# Questions tagged [heat-equation]

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

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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I am trying to find an analytical solution to the following heat equation with nonlinear Robin-type boundary condition: $$\frac{\partial}{\partial t} u(t, x) = D \frac{\partial^2}{\partial x^2} u \... 0answers 11 views ### Prove the existence of cut off function I want to prove the existence of cut off function: "a smooth "cut-off" function \Gamma such that 0\leq \Gamma(x) \leq 1, \Gamma(x)=1 for x \in B_{R}(0) and \Gamma(x)=0 for x \notin B_{2R}(0)... 0answers 14 views ### Modified 3D heat equation with a quadratic term Consider a generic 2D manifold \mathcal{M} with Lorentzian metric g_{\mu\nu} and a \xi-dependent scalar field \phi:\mathcal{M}\times\mathbb{R}\longrightarrow\mathbb{R}, where \xi is a real ... 0answers 26 views ### Show that boundary value problem of the heat equation has no solution Given the equation$$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},\ ...
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 ...