Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [heat-equation]

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

The heat equation is a particular parabolic partial differential equation used to describe the temperature or heat distribution of a system over time. It can be written most generally as

$$\frac{\partial u}{\partial t} - \alpha \nabla^2 u = 0$$

where $\nabla^2$ is the Laplace operator, and $\alpha$ is a positive constant describing thermal diffusivity (which is usually normalized to $1$).

There are a number of common solution techniques, including separation of variables and Fourier series, as well as using a Green's function to find a fundamental solution.

Reference: Heat equation.

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all solutions to the heat equation

If we solve the heat equation $u_t=u_{xx}$ by separation of variables, we assume that $u(x,t)=f(x)g(t)$, and solving 2 ordinary differential equations we can derive that $u(x,t)=e^{\omega^2t}(b\cdot \sin(\omega x)+ a\cdot \cos(\omega x))$ for some…
user56834
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Scaling transformation of nonzero initial condition

I'm trying to find the self similar solution of the fractional heat equation ,$\frac{\partial^{\alpha} u}{\partial t^{\alpha}} = \frac{\partial^{2} u}{\partial x^{2}}+ k, 0<\alpha <1, t>0,$ with initial condition $u(x,0)=f(x)$. I'm using the…
Vinoth Kesavan
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Leibniz rule; Partial Differential Equations

I'm stuck on a question :| So far I have, i) $$I'(t)=\int^L_0 2 \frac{\partial v(x,t)}{\partial x} [v(x,t)] dt$$ ii) $$I(0)=\int^L_0 [v(x,0)]^2 dx= 0$$
phamtom44
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Discrete maximum principle

\begin{align} \partial_t u(x,t) & = \kappa \partial_{xx}u(x,t), & -1 < x < 1, & \quad t>0 \quad \kappa > 0 \nonumber\\ u(-1,t) & = g_1(t) & t>0 & \\ u(1,t) & = g_2(t) & t> 0 & \nonumber \\ u(x,0) & = n(x) & -1 \leq x \leq 1…
anonym
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Heat Equation: Polynomial Inital, Neumann Boundary

Consider the heat equation solution $h(x,t)$ where $x \in [0,1]$ and $t \geq 0$ with initial condition $h(x,t=0)=f(x)$ and Neumann boundary $\frac{d}{dx} h(x=0,t)=0$ and $\frac{d}{dx} h(x=1,t)=0$. Suppose $f(x)=x^n$ where $n \in \mathbb{N}$.…
Golabi
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semi-infinite heat equation with Dirichlet BC via Laplace transforms

I am trying to solve the heat equation for a semi infinite rod with lateral surfaces insulated and $u(x,0)$ = $u_0$ for $x>0$, $u(0,t)=u_1$ for $t>0$, and the $\lim_{t\to\infty} u(x,t)=u_0$. I think I need to start…
Jackson Hart
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Why is $ g'(t)-cg(t)=0 \iff g(t)=Ae^{ct} $

Why is $$ g'(t)-cg(t)=0 \iff g(t)=Ae^{ct} $$ and how do I know that? This is a part of the heat equation that I don't understand, I must have missed this part in some other course... What do you call this and where can I read about it?
rablentain
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Heat transient problem in insulated cylinder

I try to solve the below problem: A long, hollow cylinder, $a \leq r \leq b$, is initially at a temperature of $T = F(r)$. For times $t > 0$ the boundaries at $r = a$ and $r = b$ are kept insulated. Obtain an expression for temperature distribution…
Seongqjini
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Heat conduction problem, insulated slab

A 1-D slab, $0 ≤ x ≤ L$, is initially at a temperature of $F(x)$. For times t > 0, both of the boundary surfaces are perfectly insulated. Obtain an expression for the temperature $T(x,t)$ in the slab. Clearly show the steady-state temperature in…
Seongqjini
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When a kernel is positive?

I am trying to verify that the heat's kernel is positive using the inverse Fourier transform. For this, I calculate the heat's kernel by means of a contour integral. After verifying this fact, my question arises. Let…
eraldcoil
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Heat equation on finite interval with decaying heat source

I am trying to solve the heat equation on a finite interval with a localised decaying heat source, but I am stuck. Specifically: Consider the equation $$ u_t(t,x)=\kappa u_{xx}(t,x) +q(t,x) \,,$$ where $x\in (0,L)$ and $t>0$. The boundary and…
Toffomat
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Analytical solution of 2D Laplace equation with discontinuous BCs.

I was trying to obtain analytical solution of 2D Laplace equation. I understood the standard procedure which involved the Method of separation of variable. I was trying to obtain the exact solution of Laplace equation with these BCs. u(0,y)=0,…
Prakhar Sharma
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Heat-equation problem

Determine a formal solution of the heat flow problem decribed by the following with initial and boundary values. $$ \begin{cases}\displaystyle \frac{\partial u}{\partial t}=\beta\frac{\partial^2 u}{\partial x^2},&0
Unknown
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Is there a name for this type of boundary condition of the heat equation : u(x,0)==0, u(0,t) == F(t) leading to a general solution in terms of erfc?

In an earlier contribution "The general solution to the heat equation" from Apr 27, 2018 a solution to the heat equation in terms of erfc functions was obtained for the following boundary conditions u(x,0)==0, u(0,t) == F(t) (with F(t)=…
Robert Kragler
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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 $$\lim_{t\to 0+}\int_{|x|>\delta}K_t(x)|\varphi(x)-\varphi(0)|\,dx=0?$$
eraldcoil
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