0
votes
0answers
13 views

Proving that a Sturm-Liouville problem is in the limit-point/-circle case

I would like to understand techniques anybody is able to detail to me on how one may actually prove that a particular Sturm-Liouville (S-L) problem, i.e., of the form \begin{equation} ...
2
votes
2answers
71 views

Intuition behind convolution identity for Laplace transforms

Convolutions, relatively speaking, are fairly straightforward for simple systems (from an applied perspective), but I cannot, at all, find the intuition behind the Laplace identity for convolutions. ...
0
votes
2answers
41 views

Show Uniqueness of Solution for Boundary Value Problem

Let $G \subseteq R^n$ be a simple, connected and bounded region with smooth boundary and let $f : \overline G \to \mathbb R$, $g : \partial G \to \mathbb R$ be continuous. Show that the following ...
2
votes
2answers
104 views

How to reduce this to Sturm-Liouville form?

I have the ODE $$-(1-x^2) \frac{d^2 f(x)}{dx^2} + x \frac{df(x)}{dx}+g(x)f(x)=\lambda f(x)$$ and I want to reduce it to Sturm-liouville form. The problem is that we don't have $2x$ but just $x$. ...
4
votes
1answer
133 views
+50

Uniform continuity of the function $x(t)=e^{tA}x$

Let $A$ be a bounded operator on a Banach space $X$. Consider the exponential function $x(t)=e^{tA}x:=\sum_{n=0}^{+\infty}\dfrac{t^nA^n}{n!}x$, for all $t\in \mathbb{R}$, where $x\in X$. If the ...
1
vote
1answer
42 views

Find the Green's function for an ODE

$$ Lu= (x-2)u''+(1-x)u'+u , \, u'(0)=(1)=0$$ It can be shown that ${x-1,e^{x}}$ is a fundamental set for $L$ on this interval. $$ g(x,y)= c_1(x-1)+c_2e^{x}, 0\leq x<y , c_3(x-1)+c_4e^{x}, ...
2
votes
1answer
58 views

Show solvability of ODE without explicitly calculating solution

Show that $$ u + u^{(4)} - u^{(2)} = f $$ has a solution $u \in H^4(\mathbb R)$ (without explicitly calculuting it) for every $f \in L^2(\mathbb R)$! What criteria for solvability for such ODE's ...
3
votes
3answers
47 views

Understanding the Euler operator

While reading this book I came across a differential equation $$t^5\frac{d^2y}{dt^2}+2t^4\frac{dy}{dt}-y=0$$ that was then rewritten in terms of the Euler operator, $\delta=t\frac{d}{dt}$, with the ...
1
vote
1answer
83 views

Mathieu differential equation

Given the operator $T (\psi)(x):= \psi''(x)-2q \cos(2x)\psi(x)$ with $T : D(T) \subset L^2[0,2\pi] $ I was wondering: What is the right domain $D(T)$ for this operator if we want to solve the ...
3
votes
0answers
73 views

Solving 1D telegrapher's equation by reduction to two-dimensional wave equation

The solution $w : \mathbb R \times \mathbb R_{+} \to \mathbb R$ of the Cauchy problem for the telegrapher's equation $$ w_{tt} - c^2 w_{xx} + c^2 \lambda^2 w = 0 $$ with $$ w(x,0) = 0, \qquad ...
0
votes
0answers
37 views

Green function of Sturm liouville problem

How to find the Green function of the following problem: $$\begin{cases}-(p(t)u')'+q(t)u=f(t,u), t>0\\u(0)=u(+\infty)=0\end{cases}$$ where $\displaystyle\frac{1}{p},\frac{1}{q}\in ...
2
votes
3answers
57 views

2D Heat Equation with special initial condition

I want to solve the 2 dimensional heat equation on a square $\Omega = \{ (x,y) : 0 < x < \pi, 0 < y < 2\pi \}$ with the Fourier Method \begin{align*} \partial_t u - \Delta u & = 0 ...
1
vote
1answer
49 views

An application of Banach fixed point theorem for initial value problem

Find a condition for $\beta>0$ which implies that the differential equation system \begin{align} x'(t)&=x(t)+y(t) ,\\ y'(t)&=t^{2}+tx(t) \end{align} with initial conditions $x(0)=0, ...
0
votes
0answers
24 views

Can Duhamel's Formula be applied Functional Equations?

Consider an n'th order linear Differential Equation of the form $$y^{[n]} = a_0(x) + a_1(x)y + a_2(x)y' + a_3(x)y'' \ ... \ + a_{n-1}(x)y^{[n-1]}$$ It is well known that to solve this we can create ...
2
votes
0answers
119 views

Solving an infinite non autonomous system of differential equations.

For all $\lambda\in\mathbb{R}$, let $J(\lambda)$ be the infinite matrix where $(J(\lambda))_{nn}=\lambda$, $(J(\lambda))_{n,n+1}=1$ for all $n\in\mathbb{N}$, and all other entries are $0$. This matrix ...
3
votes
1answer
35 views

Similarity transformation of a linear operator

I've seen in some books that given a differential operator $$\frac{d}{dx}$$ under a similarity transformation we get $$\frac{d}{dx}\rightarrow ...
0
votes
2answers
29 views

Inquiry about operator algebra

I've just began studying some quantum mechanics, and I'm not sure why certain rules in operator algebra are correct. For instance, in this book it is stated that ...
1
vote
1answer
33 views

Sturm Liouville problem with additional term.

