# Tagged Questions

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### Matrix representation of quadratic partial differential equations

For a particular problem, I have two following quadratic differential equations: $f_{uu}g_{u} - g_{uu}f_{u}$ = 0 $(f_{uu}g_{v} - g_{uu}f_{v}) + 2(f_{uv}g_{u} - g_{uv}f_{v})$ = 0 here, $f$ and $g$ ...
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### Questions concerning the differential operator

Consider the differential equation:- $a \phi + (bD^3 - cD)w =0$, where $a, b$ and $c$ are constants, $D$ denotes the differential operator $\dfrac{d}{dx}$, and $w$ is a function of $x$. I'm ...
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### Time-space operator order in Green's Function

When solving the heat equation, and therefore using Green's functions in general, does the operator ordering matter? It seems to me (and this is where I'm confused) that the Green's function ...
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### Question about a (relatively simple looking) differential operator and its eigenvalues

A colleague and I are interested in a specific differential operator on the reals. The differential operator L is of the form $L=-(1+x^{2})\frac{d^{2}}{dx^{2}}+c_{1}x\frac{d}{dx}+c_{2}x^{2}$ for ...
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### Convert an eigenvalue equation to ODE/s

For example define: $K=-i\frac{d}{dx}$ (non-discrete spectrum), so: $$Kf(x)=-i\frac{df}{dx}=kf(x)$$ Define $g(x,k)=kf(x)$, so: $$\frac{-i}{k}\frac{\partial{g}}{\partial{x}}=g(x,k)$$ ...
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### Composition of Differential Operators

If I have: $A=\partial_x^2+u(x)$ $B=u(x)\partial_x$ How do I compose: $AB$ and $BA$?
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### Differential operators: elliptic vs strongly elliptic

This morning a collegue of mine came to me with the following question: does there exist any elliptic operator of order $2m$ with real (variable) coefficients that is not strongly elliptic? After ...
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### Am I allowed to move around an operator like this?

Can I take this product: $$\frac{dL}{dt}\frac{d L}{d \dot{x}}$$ And factor out one of the $L$'s to get: $$L\frac{d}{dt} \left( \frac{d L}{d \dot{x}}\right)$$ Where the operator $\frac{d}{dt}$ now ...
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### Choosing boundary conditions for $(\frac{-d^2}{dx^2})^m$ on $H^m((0,1))$?

Consider the differential operator $D:$ $$Du:=\frac{-d^2}{dx^2}u$$ on the function space $$C=\{u\in C^2([0,1]):u(0)=u(1)=0\}.$$ It's not hard to find the eigenvalues and ...
Let $f$ be a real polynomial of two variables. Let $\partial_f=f\left(\frac{\partial}{\partial x},\frac{\partial}{\partial y}\right)$. Let $H$ denote the space of harmonic polynomials, i.e., ...