Tagged Questions

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Solving PDE via Fourier Transform & Uniqueness

When a PDE is solved via Fourier transform, is there already a uniqueness assertion that comes for free? For example, if we Fourier the heat equation \begin{align} \partial_t u(x,t) &= \Delta ...
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Fourier transform help for solving $u_t+u_{xxxx}+u_{xx}=0$

I just started to learn a little bit of fourier analysis in solving PDEs. I want to find a solution $u(x,t)$ to $u_t+u_{xxxx}+u_{xx}=0$. My attempt: Applying the fourier transform to both sides gives ...
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Explicit solution of the nonlinear Schrödinger equation

Consider the linear Schrödinger equation, $$(LS) \begin{cases} \partial_{t}u= i\Delta u, t\in \mathbb R,\\ u(x,0)=u_{0}(x), \end{cases}$$ $x\in \mathbb R^{n}.$ Taking the Fourier transform with ...
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intuition behind an identity related to fourier transforms

I saw the proof of this identity in a question about Fourier transforms : $F(f(−t))w=F(f(t))(−w)$ Can someone give the intuition behind it ? What I understand of Fourier transform of a function ...
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I know it s simple but how to show that $\mathcal F(f(-t))w=\mathcal F(f(t))(-w)$ ? $\mathcal F(f(-t))w=\int^\infty_{-\infty}f(-t)e^{-iwt}dt$ $if -t=x\to -dt=dx$ ...
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Using fourier analysis in order to solve differential equations.

http://www.enm.bris.ac.uk/admin/courses/EMa2/Lecture%20Notes%2009-10/LSPDE5.pdf The above PDF teaches us the separation of variables method. However, there are some things I dont understand, that I ...