# 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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### Is ‘All differential equations have infinite solutions if there aren’t any initial conditions’ wrong? [closed]

Though the answers depend on whether we are thinking about real numbers or not. I think there are counterexamples but can’t think of any Someone on stack exchange said that $f(x)^2+f’(x)^2=0$ is a ...
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### Formulating a solution ansatz for the 1D heat equation in polar coordinates to learn the PDE in a PINN setting

Hello Math Stack Exchange Community, I am working on solving a partial differential equation (PDE) with a neural network in a PINN-like fashion, and I am seeking advice on identifying an appropriate ...
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### Question about Evans’ derivation of a Green's function

At page 34 of "Partial Differential Equations" by Evans, in order to define the Green function for the set $U$, the author defines a family of functions as the solutions of the boundary ...
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### Regularity for computing the first variation

I am having a trouble understanding the regularity needed to compute the first variation for the Euler-Lagrange equation for the functional $$F(u) = \int f(u) dx$$ Suppose $u:U \to \mathbb{R}$ for ...
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### Solving 1st order PDE including convolution

I'm studying Van Kampen's "Stochastic processes in physics and chemistry" and stuck to some exercise (p.78): That is, solving \frac{\partial P(y, t)}{\partial t}=\int_{-\...
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### Confusion in understanding the definition of a linear and quasi-linear PDE.

I was recently studying Partial Differential Equations (PDE). While going through the basics, I stumbled across the definition of a linear PDE and quasi-linear PDE. The definition went as follows: A ...
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### Basic Solution to the Heat Equation

As a learning example, I am trying to derive the solution to the basic Heat Equation (https://en.wikipedia.org/wiki/Heat_equation) using Fourier Transforms. As I understand, the Heat Equation can ...
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1 vote
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### Proof of Paley-Wiener Theorem

I'm trying to understand the proof of the following version of Paley-Wiener theorem under the additional assumption $f \in L^2$: I understood the part $(2) \Rightarrow (1)$ but I couldn't follow a ...
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### Solution of the "reciprocal of the heat equation"?

I was playing around with the heat equation in one dimension and tried to guess what the solution to homogenous boundary conditions and a sine wave as initial condition on the interval $0<x<\pi$ ...
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### In this proof, how does the Fourier transform work? [closed]

I recently read the article 'E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behaviour for nonlocal diffusion equations. J. Math. Pures Appl. (9) 86 (2006), 271–291'. The Theorem 2.1. confused me....
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