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.

1607 questions
6
votes
1 answer

Ill-posedness and well-posedness

Why is the backwards heat equation an ill-posed problem? $$\frac{∂u}{∂t}=-k\frac{∂^2u}{∂x^2}$$. And what makes this heat conduction equation $$\frac{∂u}{∂t}=k\frac{∂^2u}{∂x^2}$$ well-posed?
SweetE
  • 61
3
votes
1 answer

heat equation conservation of energy

Suppose $u$ is a solution of the heat equation with the property that $|\int\limits_{-\infty}^{\infty}u(x,0)dx| < \infty$, and $u_{x}(x,t) \rightarrow 0$ as $x \rightarrow \pm \infty$. Then integrating the PDE, we…
Gabor Bakos
  • 423
2
votes
1 answer

Incorporating the initial condition

I have solved the heat equation and have gotten to the stage of getting a general solution $$u(x,t)=x+\sum^\infty_{n=1} c_n \sin(\pi n x)e^{-\pi^2 n^2 t}$$ And I have the initial condition $$u(x,0)=x+\sin(\pi x)$$ How do I incorporate the initial…
Al jabra
  • 2,331
1
vote
1 answer

Finding the heat equation (PDE)

I have the terms: $$\left\{\begin{array}{l l} u_t=u_{xx}, 00\\u(0,t)=1, u(1,t)=3, t>0\\ u(x,0)=2x+1-\sin(2\pi x), 0
rablentain
  • 645
1
vote
0 answers

How do I show that $u(x, t) = e^{-\alpha^2k^2t}\sin(kx)$ is a solution for $u_t = \alpha^2u_{xx}$.

I want to show that $u(x, t) = e^{-\alpha^2k^2t}\sin(kx)$ is a solution for $u_t = \alpha^2u_{xx}$. I did the following: $$u_x = e^{-\alpha^2k^2t}\cos(kx)k$$ $$u_{xx} = -e^{-\alpha^2k^2t}\sin(kx)k^2$$ $$u_{t} =…
GaMbiTaaaa
  • 554
1
vote
0 answers

Solve heat equation with initial condition $\delta(x-x_j)$

I am given the following heat equation: \begin{equation} u_t(x,t)-\Delta u(x,t)=0 ~~~ x \in \mathbb{R}^n,~t>0 \end{equation} with the initial condition $u(x,0)=g(x)=\sum_{j=1}^m j \delta(x-x_j)$ which I would like to solve by using the Fourier…
Andreas804
  • 139
  • 8
1
vote
0 answers

Steady state temperature in cylinder subjected to a prescribed temperature distribution

My question: Obtain an expression for the steady-state temperature distribution $T(r,φ)$ in a long, solid cylinder$ 0 \leq r\leq b, 0\leq φ \leq 2π$ for the following boundary conditions: The boundary at $r = b$ is subjected to a prescribed…
Seongqjini
  • 1,431
1
vote
0 answers

Showing that the $\Delta$ is the generator of a semigroup in the Heat's Equation.

I am studying the concept of semigroup with the example of the heat equation and I have a concern. Once you have the semigroup generated by the Laplace operator (as in the case of the heat equation). How can I explicitly show that…
eraldcoil
  • 3,508
1
vote
0 answers

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\quad \mathbb{R}^{+}\times\left[0,1\right] \\ …
Ant2198
  • 27
  • 5
1
vote
1 answer

Prove that $v(t,x) = xu_x(x,t) + 2tu_t(x,t)$ satisfies the heat equation using the fact that $u^{\lambda} = u(\lambda x, \lambda^2t)$ satisfies it

Let $\lambda \in \mathbb{R}$ and denote $u_x(x,t) = \frac{d}{dx}u(x,t).$ Let $u$ satisfy the heat equation $u_t - u_{xx} = 0$ on $\mathbb{R}\times(0,\infty).$ Prove that $v(t,x) = xu_x(x,t) + 2tu_t(x,t)$ satisfies the heat equation using the fact…
Collapse
  • 1,393
1
vote
0 answers

Question About Converting Black Scholes Differential Equation to Heat Equation

I'm reading a book about converting Black Scholes equation to heat equation and I highlighted in bold for my doubt, and really appreciate your advice on it. Let $S$,$T$,$V$ denote underlying asset price, maturity and option price separately. Here…
M00000001
  • 177
1
vote
0 answers

Can someone help me with the heat equation with a time dependent source?

It's one dimensional. 0 ≤ x ≤ π u_t = u_xx + t^2 e^-t BC: -u_x = u/2 for both ends IC: u(0,x) = 0 I already determined the eigenvalues but I can't proceed. And what if the source is te^-t x instead?
Use DuckDuckGo
  • 11
1
vote
1 answer

Is the steady state solution of the Heat Equation with Dirichlet boundary conditions always 0?

A heat equation problem with Dirichlet boundary conditions on the domain $[x_1,x_2]$ $$\frac{\delta u}{\delta t} = k \frac{\delta^2 u}{\delta x^2}$$ $$u(x_1,t) = u(x_2,t) = 0$$ Would have eigenfunctions corresponding to sines. Whereas Neumann…
hirschme
  • 163
1
vote
0 answers

heat equation with exponentially time dependent boundary condition

Consider the one dimensional heat equation on a semi-infinite bar such that $u_t=u_{xx}$ with initial condition of $u(x,0)=0$. What is the solution for all x and t if the boundaries are $u(0,t)=e^{-t}$ and $u(x,t)=0$ as x approaches infinity?
cat
  • 11
1
vote
1 answer

$u_t = t\Delta u$, solve with streching

Solve heat equation with time coefficient $$u_t = t\Delta u, u(x,0) = \delta(x)$$ The question asks using scaling and self-similarity argument, I know how to do the usual heat equation by sending $x\to\lambda x$ and $t\to\lambda^2t$. But for this…
QD666
  • 557
  • 2
  • 9
1
2 3