1
vote
1answer
61 views

Find distribution solving a differential equation

I think I have solved the following differential equation, but I am not sure of all steps are justified. Exercise: Find all distributions $u \in \mathcal{D}'(\mathbb{R})$ such that $x(u' -u) = ...
4
votes
2answers
103 views

Uniqueness for tempered distributional Cauchy problems

Question. Assume that $U\in C^1(\,[0, \infty)\to \mathcal{S}'(\mathbb{R}^n)\,)$ is a solution to the following tempered distributional Cauchy problem $$\tag{CP}\begin{cases} \frac{ d U}{dt} = f ...
2
votes
2answers
59 views

$xT' = 1$ in $D'(\mathbb{R})$

I need help solving the following problem: I want to show that all solutions of $$xT' = 1\ , T \in D'(\mathbb{R})$$ take the following form: $c_{1} + c_{2}1_{[0, \infty)} + ln|.|$ What I tried so far ...
1
vote
1answer
103 views

Fundamental solution of nonlinear PDE

A fundamental solution of a linear PDE (in sense of Schwartz), $Lu=0$ is defined as a distribution $E$ such that $LE=\delta$. Now I wish to find fundamental solution of nonlinear PDE, such as the ...
3
votes
2answers
81 views

Constructing a Distributional Solution to the Inhomogeneous C.R. Equations

The question is to find a fundamental solution to the system of equations in $\mathbb{R}^{2}$ \begin{array}{l} u_{x}-v_{y}=f\\ u_{y}+v_{x}=g\end{array} and to express the answer as a $2\times2$ ...
1
vote
0answers
42 views

Neumann boundary condition for smooth function defined on the interior

Let $\Omega\subset\mathbb{R}^n$ be open and let $f\in C^\infty(\Omega)$ be a smooth function. What examples can one come up with that distinguish the 3 criteria below? 1: f satisfies the Neumann ...
3
votes
1answer
191 views

Exercises about Distributions

I'm looking for references (books or pdf) about the following themes (especially the first two) : Fourier Series of Distributions. Distributional solutions of ordinary differential equations. ...
3
votes
1answer
58 views

Schwartz kernel theorem in the case the distributions are induced by smooth functions..

how can I show that if $A:C^\infty(\mathbb T^n)\rightarrow C^\infty(\mathbb T^n)$ is a continuous linear operators then there is a unique linear and continuous operator $K_A: C^\infty(\mathbb ...
0
votes
1answer
114 views

meaning of fundamental solution

i would like to understand what is a mathematical,even physical meaning of fundamental solution,let us consider following problem from Wikipedia $Lf=sin(x)$ where $L$ is operator of second ...
8
votes
2answers
177 views

A Differential operator.

What are the fundamental solutions for the operator $$\mathcal D=a{\partial^2\over\partial x_1^2}+b{\partial^2\over\partial x_2^2}$$ on $\Bbb R^2 $ with standard cordinates $(x_1,x_2)$. Here ...
3
votes
1answer
80 views

Differential equation on $\Bbb R$

We have a differential equation on $\Bbb R$ of the form $$\frac {d^2}{dx^2}u = \chi_{[0,1]},$$ where $\chi_{[0,1]}$ is the characteristic function of the interval $[0, 1] ⊂ \Bbb R$. I want to find a ...
6
votes
1answer
99 views

Uniqueness of Ordinary Differential Equations in $D^{'}$, the space of Schwartz distribuitions

Let $m \in \mathbb{N}$. For $k=1,...,m$ let $a_k : \mathbb{R} \rightarrow \mathbb{C}$ be a $C^{\infty}$ function. And suppose that: $a_m(x) \neq 0 \; \forall x \in [x_0, \infty[$ And let P be the ...
4
votes
1answer
314 views

Solving distributional differential equation

How to solve differential equation in $\mathcal D'(R)$: $$u''+u=\delta'(x),$$ where $\delta$ is Dirac Delta function? Solution of homogeneous problem is $C_1\cos{x}+C_2\sin{x}$, so using the ...
1
vote
1answer
117 views

Differential equation - distributions

How to find solution to the following problem (in $D'(R)$): $$u''+3 u=1+\delta (x)\text{ ?}$$ Thanks in advance.
2
votes
0answers
140 views

Solving $ T' = 0 $ for distributions in $\mathbb{R}^n$

Denoting $ T \in \mathcal{D}'(\mathbb{R}^n) $ as distributions with $ T_f(\varphi) = \int_{\mathbb{R}^n} f\varphi\ dx $, I wish to prove the distribution solution of the equation $ T' = 0 $ ...
1
vote
1answer
42 views

$L$ elliptic diff op $\implies$ singsupp$(u)\subseteq$singsupp$(Lu)$ for distributions $u$?

If $L:D'(\mathbb{R}^n)\to D'(\mathbb{R}^n),n\in\mathbb{N}$ is a weakly elliptic, linear differential operator with constant coefficients then for every $\Omega\subseteq\mathbb{R}^n$, and for all $u\in ...
4
votes
1answer
90 views

Distinction between “measure differential equations” and “differential equations in distributions”?

Is there a universally recognized term for ODEs considered in the sense of distributions used to describe impulsive/discontinuous processes? I noticed that some authors call such ODEs "measure ...
1
vote
1answer
240 views

differential equation with distributions

I'm stuggeling with this differential equation: $T'+T=0$ Where $T$ is distribution. I found solutions in form: $\sum_{n\in A} \frac{d^n}{dx^n}\Lambda_{c_n e^{-x}}$. This can be simplified to ...
4
votes
1answer
116 views

Distributional differential equation, somehow related to compact support distributions

I've been mulling over a problem from Friedlander's Introduction to Distribution Theory for a few days now: in Chapter 3 (on distributions with compact support), it asks to solve the differential ...
2
votes
2answers
95 views

particular solution by variation of constants

I have this ODE : $y^{''}(x)-Ay(x)=Bx \delta_{0}(x)$ where $A,B$ are constants and $\begin{equation} \delta_{0}= \begin{cases} \infty & \text{if $x=0$}, \\ 0& \text{else}. \end{cases} ...
2
votes
1answer
62 views

Non-regularity of non-elliptic operator

Let $\Omega\subset\mathbb{R}^d$ be open,and $P(D)=\operatorname{\sum_{|\alpha|\le N}}f_{\alpha}D^{\alpha}$ be an elliptic differential operator. Rudin proves in Functional Analysis Part II the ...
1
vote
1answer
254 views

What is good about homogeneous functions?

Given $r>0$ and $f:\mathbb{R}^n\to \mathbb{R}$, $d_rf$ is the function defined by \begin{equation}d_rf(x_1,x_2,\dots,x_n)=f(rx_1,rx_2,\dots,rx_n)\end{equation} and is called the $r$-dilation of ...
2
votes
1answer
105 views

Differential equation for distribution

Consider a distribution $T \in D'(\mathbb{R})$ such as (E) : $T' + gT = 0$ with $g \in D(\mathbb{R})$. Could you prove that $T$ is a strong solution of (E) ? I know that we must use the ...
5
votes
1answer
298 views

Laplace transform and differentiation

Let $F(s)$ be the Laplace transform of $f(t)$: $$F\left(s\right)=\int_{0}^{\infty}e^{-st}f\left(t\right)dt$$ It then follows that $f(t)$ can be recovered from $F(s)$ by the inverse Laplace ...