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Questions tagged [differential-operators]

In mathematics, a differential operator is an operator defined as a function of the differentiation operator.

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How to derive these Lie Series formulas

Relates issues: How to properly apply the Lie Series Exponential of a function times derivative In my old notes about Lie groups and/or operator calculus, I've encountered the following formulas: $$ ...
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2answers
3k views

Applications of Pseudodifferential Operators

I am very interested in just about anything that has to do with PDE's, and inevitably pseudodifferential operators comes up. Its obvious that such a novel way of looking at PDE's would be important, ...
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How to properly apply the Lie Series

I am trying to solve this problem from Symmetry Methods for Differential Equations A Beginner's Guide (Peter E. Hydon). Use the Lie Series $$F(\hat{x},\hat{y})=\sum_{j=0}^{\infty}\frac{\varepsilon^j}{...
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4answers
492 views

Determining the action of the operator $D\left(z, \frac d{dz}\right)$

This question was motivated by a question by Tobias Kienzler and its wonderful answers. I begin as in the linked question... Using the Taylor expansion $$f(z+a) = \sum_{k=0}^\infty \frac{a^k}{k!}\...
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What is the significance of differential operators over other operators in group theory?

When studying the representation of the group $SO(3)$ on the space $L^2(\mathbf{S}^2)$, we have the spherical functions as its basis: $$L^2(\mathbf{S}^2) = \text{span} \left \{ Y^l_m, l \in \mathbf{N}...
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Definition of the principal symbol of a differential operator on a real vector bundle.

I'm trying to understand the construction of the dirac operator on a manifold, but actually I guess that doesn't really matter for the question at stake. I'm interested in understanding a definition ...
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1answer
271 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., ...
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1answer
226 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 ...
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1answer
951 views

Simple proof of Chain Rule through $\frac{\Delta y}{\Delta x} = \frac{dy}{dx}\biggr|_{x=x_1} + k$

In an online lecture (link to Youtube), the professor proves the Chain Rule using the following statement: $$\frac{\Delta y}{\Delta x} = \frac{dy}{dx}\biggr|_{x=x_1} + k$$ $$\Delta y = \frac{dy}{dx}\...
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2answers
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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 ...
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2answers
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What is the meaning of $1/(D+a)$, where $D$ is the derivative operator?

Today I read the answer to this post, in which the poster integrates $x^5e^x$ by making these manipulations with the differential operator $D$: $$\frac1Dx^5e^x=e^x\frac{1}{1+D}x^5=e^x(1-D+D^2+...)x^5$$...
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Logarithm of differential operator

Using the Taylor series expansion we have (for a sufficiently regular function $f$): $$ f(x+a)=\sum_{k=0}^n \frac{f^{(k)}(x)a^k}{k!} $$ So, defining the differential operator $D=\frac{d}{dx}$ and ...
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2answers
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The minus Laplacian operator is positive definite

In a textbook of functional analysis I found this equation derived from Green's first identity $$\int _{ \Omega }^{ }{ u{ \nabla }^{ 2 }ud\tau } =\int _{ \partial \Omega }^{ }{ u\frac { \...
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2answers
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Relationship between divergence operators defined with respect to two different volume forms.

Let us assume that you have a volume form $\mu$ defined on a manifold $\mathcal{M}$. Then you can define the divergence operator with respect to this metric, such that the following relationship holds ...
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Given $g$ find an $f$ which is solution for $L f = g$. How do I do this?

I am learning about Stochastic processes. To characterize uniqueness of solutions to a given Stochastic differential equation, I need to find for each continuous function $g :\Bbb{R}^2_+ \to \Bbb{R}$ ...
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2answers
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Find $a_{n,i,j}$ in the expansion $(x + D)^n = \sum\limits_{i,j} a_{n,i,j} x^i D^j.$

This is problem 47c. in Stanley's Enumerative Combinatorics Vol. 1. Background: Let $D$ be the operator $\frac{d}{dx}$. Part (a) asks to prove $$ (xD)^n = \sum\limits_{k = 0}^n S(n,k)x^k D^k $$ ...
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Besov–Zygmund spaces and the Inverse Function Theorem, is the Inverse Zygmund?

