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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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Bounding the symbol of a differential operator

Consider a bundle map $P$, between the bundles $\Bbb R^m \times \Bbb R^n \rightarrow \Bbb R^m$ and $\Bbb R^m \times \Bbb R^k \rightarrow \Bbb R^m$. On each fiber above $\xi \in \Bbb R^m$, $P$ acts as ...
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Solution of $\left(D^2+1\right)y=\sin x$ by partial fractions and direct formula

To find the particular integral of the equation $\left(D^2+1\right)y=\sin x $,I converted this equation to its inverse $y=\frac{1}{D^2+1}\sin x$ . The solution to this can be obtained by taking the ...
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PDE Linearity Textbook Problems: Deducing Correct Operators For Determining Linearity

I am doing PDE textbook problems that require me to determine whether an operator is linear or not. In a previous problem, I had $\mathscr{L} u = u_x + u_y + 1$. I determined that the operator is $\...
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Partial Derivatives of y and y'

In this application of the Euler-Lagrange equation, it is said that there is no $y$ in the function $\sqrt{1 + (y')^2}$. I see that the algorithm in progress treats $y'$ as unusually autonomous, as ...
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Showing any function in $L^p(\mathbb{R}^n)$ is a tempered function.

Definition. Let $f$ be a measurable function defined on $\mathbb{R}^n$ such that $\int_{\mathbb{R}}\frac{|f(x)|}{(1+|x|)^N}dx<\infty$ for some positive integer $N$. Then we call $f$ a tempered ...
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Proposicion 4.2. of the book "An introduction to pseudo differential operators by Wong

Hi! Why $$(2\pi)^{-n/2} \int e^{-ix\xi}(D^{\alpha}\varphi)(x)dx=(2\pi)^{-n/2}\int \xi^{\alpha}e^{-ix\xi}\varphi(x)dx?$$ I tried using part integration with $u=e^{-ix\xi}$ and $D^{\alpha}\varphi(x)=dv$...
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Finding the symbol of an differential operator.

This problem is of the book "An Introduction to pseudo differential operators" by Wong. Find the symbol of each of the following partial differential operators on $\mathbb{R}^2.$ $\frac{\partial^2}{\...
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Smoothness of fundamental solution of a hypoelliptic operator.

Let $p$ be a polynomial such that $p(D)$ is a hypoelliptic operator, i.e. if $u$ is a distribution satisfying $p(D)u = 0$ then $u$ is smooth. Here, $D = -i\partial$. Let $E$ be a fundamental solution ...
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High order derivatives on manifold

Suppose I have a Riemannian manifold $M$ and a smooth function $f:M \to \mathbb{R}$. I denote the gradient of $f$ with $\nabla f$. What is the meaning of $$ \nabla^N f$$ with $N \ge 3$ integer? Is it ...
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Raising a differential operator to a power

I've been reading about the shift operator $E=e^{\frac{\mathrm{d}}{\mathrm{d}x}}$, which can be represented as $$e^{\frac{\mathrm{d}}{\mathrm{d}x}} = I + \frac{\mathrm{d}}{\mathrm{d}x} + \frac{1}{2!} ...
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How do I apply the derivative operator $dT$?

I have the following equation for the torque exerted on a turbine blade. The paper that I am reading says, "the corresponding torque is given by..." $$dT = ρV_2ωr^22πrdr$$ My problem is that while ...
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Factoring differential operators

I need help with the proof of this fact: An ordinary differential operator in the variable $x$ can be factored $P = AB$ if and only if $\ker(B) \subset \ker(P)$. The forward direction is obvious, ...
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Question About Derivative Operator's Usage

Very simple question, I didn't understand the way we are using derivative operator(dy/dx) when we want to derive multiple times such as $d^{10}f(x)/dx^{10}$ for a function such as; $$f(x)=x^{12}-4x^{...
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What is the funciton of $dx'^2$?

Let's say same point in two co-ordinate system has the following relation from partial derivatives, $$dx'=\frac{\partial x'}{\partial x} dx + \frac{\partial x'}{\partial y} dy$$ and $$dy'=\frac{\...
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Specialized theory for differential operators structured in a matrix?

