# Tagged Questions

Questions on "Partial Differential Equations", as opposed to "ordinary differential equations".

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### Kernel of Helmholtz Equation on a plane

On the $z=0$ plane I have the boundary conditions $V=\delta(x)\delta(y)$ I want to solve for $z>0$. Helmholtz equation is $\nabla ^2 V +k^2 V=0$ I though that spherical harmonics would be useful. ...
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### Gårding's inequality for $\mathbb R^n$ implies that for bounded smooth domains?

We are given (weak) Gårding's inequality for elliptic pseudodifferential operators: Given $a\in S^m$ such that $\operatorname{Op}(a)$ is an elliptic operator, namely $\exists c,R>0$ such that ...
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### Find the Equation of the Envelope of a Family of Line (Plane) Segments

Consider the first quadrant in the $OXY$ plane in $\mathbb{R}^2$. Point $O$ is the origin and the points $P$ and $Q$ are chosen on the $y$-axis and the $x$-axis, respectively as it is showed in the ...
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### Solution of wave equation in first quadrant is positive?

My question: Consider $u_{tt}-u_{xx}=F(x,t)$ in the first quadrant ($x,t>0$) with boundary conditions $u(x,0)=f(x)$, $u_t(x,0)=g(x)$ and $u(0,t)=0$. Does it necessarily follow that $u\geq 0$? My ...
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### General solution to Laplace Equation

Show the general solution to the Laplace equation, $$\frac{\partial^2\phi}{\partial x^2}+\frac{\partial ^2\phi}{\partial y^2}=0$$ is $\phi(x,y)=f(x+iy)+g(x-iy)$. The only thought I have is let $x+iy$ ...
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### Show the real part of a complex function is a solution to the Airy Equation

I am considering a function $v=$ Re$\left(e^{i(Nx + tN^3)} u\left(t,\frac{x + 3N^2t}{\sqrt{3N}}\right) \right)$, where $u$ is a smooth solution to the free Schrodinger equation, $i u_t = -u_{xx}$, and ...