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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How to calculate the derivative of logarithm of a matrix?

Given a square matrix $M$, we know the exponential of $M$ is $$\exp(M)=\sum_{n=0}^\infty{\frac{M^n}{n!}}$$ and the logarithm is $$\log(M)=-\sum_{k=1}^\infty\frac{(I-M)^k}{k}$$ The derivative of ...
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Link to question regarding treating differential operator as a ratio [duplicate]

I have attempted to find the post which provides an explanation as to the circumstances in which we can treat $\frac{dy}{dx}$ as a ratio which appears to be used in solving separable DE's, but I have ...
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Solving the Sturm-Liouville problem using Green's function and Spectral Theorem.

I am reading a paper that deals with the solution of the Sturm-Liouville problem: $u''(t) + \rho (t) u + \lambda ^{-1}u= -f $ $ u(0)=u(1)=0$ For $\rho(t) \leq 0 $. First it is solved the ...
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Elliptic operators for a Laplacian transform

I need to show that the operator: $L[u]=(1-x^2)\frac{\partial^2 u}{\partial x^2}+2xy\frac{\partial^2 u}{\partial x \partial y}+(1-y^2)\frac{\partial^2 u}{\partial y^2}$ Find the transformation of ...
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Sturm-Liouville operator, basis functions.

When dealing with a Sturm-Liouville operator: $$\hat A=\frac{1}{w(x)}\{\frac{d}{dx}(p(x)\frac{d}{dx})+q(x)\}$$ It is Hermitian if its functions satisfy the boundary conditions: $$[f^*p ...
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Reference for Fractional calculus and Differential Operators

I`ve been struggling with Fractional calculus and differential operators while studying special functions, and got to the conclusion that I need some references for them. So I ask for as many ...
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28 views

Fokker-Planck derivation. Path integral?

I am trying to understand the development of Fokker-Planck equation as is described here. Unfortunately, I cannot understand how the first equation on page 4, \begin{multline} \frac{1}{2} ...
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Can anyone prove this equation? (Eq. with operators)

I am trying to understand the last equation from page 2 of this pdf http://physics.gu.se/~frtbm/joomla/media/mydocs/LennartSjogren/kap7.pdf but I am not being able to develop as here it says. Could ...
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What are the eigenfunctions of the D'Alembert operator on pseudo-Riemannian manifolds?

Consider the operator $\Box=g^{\mu\nu}\nabla_\mu\nabla_\nu$ acting on a function space $\mathbf{F}(M)$, given by the set of functions $\phi:M\to\mathbb{R}$ whose values go to zero at infinity (at the ...
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Doubts related to Differential Operator of Infinite order

Let $$f(s)=\sum_0^\infty c_vs^v$$ be some entire function. We say that the differential operator $f(d/dx)=\sum_0^\infty c_vd^v/dx^v$ is defined in some fundamental space $\varPhi$, if for any $\varphi ...
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Finding an matrix for an operator.

I was attempting to find a matrix for the function $x\frac{d}{dx}$ in the span of the set $\{1,x,x^2\}$ for the the standard dot product. Could someone guide me in how to do this?
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How to find the differential with respect to the supremum norm and $L^1$ norm

I'm given a function $F:C([0,1])\rightarrow C([0,1])$, $F(f)=f^2$ (where $C([0,1])$ is given the supremum norm) and I want to find $D_g F(f)$ for any $f,g\in C([0,1])$. I find that $D_g F(f)= 2fg$, ...
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41 views

Closure of a differential operator

Consider $A:\mathcal{D}(A)\subset L^{2}[0,1]\to L^{2}[0,1]$ given by $$A:=-\frac{d^{2}}{dx^{2}},\qquad\mathcal{D}(A):=C^{2}_{0}(0,1)$$ Now, I assume that the closure of $A$ is its extension defined ...
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Kronecker delta representation of a matrix (Quantum raising / lowering operators)

The Kronecker Delta is commonly used to represent a diagonal matrix: $$ a_i \delta_{ij}=\left( \begin{array}{ccc} a_1 & 0 & 0\\ 0 & a_2 & 0\\ 0 & 0 & a_3 \end{array}\right) $$ ...
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Proving the principle symbol is globally defined

I want to prove the principle symbol is globally defined as an element \begin{align} \Gamma(T^* M / \{0\}, \textrm{Hom}(\pi^*E, \pi^* F) ) \end{align} To more specify, let me explain the definition ...
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24 views

Linear Differential Operator

Let $\phi({D})$ be a differential operator defined by $$D^n+a_{n-1}D^{n-1}+a_{n-2}D^{n-2}+...+a_0=\sum_{j=0}^{n}a_jD^j$$ with constant coefficients $a_j$ (setting $a_n=1$) and the derivative ...
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k-symbol differential opeartor L, and its independent of choices.

