# Questions tagged [differential-operators]

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

455 questions
Filter by
Sorted by
Tagged with
3 views

### Scaling as a repeated shift

I was wondering if the following derivation of the scale operator starting from the shift operator is good. Everybody knows that an operator $T_a$ acting on a function $f$ as $$T_af(x)=f(x+a)$$ can ...
24 views

### Normal intersect the $xy$ plane

I am so stuck in this problem any help will be apreciated: show that the point where the normal to the surface $\mathcal S:\:x^2 + y^2 + z^2 = xf(y/x)$ intersects the $xOy$ plane is at the same ...
20 views

### Differential Operator: Solving Simple ODEs

This will likely be an exceedingly simple question, but the text's use of the operator is confusing me. To preface, the operator $D$ is defined as: $$\frac{d^{n} y}{d x^{n}}=D^{n} y$$ and can be ...
28 views

6 views

### Bi-differential operators are necessarily tensor products

When working with deformation quantisations and Moyal products, the notion of bi-differential operators comes up very often. For example, see here: https://en.m.wikipedia.org/wiki/Moyal_product My ...
22 views

### I do not know what this series expansion is.

I am reading my differential equations book and it is going over the differential inverse operator and more specifically this case where $y_p=\frac{1}{D-a_0}(bx^k)$. So then they do these two steps ...
26 views

### Grunwald fractional derivative [duplicate]

I came across this maths statement \begin{align*} f(t+h) &= f(t) + h\frac{d}{dt}f(t) + \frac{h^2}{2!}\frac{d^2}{dt^2}f(t) + \frac{h^3}{3!}\frac{d^3}{dt^3}f(t) + \cdots \\ &= f(t) + hDf(t) + \...
8 views

### Discretization of Differential Operators in Fourier Spectral Method

I am currently implementing a Fourier psuedospectral method to solve some rather complicated nonlinear PDEs where I have operators of the form $[y^3y''']'$ and $[D(x)y']'$ for some real unknown $y(x)$....
31 views

### A bound for a differential operator in Sobolev norms

Let $s$ be an integer and $L$ a periodic linear partial differential operator $L=\{L_{ij}\}$ of order $l$ on $\mathcal{P}$, the space of $2\pi$-periodic functions $R^n\longrightarrow C^m$. The sobolev ...
33 views

### Differential operator as sum of Dirac delta functions

For $x \in [0,1]$, let $$\tilde{G}(x) = \frac{1}{2(e - 1)}(e^x + e^{1-x})$$ And let $G(x)$ be the 1-periodic extension of $\tilde{G}(x)$ to all of $\mathbb{R}$. Show that, in the sense of ...
41 views

36 views

Verify that $y=x^2-6$ is a solution of $$y''+y'-2y=14+2x-2x^2$$ I have tried like this: \begin{align} P.I. &=\frac{14+2x-2x^2}{D^2+D-2}\\ & =\frac{-2(x^2-x-7)}{-2\left(1-\frac{D^2+D}{2}\... 1answer 23 views ### D-operator method Evaluate a particular value of\frac{1}{D^2+4D}\sin 2x$$I know the complementary function(C.F.) & particular integral(P.I.)....but I can't understand what is particular value, please explain 1answer 40 views ### Definition of an elliptic operator with measurable coefficients Let \mathbb{k} be one of \mathbb{R} or \mathbb{C}. Say I'm given an m-th order linear partial differential operator L in the form of a \mathbb{k}-linear operator$$ L = \sum_{\substack{\...
34 views

I am a little confused. Let $g(x)$ is a nontrivial solution of the equation $-y'' + q(x)y = \mu y$. Then define (in one book) the operators $$A = g(d/dx) (1/g) \quad and \quad A^* = -(1/g) (d/dx)g$$ (...
64 views

### Exponential of the differential operator $x \frac{d}{dx}$

I am studying conformal field theory, and I have run into an problem with calculations involving the quantum-mechanical translation and dilatation operators. It actually boils down to an issue about ...
49 views

82 views

### Is $\int \frac{{dx}^2}{dy}$ valid?

It appears to me that $\int \frac{{dx}^2}{dy}$ can be rewritten as $\int \frac{dx}{dy}\cdot{}dx$ which in turn can be rewritten as $\int f^{-1}{^\prime}(y)\cdot{}dx$. (it is assumed that $y=f(x)$ ...
26 views

### Is there a name for the differential operator $\nabla\otimes$

There seems to be quite a few names for differential operators involving $\nabla$: gradient: $\nabla$ divergence: $\nabla\cdot$ curl: $\nabla\times$ laplacian: $\nabla^2$ I was wondering whether ...
103 views

### Example of an additive map $\mathcal{O}_X \to \mathcal{O}_X$ that is not a differential operator on a Scheme $X$.

I'm looking for a $S$-Scheme $X$ (with structural morphism $f$) and an additive (or rather $\newcommand{\O}{\mathcal{O}}f^{-1} O_S$-linear) endomorphism of sheaves $D: \O_X \to O_X$ which is not a ...
Suppose that $f : \mathbb R^6 \rightarrow \mathbb R$ is a function with the following two properties: $f(0) = 0$, and at at any point $c \in \mathbb R^6$ and any increment $h$, \$\Vert Df(c)(h)\Vert \...