In mathematics, a differential operator is an operator defined as a function of the differentiation operator. (Def: http://en.m.wikipedia.org/wiki/Differential_operator)

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Is this entity with operators correct?

Let define the operators $A = \frac{1}{\sqrt{2}}(x+\partial_x)$ and $B = \frac{1}{\sqrt{2}}(x-\partial_x)$. I am suppossed to check the identity $AB-BA=1$ but I cannot proof it. Is the identity ...
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How to apply this operator?

Let $A$ be the operator $2(x+\partial_x)$. Suppose we have a function $f$ and that we apply the operator to this function. How this operator is applied? $2xf+\partial_xf$ or $2x+\partial_xf$? I guess ...
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Decomposition of a differential operator

Let $\mathcal{O}$ be the ring of holomorphic functions on the unit disk deprived of the non-negative real numbers. Let $\mathcal{D}$ be the ring of differential operators on the same space, $\alpha$ a ...
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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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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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Laplacian Smoothing Irregular Initial Data

Apparently for many parabolic and elliptic PDEs the (ir-)regularity of initial data does not have any significant impact on the regularity of (weak) solutions. Very often people when people talk ...
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Linear Second order Differential operator proof questions

I have 3 proof questions from my book that I have tried and I would like to see if my solutions are valid and/or there is a simpler way to prove them. Firstly, the notation $ker(L)$ means all $f$ ...
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Is the differential operator defined on the natural log of x when no bounds are specified and why or why not?

D_x (f) = d/dx [f]. C[a,b] is the set of all continuous functions on the interval [a,b]. If D_x defines a linear transformation from C'[a,b] to C[a,b], where C'[a,b] is the set of all functions whose ...
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Is this second order differential operator Hermitian (2 variables)?

Let $ T= \frac{ \partial^{2}}{\partial _{x} \partial _{y}}+iay\frac{ \partial}{\partial _{y}}+i(1-a)x \frac{ \partial}{\partial _{x}}+ \frac{i}{2}$ be the second order differential operator, where $ ...
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Linear operator differentiation on a torus

I'm trying to analyze this article about area-preserving diffeomorphisms and don't quite understand a sentence. 4.1. Linear involutions. We start characterizing the linear involutions $R \! : ...
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Metric tensors and linear (differential) operators defined on a manifold and its osculating sphere

Given a 1D Riemannian manifold $\Gamma$ embedded in 2D Euclidean space (e.g. a parametric curve on a plane $\mathbb{R}^{2}$ ), and point $x_{0}\in \Gamma$, we denote $S^{1}(x_{0})$ the circle ...
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Complex differential operators

Consider the differential operators $\dfrac{\partial}{\partial z}$ and $\dfrac{\partial}{\partial \bar{z} }$ defined by $\frac{\partial}{\partial z} = \frac {1}{2} (\frac{\partial}{\partial x} - ...
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Higher Order ODE with Differential Operators

I am trying to solve an ODE problem involving higher order. Let $p(s) = s(s^2-s+1)(s-1)$ and $D = d/dt$. Solve the initial value problem $$p(D)x = t + e^t,$$ $x'''(2) = 1$, $x''(2) = 1$, and ...
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How to show that differential operator can be defined in terms of certain commutator operators

Let $U$ be any open subset of $\mathbb{R}^n$ (or, more general, of some smooth manifold). Define $\mathcal{D}_{-1}(U):=\{0\}$. For any two linear operators $A$ and $B$, the commutator operator $[A,B]$ ...
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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} + ...
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Not understanding why equation 2 and equation 3 has to be multiplied with $v(x)$

The discrete differential operator $L_h$ is defined as: $(L_h v)(x_j)=-\frac{v(x_{j+1})-2v(x_j)+v(x_{j-1})}{\Delta x^2}$ (Equation 1). The contineous problem had solution $v(x_j)=sin(\beta x_j)$. It ...
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Relation between a function and its norm

While reading up on Sturm-Liouville system theory, I came across something I didn't fully understand. At one point, in the midst of proving the existence of solutions to the Sturm-Liouvill problem, ...
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1answer
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Struggling with Differential Operators?