This material is in O. Well's Differential analysis on complex manifold, page 115. Let, $(x,v) \in T'(X)$ ($T^*(X)$ with deleted zero section) and $e \in E_x$, Find $g\in \epsilon(X)$ and $f\in ...
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$\forall$ $\epsilon > 0$, $\exists \ \delta > 0$, $\|s\|, \|t\| < \delta \implies$ $|f(x_0 + s) - f(x_0 + t) - Df(x_0)(s-t)| < \epsilon \|s - t \|$

$E$ is a normed vector space and $\Omega \subset E$ is open. Let $f: \Omega \to \Bbb R$ be Fréchet differentiable on $\Omega$. Let $Df$ be the derivative map of $f$; $Df: \Omega \to \mathcal L(E, ...
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Different solutions to the Hermite equation

The Hermite differential equation is given as such $$ y'' - 2xy'+2\lambda y=0 $$ writing this in strum-liouville form you get $$-(\exp(-x^2)y')'= 2\exp(-x^2)\lambda y $$ However, in order for it to ...
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Differential Operators and Coefficients

First question on Math StackExchange here. I have been staring at this for a bit, but wasn't quite able to get the hang of it. Here it goes. We are given \begin{align} \frac{\partial}{\partial x} = ...
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Operators in polar coordinates in n-dimensions

I want help on converting differential operators such as the reduced wave operator (L=Δ+c) and the biharmonic operator (L=Δ^2) from Cartesian to spherical coordinates in n-dimensions. For example I ...
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Divergence in Definition of Laplace-Beltrami Operator

I am trying to derive an explicit formula for Laplace-Beltrami operator in global Cartesian coordinates for a special case of plane curve. I have found this article, and I would like to match their ...
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Consider a first-order operator

Consider the first-order linear differential operator L = D + 5 a) do all solutions of lx(t)=0 a subspace of F b) find these solutions c) do all solutions of lx(t) = 1 form a subspace of F I'm no ...
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Trial Solutions to Non-homogeneous Differential Equations

I'm having trouble finding information on something in my Differential Equations & Linear Algebra class. When you're trying to find the general solution to an nth order linear non-homogeneous ...
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The domain of a root of a self-adjoint operator associated with an interpolation space

We now that $V$, $H$ are separable Hilbert spaces such that $V$ is dense in $H$ and $V\hookrightarrow H$ continuous, by representation theorem exists $A: D(A)\subset V\rightarrow H$ self adjoint e ...
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Surface gradient definition

Let $\Omega$ be a bounded domain with $C^2$ connected boundary $\partial\Omega$. For a function $p\in H^1(\partial\Omega)$, we define the surface gradient $\nabla_{\partial\Omega}$ as $$ ...
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Approximating an element in the domain of an unbounded operator by a sequence in a dense subset of the domain.

Let $T$ be a closed unbounded (in my case also symmetric) operator on a Hilbert space $\mathcal{H}$ with dense domain $\mathcal{D}(T)$, and let $f\in \mathcal{D}(T)$. Suppose there is a dense ...
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Classifying a 2nd order linear partial differential operator.

As a follow up to: Fourier Transform of a PDE in 2 spatial variables. I wish to classify the right side of the equation $\partial_t u = \partial_x^2 u + x \partial_y u,$ viewed as an operator. From ...
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Differential operators with arbitrary functions?

By Taylor expansion, one has $$f(x+t) = \sum_{k=0}^∞ \frac{D^k}{k!}f(x)([x+t]-x)^k = \sum_{k=0}^∞ \frac{(Dt)^k}{k!}f(x)$$ and hence one could say $e^{Dt}$ is translation by $t$. But this isn't a ...
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Mean value operator on Riemannian manifold

Let $(M,g)$ a Riemannian manifold. Further $M$ should be a harmonic space, that is $M$ is a symmetric and simply connected space of rank 1. (Example: Spheres $S^n$) Consider the mean value operator, ...
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What is the purpose of representing a (partial) differential equation with linear operators?