I'm wondering if there are any special considerations/ideas/treatments of equations of the following sort, where if $\boldsymbol{x} = (x_1,x_2,x_3)$: $$\begin{bmatrix} \partial_{x_1} & \left(\...
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Exponential differential operator

Consider the operator $D= e^{ax*d/dx} $ operating on an infinitely differentiable function f(x). My approach: $Df(x)= f(x) + ax*df(x)/dx + (ax)^2*d^2f(x)/dx^2 + ... $ $=f(x+ax)$ But this does not ...
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Convergence to the differential operator

This is not an accurate question. I am not so sure about unbounded operators on a Hilbert space. Let $D$ be the differential operator on $L^2(0,1)$. Well, to somewhat, we can extend $D$ to a normal ...
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$\exp(a^2\partial_x^2)f(x) = ?$

I can prove that $\exp(a\partial_x)f(x) = f(x+a)$, but what happens for second derivatives? To be more precise, what is the right-hand side of $\exp(a^2\partial_x^2)f(x)$? The above operator has an ...
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Exponential of the product between $x$ the derivative operator of $x$ acting in a $f(x)$

The question I'm stuck here trying to figure out how to compute and prove, the following operator action in a function: $\exp(\varepsilon x \partial_x) f(x) = f(x \exp(\varepsilon) )$ where $\...
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Large radius limit for a differential operator on a circle

Let $(A_r)_{r\geq0}$ be a one-parameter family of linear operators, with $A_r$ being the (weak) first derivative operator on $L^2(S_r)$, $S_r$ being the one-dimensional circle, with a multiplicative ...
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Graphing Lie transport of a function

I am relatively new to differential geometry. I am studying it from Fecko Textbook on differential geometry. As soon as he introduces the concept of lie derivative,he asks to do exercise 4.2.2 in ...
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Leibniz rule for exterior derivative of wedge product

I'm trying to show $\text{d}(\alpha\wedge\gamma)=\text{d}\alpha\wedge\gamma+(-1)^p\alpha\wedge \text{d}\gamma$ for all $p$-forms $\alpha$ and $\gamma$, $\text{d}$ is exterior derivative. I want to ...
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Vector Fields as Differential Operators

Given a manifold $\mathcal{M}$, the notion of a vector field $\xi$ on $\mathcal{M}$ can be interpreted as a collection of arrows on the manifold. In the book The Road to Reality by Roger Penrose, ...
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Understanding Partial derivative - how they work in this example

Can someone please help me in understanding why the following equation is true? $[(r\frac{\partial}{\partial r})^2+ r\frac{\partial}{\partial r}] = \frac{\partial^2}{\partial r^2} + \frac{2}{r}\frac{\...
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suggest a course on differential operator and Weyl

Can someone suggest me a course in pdf or textbook about differential operator and Weyl algebra. let $ R$ a commutative algebra over a field $k$ let M, N two R-modules. we define the set of ...
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Differential operator commuting with Euclidean transformations

Why is it true that a differential operator $S$ on $\mathbb{R}^n$ commuting with translations and rotations must be of the form $$S = \sum a_j \Delta^j $$ where the $a_j$ are constant coefficients ?
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Projective-invariant differential operator

This question has been cross-posted to MathOverflow. Suppose we want a differential operator $T$ acting on functions $\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that \begin{align*} &T(g) = 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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Proving 2nd ode [closed]

If $x=e^t$, can someone give me proof that $$ \frac{d^2}{dx^2}=\frac{1}{e^{2t}}\left(\frac{d^2}{dt^2}−\frac{d}{dt}\right). $$ Thank you
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Definition of the weight $k$ hyperbolic Laplacian

I saw two different definitions for the weight $k$ non-Euclidean Laplacian. First, in Daniel Bump's book Automorphic Forms and Representations, the following definitions are given for smooth $\mathbb ...
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How to solve linear differential-difference equation?

Given a linear differential-difference equation: $$A_{n+2}+\partial A_{n+1}+\partial^2 A_n=0,$$ where $A$ is a function of $n$ and $x$, and $\partial$ represents the derivative about $x$. How to ...
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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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Exercise about the Sturm-Liouville problems

Under what condition on the constants $c$ and $c'$ are the boundary conditions $$f(b)=cf(a)$$ and $$f'(b)=c'f'(a)$$ self-adjoint for the operator the operator $$\mathcal{L}f=\frac{d}{dx}\left(p_0(x)\...
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Formula for the application of a linear differential operator to the product of exponential and polynomial functions

In the context of linear differential equations, I've stumbled upon the following identity for an arbitrary pair of polynomials $P$ and $Q$ with real or complex coefficients: $$ P\left(\frac{d}{dx}\...
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Asymptotic expansion of heat operator $e^{-\Delta{t}}$ and $e^{-\mathcal{D}t}$ of Dirac operator

For a closed Riemannian manifold $M$ of $n$-dimension, we consider the Laplace-Beltrami operator $\Delta$. It is known that we have an asymptotic expansion for the trace of heat operator $e^{-\Delta{...
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Is this differential operator Hermitian?