I'm taking a basic linear algebra/Differential Equations class hybrid (weird right?), and we're currently learning about differential operators. Am I correct in saying that a differential operator is ...
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34 views

How can I obtain these differential operators for this transformation?

I have transformation as the following form \begin{eqnarray} \begin{split} &u \longrightarrow \bar{u}=(ax+by+\eta)^{-3} u,\\ &x \longrightarrow \bar{x}=\frac{\alpha x+\beta ...
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Localization of differential operators

Let $R=k[x_1,...,x_n]$ with $k$ a field of characteristic 0. And let $S$ be some multiplicitive set. Or, even, just powers of a monomial. I'm having trouble seeing how to show that ...
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Elliptic and Fredholm partial differential operators

As I learn from the comments to this question a non elliptic operator on a compact manifold can not be a fredholm operator. On the other hand it seems that the following operator is fredholm(At ...
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different between (-(a.b)) and ((-a).b)

One of our teachers said that there is just one example that there is different between $(-(a\cdot b))$ and $((-a)\cdot b)$. He said by using "twice complement," you can find one. I am trying to find ...
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Question on a special Derivative

I have this functional defined from a Hilbert space $H$, $J\colon H\rightarrow \mathbb{R}$ defined by: $$ J(u)=\frac12 \|u\|^2-\int_0^1(A(su),u) ds $$ where $A\colon H\rightarrow H$ is a potential ...
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Linear algebra references explaining matrix form of linear differential and integral operators

Some years ago I was in a lecture where I met for the first time the matrix representation of some differential and integral operators (once discretized). Back then, someone mentioned me that every ...
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Do spectrum and Eigenvalues of $Af=-f''$ concide (under dirichlet boundary conditions)

I am asked to show that for the operator $$ Af = -f'' $$ with $D(A)=\left\{f\in H^2(0,1), f(1)=f(0)=0 \right\} \subset L^2(0,1)$ is self Adjoint in $L^2(0,1)$ (This part is solved). I cannot see ...
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Is it possible to construct a 1-D linear differential operator with given spectrum $0\leq\lambda_0\leq \lambda_1\leq\dots\leq\lambda_n\le\dots$?

Suppose one is given with a sequence $S$ of non-negative real numbers $0\leq\lambda_0\leq \lambda_1\leq\dots\leq \lambda_n\leq\dots$. Under what conditions on $S$, is it possible to construct a Linear ...
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Elliptic differential operator

I am given the differential operator $D(f):=-(fg)'+hf$ and $D^* (f) = g \cdot f' + hf$ where $h,g$ are some smooth functions and want to find out under which conditions, these two operators are ...
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Linearity of a differential equation

The following is the general form of a linear ODE, where $t$ is the independent variable and $y$ is the dependent one: $a_n(t) \frac{d^ny(t)}{dt^n} + a_{n-1}(t) \frac{d^{n-1}y(t)}{dt^{n-1}} + \dots + ...
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Raising and Lowering Through Differentiation

I'm calculating the Christoffel symbols of the second kind which is of course defined as multiplying the symbol of the first kind multiplied by the contravariant metric. I was thinking of how to make ...
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The resolvent of a differentation operator on $C[a,b]$

Consider a densely defined operator $A : C[a,b ]\rightarrow C[a,b ]$, $$Au=u^{\prime}$$ with domain $$D(A)= \{ u\in C^1[a,b]: u(b)=ku(a) \}$$ for some $k>0$. I have to find $R_A(\lambda)$ for ...
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Resolvent operator

Let's consider the following operator on $L^2(\mathbb{R}^3)$ $$A(t)=\Delta+b(t,x)\cdot\nabla$$ where $\Delta$ is the Laplace operator and $b(\cdot,\cdot)$ a smooth vector field. How to compute the ...
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Determine adjoint operator

Let $K\in D\{(x,\xi)\in \mathbb {R}^2 : x > 0, \xi > 0\} $ and $L (\phi)(x)=\int_0^x K (x,\xi)\phi (\xi)d\xi $ for $\phi \in D (\mathbb {R})$ $D $ is the space of testfunctions I know that ...
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Distance between differential operators