One of the first things that is covered in a PDE class (and linear algebra, of course) is the concept of linearity and linear operators, i.e. an operator $L$ such that ...
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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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Integral of the Laplace-Beltrami Operator multiplied by a function

I have the following problem: Let $\mathcal{M}$ be a $2D$-manifold in $\mathbb{R}^3$ and let $g$ denote its metric. Furthermore it is known that $\mathcal{M}$ is a closed manifold (i.e. it has no ...
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What's an example of algebra where differential operators aren't generated by order 1?

For any commutative algebra $A$ over a field $k$, one can define its algebra of differential operators $\def\Diff{\operatorname{Diff}} \Diff_*(A)$, which has a filtration by order. In many cases the ...
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Dot product with Del operator in Cylindrical Coordinates?

It's not hard to derive the equation for the Del operator in cylindrical coordinates from the Del operator in cartesian coordinates. From $$\nabla = \hat{\bf{x}} \frac{\partial}{\partial x} + ...
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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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Sobolev spaces and symmetric operators

I am slightly confused with regards to the way one obtains self - adjoint differential operators in spectral theory. The aspect that I'd like to understand better is the following: Suppose we are ...
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Name of operators somewhat similar to differential operators returning “pace” of functions

I have a set of operators with specific properties, and I believe that somebody must have studied (and baptized) them before. The operators remind me of differential operators, however-as far as I ...
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Fundamental solution of the frozen opearator

Let $L$ be some differential operator of the form $$ Ly = y^{(n)}(x) + a_{n-1}(x) y^{(n-1)}(x) + \dots + a_0(x) y(x) $$ with all $a_k(x)$ being smooth. Let also $M$ be the frozen at $x=0$ operator ...
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Newtonian gravity gradient

Could someone explain to me why the gradient operator in $x$ below "consumes" the square of the norm from the denominator and minus sign? How are the two expressions equivalent? ...
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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 ...
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Why is n-th Fréchet derivative symmetric?

Let $V,W$ be nonzero normed spaces over $\mathbb{K}$. Let $E$ be open in $V$ and $f:E\rightarrow W$ be a twice Fréchet-differentiable function. Then, $D^2 f: E\rightarrow \mathscr{L}_2(V^2,W)$ is ...
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Finding series representation of $\frac{1}{P(D)}$ through ordinary division

I am studying ODEs from ordinary differential equations by Tenenbaum and Pollard. The book in its fifth chapter explains inverse operators for finding the particular solution of a constant coefficient ...
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Show for the Hamilton's operator $H$ that $\overline{(H, C_0^{\infty}(\mathbb{R}))} = (H, W_2^2(\mathbb{R}))$ using Fourier transform

Let $V \in C_{b}^{1}(\mathbb{R}, \mathbb{R})$ be a differentiable real-valued function defined on $\mathbb{R}$ bounded with its first derivative. Consider the Hamilton's operator $H$ such that: ...
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Taylor expansion of $\frac{1}{1-D}$, where $D$ is the differential operator

I understand that we can represent $e^D$ simply as a power series of D. But what about functions of D which are not entire on the complex plane? What if the function has no taylor expansion, or if it ...
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Continuous dependence on initial conditions for second order eigenvalue problem

Consider the Schrödinger eigenvalue problem in one dimension $$\phi'' - V\phi + \mu \phi = 0$$ on $[0,a]$ with boundary $\phi(a) = c$. Suppose that I already have the existence of ...
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Notation of the differential operator

I see the differential operator both with upright and italic d in different books/articles. So I'm curious about $$ \int x^2 \, dx \quad \text{vs.} \quad \int x^2\, \mathrm{d}x,$$ and ...
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What type of self-adjoint operator does $\hat{P}$ has to be for Green's function to result in a radial exponetial $e^{-\| x-t \|^2}$

I was reading the following paper on hyper basis function (HBF) (similar to radial basis function RBF network) and was trying to understand when is it the case that the network has radial basis ...
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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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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 ...