The operator is $$\hat{A} = -i \left(x \frac{d}{dx} + \frac{1}{2} \right).$$ Is it true that $$\langle \hat{A} \psi_1(x)|\psi_2(x)\rangle = \langle \psi_1(x)|\hat{A}\psi_2(x)\rangle\ ?$$ Here, $\...
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Generalized linear transport equation

I stumbled upon a transport equation of the form $$u_t(x,t)=u_x(x,t) + u_x(1,t).$$ Since I can write it in the form $u_t(x,t) = Lu(x,t)$ where L is some linear operator I thought that there must be ...
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Calculating Curl of a vector field using properties of $\nabla$.

So i need to find the curl of a vector field $v=(a\cdot r)a\times r$ where $a$ is some constant vector and $r=(x,y,z)$ is the position vector. So i know the curl is given by the cross product $$\...
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Elliptic boundary condition eigenvalue problem

In Chapter 1, section 1.5 of "The Dirac Spectrum" by Nicolas Ginoux, different elliptic boundary conditions for Dirac operators are introduced. On page 24, there is the following theorem Theorem 1....
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A functional recurrence relation with differentiation, closed form

I'm interested in the way of solving the following recurrence relation: $$a_{n+1}=a_n'+a_1 a_n-b_1 b_n \\ b_{n+1}=b_n'+b_1 a_n+a_1 b_n$$ Where all $a_n,b_n$ are functions of $x$, and $'$ means ...
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Method to factorize/diagonalize differential operators with field redefinitions.

Assuming that $h_{ab}$ is some metric perturbation and $h_{<ab>}$ means the traceless part, I have the LHS of the next equation and I want to find a field redefinition $t_{ab}$ as a function of ...
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Does the formula $\frac{1}{D + a} f = \frac{1}{a}\left(f - \frac{1}{D + a} f'\right)$ have a name?

Using integration by parts, we can show that: $$\frac{1}{D + a} f = \frac{1}{a}\left(f - \frac{1}{D + a} f'\right)$$ where $a$ is a real number, $D$ is differentiation, and $D+a$ is the corresponding ...
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Are there good tutorial ressources to train myself on vector field equations

I've been interested for years in the formalism of maxwell equations or general relativity, but i cannot figure out how those equations including differential operators like gradient, divergence, ...
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Determining domain of differential operators in a concrete semigroup problem

Let s>3/2, $H^s=H^s(\mathbb{R})$ be the Sobolev space of order $s$, $B$ be the set of bounded operators from $H^{1/2}$ to itself, $u\in H^s$ and $A(u):=u\partial_x$ an operator. How can I determine ...
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Representation of ordinary differential operators in terms of a given regular operator

I'm trying to understand Lemma 3.2 from p. 355 of the paper R. C. Carlson and K. R. Goodearl, Commutants of Ordinary Differential Operators, Journal of Differenial Equations 35 (1980), 339–365. ...
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Derivative of $\frac{d}{dt} f(\gamma(t))$ with differential operators $\frac{\partial}{\partial z}$ and $\frac{\partial}{\partial \overline{z}}$

Let $f: \mathbb{C} \rightarrow \mathbb{C}$ a $C^1$ function (i.e. real and imaginary part $f_1, f_2$ are continuously differentiable, where $f=f_1 + i \cdot f_2$) and let $\gamma: \mathbb{R} \...
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$\sum_{i=1}^nx_i\frac{\partial}{\partial x_i}$ is invariant under all matrices

Consider the differential operator $D:=\sum_{i=1}^nx_i\frac{\partial}{\partial x_i}$. Then I claim that $D(u\circ A)(x)=D(u)(Ax)$. The proof is as follows: First note that $Du(x)=\langle x, \nabla_x u(...
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Eigenvalues of the Divergence Operator

I am researching the spectrum of eigenvalues for the divergence operator on Riemannian manifolds and how they deform tensor fields. This is mainly motivated by trying to understand dynamical systems ...
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Find Adjoint of Product & Sum of Differential Operator

I am asked to find the adjoint of the following differential operators: \begin{equation} L_{1} = a(x)\frac{\partial}{\partial x}b(x)\frac{\partial}{\partial x}, \end{equation} and \begin{equation} ...
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General formula for Differentiation Operator

I was considering the Operator $$ x\,\frac{\rm d}{{\rm d}x} $$ and applying it $n$ times to an arbitrary function $f(x)$. Is there a general formula for it? I started with the first few \begin{align} ...