Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [differential-operators]

In mathematics, a differential operator is an operator defined as a function of the differentiation operator.

In mathematics, a differential operator is an operator defined as a function of the differentiation operator. Reference: Wikipedia.

It is helpful to consider differentiation as an abstract operation that accepts a function and returns another function (in the style of a higher-order function in computer science).

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What does it mean for the leading symbol of a differential operator to be scalar?

I would like to better understand what it means for the leading symbol of a differential operator to be scalar. Concretely, I am currently looking at the Laplace - Beltrami operator on an n-dimensional Riemannian manifold $(M,g)$.…
harlekin
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Differential Operator squared

If I have the differential operator $$L = \dfrac{d}{dx} + c v(x)$$ what will be equal to $L^2$? To $L^2 v(x)?$ What is the meaning of $L^2$? What are my issues to compute it? In my task I have the formula $$L^N v(x) + \sum\limits_{j=1}^N a_j(x)(L^{j…
Andrew
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Exponential differential operator

Consider the operator $D= e^{ax \frac{d}{dx}}$ operating on an infinitely differentiable function $f(x)$. My approach: $$ Df(x)= f(x) + ax \frac{df(x)}{dx} + (ax)^2\frac{d^2f(x)}{dx^2} + \cdots =f(x+ax) $$ But this does not seem to be the answer.…
user354595
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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 given by $L=\frac{d^2}{dx^2}-F'(\varphi(x))$ where…
MeLoco
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Changing variables in a differential operator

Given the following differential operator, i am asked to rewrite it in polar coordinates $\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + 2\frac{\partial^2 f}{\partial x\partial y}$ I know how to get $\frac{\partial…
Fireeyes
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Where do the constants in this formula come from? I do not exactly know what they are.

I am reading about inverse operators and the book is going over this one. After proving some stuff about this operator it finally says if…
Angel Borge
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$\frac{1}{D+1} e^x$

How do I evaluate $$\dfrac{1}{D+1} e^x$$ where $D$ is the differential operator? I have tried using series expansion, but it just doesn't seem right to me: $$\sum^{\infty}_{k=0}(-D)^ke^x$$
Max Wong
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Finding the symbol of an differential operator.

This problem is of the book "An Introduction to pseudo differential operators" by Wong. Find the symbol of each of the following partial differential operators on $\mathbb{R}^2.$ $\frac{\partial^2}{\partial {x_1}^2}+\frac{\partial^2}{\partial…
eraldcoil
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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? $$\frac{d^2x}{dt^2}=-G\int…
almighty_dollar
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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 correct? My attempt: $$AB =…
DOMiguel
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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 is the last one but I am not sure. By the way,…
DOMiguel
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Exponential Operator Representing Solution to Autonomous First Order Differential Equations

I am studying Dominic Edelen's Applied Exterior Calculus Section 1-4 as a start on understanding derivatives in differential geometry. He uses an exponential function containing a derivative operator as a way of "representing" the solution. My…
David Park
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Change of variables in a differential operator.

I would like to know how could I change the coordinates to cilindrical coordinates of the following differential operator. $y\frac{\partial f}{\partial x} + xy^2z^5\frac{\partial f}{\partial y} + x^3yz^2\frac{\partial f }{\partial z}$ EDIT:…
Fireeyes
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D-operator-methods

Solve the following differential equation: $$(D^2-2D+1)y=x^2e^{3x}$$ I found the $C.F.=(c_1+c_2x)e^x$ $$\begin{align} P.I. & =\frac{x^2e^{3x}}{(D^2-2D+1)}\\ & =e^{3x}\frac{x^2}{(D+3)^2-2(D+3)+1}\\ & =e^{3x}\frac{x^2}{D^2+6D+9-2D-6+1}\\ &…
Ankita Pal
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$\exp(a^2\partial_x^2)f(x) = ?$

I can prove that $\exp(a\partial_x)f(x) = f(x+a)$, but what happens for second derivatives? To be more precise, what is the right-hand side of $\exp(a^2\partial_x^2)f(x)$? The above operator has an integral representation $$\exp(a^2\partial_x^2)f(x)…
Jens
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