Given two differential operators say $D_1$ and $D_2$ is there any meaningful way to define distance between them, does there exist some metric $d(D_1,D_2)$ that satisfies all the necessary properties? ...
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Inverse of a particular operator

I need help finding the inverse of the following operator. I am not sure about how to start. Any help would be hugely appreciated. Operator: $( I + \frac{\partial^2}{\partial x^2})$ Edit: I ...
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References for Second order differential operator

I'd like to study the operator $-\partial_{xx}+\partial_{yy}$ in ${\bf R}^2$. In particular I'm interested in finding a characterization of its domain of (essential) self-adjointness. Does this ...
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Del Dot Expansion

I have an expansion question using the $\nabla$. If I have this equation: $\nabla \dot\ (V \dot \ \nabla V)$. Where $V = x$ and $y$ components of velocity. How does this expand to $(\frac{du}{dx})^2 ...
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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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Selfadjointness of the differential operator in a singular potential

The free Dirac operator is the differential operator of the following form $$ T_0 = i \alpha \nabla + \beta,$$ where $\alpha$ and $\beta$ are Hermitian $4 \times 4$ matrices, and $T_0$ is selfadjoint ...
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Is $L^{*}L$ a real operator?

let $(M,h)$ be a compact complex manifold with a hermitian metric $h$. Let $L$ be a $\mathbb{C}$-linear differential operator with smooth coefficients \begin{equation} ...
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Derivative of a function which is treated as a variable

I have got a function $f=f(x)$. The derivative is $\partial_xf$. There are applications in which it is reasonable to treat $f$ as another variable in a larger context. In my application I now need an ...
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put $-\frac{d^2 u}{d x^2} +\frac{d u}{d x}=\exp (x)$ on $[0,1]$ and BC $B_0u = u(0) = 5$, $B_1u=u_x(1)+u(1)=2$ in bilinear form $a(w,u) = F(w)$

Consider the elliptic problem $Lu=\exp (x)$ on $[0,1]$ with $Lu = -\frac{d^2 u}{d x^2} +\frac{d u}{d x} $ and boundary contidions $B_0u = u(0) = 5$ and $B_1u= \frac{du}{dx}(1)+u(1)=2$. Show that the ...
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What is the mathematical truth behind the Leibniz notation in differentiating twice or more?

So $f: \mathbb{R} \to \mathbb{R}$ is $n>1$ (or more) times differentiable. The notation of the first derivative makes perfect "sense" with regard to what's going on: $$\lim_{h \to 0} ...
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Meaning of (generalized?) differential operator

I am currently reading this paper which makes use of generalized differential operators. As I understood it, the operator $D_x$ works like this: If $F$ is a continuous function on $[a,b]$ and $G$ an ...
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Is there any difference between formally symmetric and formally self-adjoint differential operators?

I work with the well known book of Dunford/Schwartz "Linear Operators (Part II)". At first I should mention that the general difference between self-adjoint and symmetric operators is obvious to me. ...
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Formal definition of “node” with respect to eigenvalues and functional analysis.

I'm concerned with a special problem of spectral analysis for a certain Sturm-Liouville-differential-operator, that is to say $L:=\frac{d^2}{dx^2}-q(x)$ and the spectrum $\sigma(L)$. While reading an ...
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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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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
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Write out the operator (A)^2 for A = (d/dx + x)

I am having difficulties understanding how this operator is multiplied out. I have the answer, but do not know why (see below) it is what it is. Imagine there is a carat (^) above the "A" for correct ...
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Decomposition of Laplacian into tangental and normal components w.r.t. submanifold

If I have the covariant Laplacian operator acting on a tensor e.g. $\nabla^2 h_{\mu\nu}$ on a (pseudo-Riemannian) manifold, and I have (say a codimension-2) submanifold, how can I "decompose" the ...
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self-adjoint differential operator on $C^{0}([a,b])$?

I've got problems in understanding the way a special (self-adjoint) differential operator is acting on the domain and the range. So, I try to explain my difficulties: The differential operator is ...