Feb 8, 2015 · Then what you call "energy" corresponds to the total mass of particles, that is, $$ M[u]=\int_{\Omega}u(x)\mathrm{d} x, $$ and if you consider a closed system, which means we impose Neumann boundary conditions, then the conservation of "energy" translates to the conservation of total mass, that is, solutions of the heat equation satisfy $\frac ...

May 4, 2016 · The initial conditions only enter if one wants to determine the total energy, not in showing that it's constant. $\endgroup$ – Semiclassical Commented May 3, 2016 at 23:37

Jan 18, 2018 · On Duhamel's principle for the inhomogeneous heat equation with irregular data and regularity of solution Hot Network Questions Going from graduate course knowledge to Research Knowledge and associated career paths?

May 30, 2023 · Reading the Wikipedia page on the heat equation, I've found the following passage: An additional term may be introduced into the equation to account for radiative loss of heat. According to the St...

Aug 4, 2021 · I been looking at the papers 1) B Gaveau,Principle de Moindre action,propagation de la chaleur,et estimees sous,elliptiques sur certains groupes nilpotents 2) A Hulanicki,The Distribution of energy in the Brownian motion in the Guassian Field and analytic hypoellipticity of certain subellitpic operators

Jan 17, 2019 · In other words, the Heat Eqn. is a special case of the Poisson Eqn. in which the "thing to solve for" is the rate of change of heat. The Poisson Eqn. is a generalization that could solve for many kinds of things, including gravity (f=gravitational force=4 pi G), electrostatics (f=charge=-p/e), fluid pressure (f=pressure), or heat (f=rate of ...

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Coming back to the physical interpretation: you see that Wikipedia is being a bit imprecise: it is not the adding up of solutions that matters really; what drives Duhamel's principle is the fact that you are adding only a limited number of solutions.

Mar 11, 2021 · I cannot understand the derivation of the formula for potential energy ($\frac{1}{2}mv^2$).

May 27, 2023 · Like said earlier, by far the easiest method is to use potential energy at the extreme position. I'll be solving this as a general SHM, not using any properties of the pendulum case given I consider ω as maximum angular speed and ω' as angular frequency Maximum energy = 0.5 * I * ω * ω. ω = A * ω'