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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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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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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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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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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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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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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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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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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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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2answers
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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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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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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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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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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 ...
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Is there any significance to this matrix/operator?

I am working on a problem involving the the polarized Hessian covariant in Cartesian coordinates on $\mathbb{R}^2$ $[a,b] = \frac{1}{2} \frac{\partial ^2 a}{\partial x ^2} \frac{\partial ^2 ...
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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 ...
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right definition of correct space of domain and range for a self-adjoint Operator

at first, I'm quite sure it's not necessary to pay too much attention to the way the Operator is defined, it's rather important which spaces to choose to obtain a self-adjoint operator I've got a ...
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Question about differential operators

Say $N = ab$. How can I express $\frac{d}{dN}$ in terms of $\frac{d}{da}$ and $\frac{d}{db}$?
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del operator question regarding commutativity with a scalar

Is the following true if $f$ is a scalar? $f\,(\nabla\circ\textbf{B})=f\frac{\partial B}{\partial x}+f\frac{\partial B}{\partial y}+f\frac{\partial B}{\partial z}=\frac{\partial B}{\partial ...
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Squaring an operator

There is an excercise of squaring an operator in my book of quantun mechanics. The operator is $$\hat{A}=\frac{\mathrm{d}}{\mathrm{d}x}+x$$ And I should compute $\hat{A}^2$. He gives me a result ...
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Lie algebra (su(1,1)) from legendre polynomials; question regarding http://arxiv.org/abs/1205.6353

Apologies if this question is a duplicate. OK, so my question heavily involves the paper http://arxiv.org/abs/1205.6353 which nicely details the Lie algebra su(1,1) coming from the Laguerre $L_n$, ...
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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 = ...
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Continuity domain for momentum operator

The momentum operator in one dimension in quantum mechanics is $P=-i\frac{d}{dx}$ (with $\hbar=1$). Consider it as an operator on $L_2(0,2\pi)$, the space of square-integrable functions on $(0,2\pi)$. ...
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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 ...