# Tagged Questions

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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### 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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### 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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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 ...
### 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 $$(xD)^n = \sum\limits_{k = 0}^n S(n,k)x^k D^k$$ ...
### 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 ...