# Can the Solution of a Heat Equation Only Get Smoother as Time Increases?

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• feynman1
In summary, you are asking how to show or prove that the shape of the solution of a heat equation can only go smoother and smoother but not the opposite as time increases.
feynman1
how to show or prove that the shape of the solution of a heat equation can only go smoother and smoother but not the opposite as time increases?

Last edited:
feynman1 said:
how to show or prove that the solution of a heat equation can only go coarsened/smoother as time increases?
Sorry, my Google search of your term coarsened seems to say the opposite of smoother. Could you please expand on your question, and post links to relevant articles?

berkeman said:
Sorry, my Google search of your term coarsened seems to say the opposite of smoother. Could you please expand on your question, and post links to relevant articles?

graduate school / PhD level is the right category.
maybe coarsen means differently in different fields, so let's forget about coarsen and just consider a solution becoming smoother and smoother.

feynman1 said:
how to show or prove that the solution of a heat equation can only go coarsened/smoother as time increases?

berkeman said:
sorry but if there was such a link there'd be explanations then I wouldn't have posted here. It's just about time irreversibility of heat equations.

hutchphd
feynman1 said:
time irreversibility of heat equations
So you're really going to make us Google search that phrase? Please do that search and tell us what you don't understand. Seriously.

PhDeezNutz, jim mcnamara and hutchphd
Consider the simplest situation: boundary conditions are ##2\pi-##periodic
$$u_t=u_{xx},\quad u\mid_{t=0}=v(x)=\sum_{k\in\mathbb{Z}}v_ke^{ikx}$$ then the solution
is
$$u(t,x)=\sum_{k\in\mathbb{Z}}v_k e^{-|k|^2t}e^{ikx}.$$ It is it.

feynman1 and jasonRF
A similar way to look at this is to consider the same problem as Wrobel did, ##u_t = u_{xx}## with ##u(x,t=0) = v(x)##, but over the entire real line. You should be able to derive the kernel ##g## such that the solution is ##u(x,t) = \int g(x-y, t) v(y) \, dy##.

feynman1
thanks for the suggested explanation, however is there any approach of not using a kernel/Green's function/fourier expansion, since an explanation via a summation of functions isn't the most convincing because one needs to assess the shape along the whole curve?

feynman1

## 1. What is a heat equation?

A heat equation is a mathematical model that describes how heat is distributed in a given region over time.

## 2. Can the solution of a heat equation only get smoother as time increases?

Yes, the solution of a heat equation is known to become smoother as time increases. This is because as time passes, heat is gradually distributed and diffused throughout the region, resulting in a more uniform temperature distribution.

## 3. What factors affect the solution of a heat equation?

The solution of a heat equation is affected by various factors such as the initial temperature distribution, the material properties of the region, and the boundary conditions.

## 4. Is the solution of a heat equation always a smooth function?

No, the solution of a heat equation may not always be a smooth function. It depends on the initial conditions and boundary conditions of the region. In some cases, the solution may have discontinuities or sharp changes.

## 5. How is a heat equation used in real-world applications?

The heat equation is used in various fields such as physics, engineering, and economics to model and understand heat transfer processes. It is commonly used in designing and optimizing heating and cooling systems, predicting temperature changes in materials, and analyzing heat flow in different scenarios.

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