Preliminary Definitions Let $\Omega \subset \mathbb{R}^n$ be open. We define the Zygmund spaces $C^r_{*}(\Omega)$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are ...
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1answer
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Differential Operator Issue

Let us consider the differential operator $H:= x \frac{d}{dx}$ and let us define \begin{equation} \hat{\mathcal{O}_n}= \frac{1}{n!}(H+n)(H+n-1)\cdots(H+2)(H+1). \end{equation} I proved - by induction ...
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2answers
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Transforming the Laplace operator from Polar to Cartesian coordinates

I'm trying to find the error in my logic here. Let's say we are given the Laplace operator in polar coordinates: $$ \frac{\partial^2}{\partial r^2} + \frac{1}{r}\frac{\partial}{\partial r} + \frac{...
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1answer
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Does the left shift operator, as defined here, satisfy an analogue of the product rule, and if so, what?

Define the left shift operator as follows: $$\lambda_x y = \frac{y-(x:=0)y}{x}$$ where $(x:= 0)$ means "replace every copy of $x$ by $0$," or equivalently "evaluation at $0$." For example, $$\...
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1answer
826 views

On the propagation of singularities in PDE

This question might be a little generic, but i wanted to get some idea on the concept of propagation of singularities in PDE. Searching the internet i only found very complicated things about the ...
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1answer
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Definition of Differential operator

Definition 2.2, page 19 Let $M$ be a smooth manifold and $E_i \rightarrow M$ be two smooth vector bundles. A PDO $P:\Gamma (M,E_0) \rightarrow \Gamma(M,E_1)$ of order $k$ is a a linear map which ...
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Is the kernel of the product of two commuting differential operators the sum of the kernels?

Consider the wave equation in one dimension: $$\frac{\partial^2 u}{\partial t^2}-\frac{\partial^2 u}{\partial x^2}=0.$$ The most general solution of this can be written as $F(x-t)+G(x+t)$ for ...
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2answers
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How to prove convergent function imply its derivative equals to zero?

Let $f\colon (0,\infty) \to\Bbb R$ be differentiable and let $A$ and $B$ be real numbers. Prove that if $f(t) \to A$ and $f′(t) \to B$ as $t \to \infty$ then $B = 0$.
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Computing a derivative through Lie series

Consider the $N$-dimensional autonomous system of ODEs $$\dot{x}= f(x),$$ where a locally unique solution $x(t)$, starting from the initial condition $x$, is denoted as $x(t)=\phi(t,x)$. Assume ...
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2answers
285 views

Differentation Operator

having trouble completing the proof for this question Let $D:\mathbb{R}[X] \to \mathbb{R}[X]$ be the differentiation operator $D(f(X))=f'(X) .$ Prove that $e^{tD}(f(X)) = f(X+t)$ for $t \in \mathbb{...
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1answer
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Closure of the Hamilton's operator $(Hf)(x)=\frac{1}{2}f''(x)-V(x)f(x)$ with $C_c^\infty(\mathbb{R}, \mathbb{C})$ domain

Let $V \in C_{b}^{1}(\mathbb{R}, \mathbb{R})$ be a differentiable function bounded with its first derivative and $H$ be a Hamilton's operator such that: $$(Hf)(x)=\frac{1}{2}f''(x)-V(x)f(x)$$ The ...
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Show that the Hilbert transform is a pseudo-differential operator of order 0

I have tried to solve the following exercise but I don't know if I have missed anything. Show that the Hilbert Transform $Hf = \mathcal{F}^{-1}(\operatorname{sgn}(\xi)\hat{f}(\xi))$ is a pseudo-...
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1answer
23 views

About peakon kernel

Do you have any reference which explain following For differential operator $L=(I-\partial^2_x)$ and peakon kernel $Q=\frac{e^{-|x|}}{2}$ $$L^{-1}f(x)=Q*f=\int_\mathbb{R}Q(x-y)f(y)dy$$
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1answer
896 views

Change of variables in a differential operator.

I would like to know how could I change the coordinates to cilindrical coordinates of the following differential operator. $y\frac{\partial f}{\partial x} + xy^2z^5\frac{\partial f}{\partial y} + x^...
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Find eigenfunctions of the integral operator with kernel $\sum\limits_{n=1}^\infty \frac{1}{n^2} \sin((n+1)x)\sin(ny)$

Find the eigenvalue and eigenfunctions of the integral operator $Ku=\int_0^\pi k(x,y)u(y)dy$. $k( x,y) = \sum\limits_{n=1}^\infty \frac{1}{n^2} \sin\big((n+1)x\big)\sin(ny)$. This is how I ...
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Laplacian and isometry group

Consider the Laplacian on $\mathbb R^n$, $$ \Delta=\partial_i^2 $$ I want to prove that the most general differential operator that commutes with rotations and translations is of the form $$ \sum_k ...