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I am pretty aware of the interpretation of the equation itself: $u_t=\lambda u_{xx}$ along with the boundary conditions $u(0,t)=u(l,t)=0$ and $u(x,0)=T(x)$, with $0<x<l$ (e.g., Gyu's answer Interpretation of the heat equation is an excellent explanation).

My question is something about the Fourier series that describes the initial temperature $T(x)=u(x,0)$. When solving the equation (along with the boundary conditions), one realizes that the function $T(x)$ can be described as a Fourier series:

$$T(x)=\sum_{n=1}^{\infty} A_n \sin(\frac{n\pi x}{l}),$$

with $0<x<l$. The $A_n$ can be completely described by integrating $\frac{2}{L}\int_0^l f(x)\sin(\frac{n\pi x}{l})dx$, which can be used to described the final solution of the PDE, namely,

$$u(x,t)=\frac{2}{l}\sum_{n=1}^{\infty} \bigg(\int_0^l f(x)\sin(\frac{n\pi x}{l})dx\bigg)e^{-\lambda((n^2\pi^2)/l^2)t}\sin(\frac{n\pi x}{l}).$$

Is there any physics interpretation of the $A_n$ in terms of this specific problem? Moreover, is there any physics interpretation of the functions $A_n \sin(\frac{n\pi x}{l})$.

My questions are based on the idea that if we have a function that describes some vibration (like some sound) then its Fourier series can be understood as a decomposition of vibrations that caused the vibrations of $f(x)$ by overlapping (and then I can study the lower/higher frequencies). This idea is used as an application for several problems, so I was wondering if there is something similar to it in the heat equation.

Thanks

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  • $\begingroup$ You're probably reading more into the expression than you really should. The main idea is that the heat equation smooths out sharper gradients (which correspond to high frequencies) faster than less sharp gradients. That's about it. There isn't really a strong physical connection to vibration. $\endgroup$ – Ian Aug 27 at 1:00
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In a first place the Fourier series is only a useful mathematical tool to analytically solve or approximate a partial differential equation (PDE). One should be careful to attribute too much physical meaning to these mathematical expressions.

Next you can regard the Fourier series as a trigonometric interpolation, which is an exact approximation of the PDE in the steady state $T(x)$ for $n\rightarrow\infty$. Here each individual summand can be intepreted as mode with a prefactor $A_n$ as weight. For example the first mode $A_0$ is the integral mean.

Note that the polynomial representation with a trigonometric basis is only one of many modal representations. You could also use other arbitrary polynomials. The interpretation of the weights would change each time.

Regards

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