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I am currently reading some material on the Fourier series. The main motivation of course is to write a periodic function as a series involving cosine and sine functions, which we understand much about. It turns out that we have some nice convergence results regarding the Fourier series and the function which we wish to compute the series for.

Now the texts I am reading show that it is possible to derive solutions for PDE's by applying Fourier series methods, however, the text does not mention when such an application will work. So I am curious to find out when can someone use Fourier series methods to find solutions of PDE's? Also given a PDE is it always possible to re-write the PDE for the Fourier series?

For example, suppose we consider $-\Delta u+u=f$ in $\Omega\subset\mathbb{R}^{N}$ with zero boundary conditions. What conditions must be satisfied so that $\hat{-\Delta u}+\hat{u}=\hat{f}$ in $\Omega\subset\mathbb{R}^{N}$ makes sense, with appropriate boundary conditions? Is it sufficient to take the periodic extension of, $\tilde{u}$, on the appropriate domain $\tilde{\Omega}$, and hence we can always consider the "transformed" PDE? Is it necessary for the differential operator to have eigenfunctions which form an orthonormal basis of $L^{2}(\Omega)$?

I understand that for any periodic function one can always take the Fourier series. However, it is not clear to me under what conditions one can consider solutions of a PDE as a Fourier series and more specifically, when one can consider the "transformed" PDE.

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This question has my interest too. For me, all you need to satisfy so as to write your PDE's is a regularity condition on $u$, $f$, they need to be in a periodic $L^2$ space so you can use the hilbert bases made of adapted exponential. I don't think it has to be linked with the partial operator eigenvalues.

I know also that for time dependent PDE's, it will depends if the initial function condition is periodic. Historically, Fourier series were introduced to deal with the heat equation through a wall. I can give you the following exemple if it helps.

Let $u_0$ non-zero $\in C^0(\mathbb{R})$, $2\pi$-periodic , $C^1$ piecewise on $\mathbb{R}$. We are looking for $ u : \mathbb{R}^+ \times \mathbb{R} \mapsto \mathbb{R}$ such as :

\begin{align} & \forall t \in \mathbb{R}^+, \quad x \mapsto u(t,x) \text{ is 2$\pi$-periodic}\\ & u \in C^0(\mathbb{R}^+ \times \mathbb{R})\\ & u \in C^{\infty}(\mathbb{R}^{+*} \times \mathbb{R})\\ & \frac{\partial u}{\partial t}(t,x)= \frac{\partial^2 u}{\partial x ^2}(t,x) , \quad \forall (t,x) \in \mathbb{R}^+ \times \mathbb{R}\\ & u(0,x)=u_0(x), \quad \forall x \in \mathbb{R} \end{align}

Then such a function is unique and is given by $$\displaystyle u(t,x)= \sum_{n=-\infty}^\infty c_n(u_0) e^{-n^2t} e^{inx}$$ where $c_n$ are the Fourier coefficiuent sof $u_0$. In this case, we use the fact that for a fixed $t \in \mathbb{R^+}$, $x \mapsto u(t,x) \in \mathbb{L}(0,2\pi)$ (periodic) so we can develop in Fourier series.

Hope this helps a little.

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