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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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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63 views

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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159 views

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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41 views

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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45 views

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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29 views

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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182 views

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 ...
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What is the significance of the integral of the Hessian determinant?

The integral of a function over some region measures the total value of the function in that region: $$T(u)=\int u\thinspace\mathrm{d}V$$ The integral of the squared norm of the gradient of the ...
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Eigenvector and eigenvalue of the differential operator $L(x)=x''+3x'-4x$

This is a follow up question to this one. Just to summarize. I want to find the kernel, eigenvalues and eigenvectors of the differential operator: $$L(x):=x''+3x'-4x$$ In other words I want to find ...
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Continuously differentiable operator

I consider an operator $A:H^1_0\to H^1_0$ defined by $$Au(t)=\int_0^1 G(t,s) f(s,u(s))ds$$ where $$ G(t,s)=\begin{cases} t(1-s), &t\leq s\\s(1-t), &s\leq t.\end{cases}$$ I want to know what ...
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Abstract definition of a differential operator

In Kolar, Michor, and Slovak, a differential operator is said to be a rule transforming sections of a fibred manifold $Y \to M$ into sections of another fibred manifold $Y' \to M'$. Is this is a ...
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Manifold,affine connection,vector field

An affine connection on $M$ is a differential operator, sending smooth vector fields $X$ and $Y$ to a smooth vector field $∇_X Y$ , which satisfies the some conditions.I would like to know the ...
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Proof of positive definiteness

$Lu = -u'' + c u$ where c is some constant The question is when it's positive definite in square integrable on $[0; 1]$ with $u(0)=u(1)=0$ $(Lu, u) = \int^1_0 u Lu dx = -u u''+c u^2 dx = \int^1_0 ...
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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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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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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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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 ...