Questions tagged [partial-differential-equations]

Questions on partial (as opposed to ordinary) differential equations - equations involving partial derivatives of one or more dependent variables with respect to more than one independent variables.

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PDE solution using convolution and (inverse) Fourier transform

I'm studying PDEs from old course material which includes answers to most of the exercises. Given a PDE $-u''(x)+au(x)=f(x)$ and a function $g(x)=e^{-|x|}$, we should find $u(x)$ by calculating the ...
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When are the coefficients of Laplace equation based on Fourier series?

in various posts I have made recently on the Laplace equation, I have questioned the methods for finding the coefficients of the harmonic/hyperbolic functions (Y(y) and X(x) of U(x,y). One examples is ...
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Are there any $L^\infty$ bounds on the eigenfunctions of a first order elliptic dfferential operator on $\mathbb{R}^n$ in terms of eigenvalues?

The question is as in the title. All references I searched for only seem to deal with compact manifolds. So I ask for noncompact cases. For a first-order elliptic differential operator with "...
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Developing a correct Ansatz and solving the Laplace equation on a rectangle

I am trying to find a way to efficiently solve the Laplace equation, by generating a rule for making Ansatz. However, once this ansatz is found, I struggle with finding the coefficients and the ...
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