0
votes
0answers
20 views

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 $g$ ...
1
vote
1answer
88 views

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 ...
0
votes
0answers
30 views

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 ...
5
votes
0answers
70 views

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 ...
1
vote
0answers
42 views

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: $$\frac{-i}{k}\frac{\partial{g}}{\partial{x}}=g(x,k)$$ ...
2
votes
1answer
90 views

Composition of Differential Operators

If I have: $A=\partial_x^2+u(x)$ $B=u(x)\partial_x$ How do I compose: $AB$ and $BA$?
1
vote
1answer
205 views

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 ...
-1
votes
2answers
94 views

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 ...
2
votes
0answers
55 views

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 ...
1
vote
2answers
589 views

Symbol of a (non linear) differential operator

I am interested in knowing whether there is a definition for the symbol of a PDO which is NOT linear. In Wikipedia and in the book I am reading (An Introduction to Partial Differential Equations by ...
8
votes
1answer
207 views

When do Harmonic polynomials constitute the kernel of a differential operator?

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., ...