Imagine you want to solve an ODE on $[a,b] \subset \mathbb{R}$ $f''(x) + (A(x) + B(x))f(x) = \lambda_n f(x)$, where $A,B$ are some smooth functions and $\lambda_n$ the n-th eigenvalue. Furthermore, ...
4
votes
0answers
98 views

Operators such that $\langle Ax,x \rangle=-\langle x,Ax \rangle$

Let $X$ be a Banach space. We consider the differential equation: $$x'(t)=Ax(t), \ \ \ t\in\mathbb{R}$$ where $A$ is a bounded operator on $X$. If $X$ is a Hilbert space, and $x(t)$ is a solution of ...
0
votes
1answer
45 views

Positive operators in Hilbert spaces

Let $H$ be a Hilbert space. I am just asking if there's some reference which studies operators $A$ with this property: $$\left\langle Ax,x\right\rangle \geq0,$$ for all $x\in H$. And $Ax=0$ whenever ...
1
vote
1answer
18 views

Proof that solution of $\lambda$-affine, linear ODE is entire in $\lambda$

Suppose $F(\lambda)~(\lambda\in\mathbb{C})$ is a linear ordinary differential operator (with, say, domain $D$ dense in some Hilbert space), and is also affine-linear in $\lambda$. Is there a proof ...
1
vote
1answer
44 views

Optimal String Shape Problem

So here is the problem I am working on, Given a curve of length L connecting the points (0,1) and (1,0) find an expression for the equation of the curve that minimizes the area underneath it. In ...
2
votes
1answer
37 views

A soft question on the dimension of normed spaces

There's some properties such that if satisfied by a normed space, then necessarily this normed space is finite dimensional. An example is of course the compactness of closed bounded sets. Another ...
3
votes
2answers
172 views

Banach fixed point theorem and Picard-Lindelöf applied to this equation (explanation needed)

Consider the following equation which holds for all $w$ in some space, $$\langle v(t), w \rangle = \langle v(0), w \rangle - \int_0^t \langle F(s,v(s)), w \rangle$$ where $\langle F(s,v),w \rangle$ is ...
1
vote
1answer
120 views

Solving Volterra integral equation of first kind with a Gaussian diffusive evolution kernel

I am trying to solve following Voltera integral equation for $P(t|t')$ numerically: $$ \rho(1,t|0,t') = \int_{t'}^{t} dt'' \rho(1,t|1,t'') P(t''|t') $$ where $$ \rho(x,t|x',t') = ...
1
vote
1answer
53 views

Does bounded and continuous implies Lipschitz?

If a function $f : \mathbb{R} \rightarrow \mathbb{R}$ is integrable, bounded and continuous, is it also Lipschitz continuous?
0
votes
1answer
22 views

Increasing convex-like function in Hilbert space

I am intersted with the differential equation $$x'(t)=f(t,x(t)),\ t\in \mathbb{R}.$$ Can we find an example of a Hilbert space $H$ and a function $f:\mathbb{R}\times H \to H$ which satisfy the two ...
0
votes
0answers
24 views

ODE in case of discontinuity

Consider a function $\varphi(x) : \mathbb{R} \rightarrow \mathbb{R}$ bounded and continuous and $c \in \mathbb{R}$. Question 1 Is there a unique $f : \mathbb{R} \rightarrow \mathbb{R}$, among all ...
0
votes
0answers
37 views

Impulsive Boundary value problems

I have this paper They consider this impulsive problem i dont understand this : Proof. First, suppose that $x\in E\cap C^2[J',R]$ is a solution of problem $(1.5)$. It is easy to see by ...
1
vote
1answer
44 views

Gronwall inequality for $\frac{d}{dt}u(t) \leq C_1u(t) + C_2\sqrt{u(t)}$

I have the inequality $$\frac{d}{dt}u(t) \leq C_1u(t) + C_2\sqrt{u(t)}$$ for a positive function $u$. Is there a Gronwall inequality that I can use to write $$u(t) \leq C_3u(0)?$$ or something ...
1
vote
0answers
25 views

Question about Green function

how to find the Green function of this problem : $$ \begin{cases} -(p(t)u'(t))'=\lambda f(t,u(t)) ~ \text{a.e.} ~t>0\\ u(0)=u(+\infty)=0 \end{cases} $$ Thank you
0
votes
1answer
32 views

Is the particle in a ring a regular Sturm-Liouville problem?

The problem of a particle in a ring is a well-known eigenvalue problem $$\frac{d^2}{d\theta^2} \psi(\theta) + V_0 \psi(\theta) = \lambda \psi(\theta)$$in physics and the Schrödinger equation has a ...
2
votes
1answer
103 views

Properties of this Schrödinger equation / Sturm-Liouville problem.

Given the ODE $$\Psi''(\theta) + \eta \cos(\theta) \Psi(\theta) + \xi \cos^2(\theta) \Psi(\theta)= \lambda \Psi(\theta),$$ where $\theta \in [-\pi,\pi]$, $\Psi(-\pi)= \Psi(\pi)$ and $||\Psi ||_{L^2} = ...
0
votes
0answers
25 views

Second Level Operators:

What would be an example of an Operator $$H$$ such that for any and all explicit functions U $$H[u] = I$$ where I is some other function However, for some other Operator W ex: [d/dx] ...
0
votes
3answers
56 views

Matrix norm in Banach space

How can I calculate the following matrix norm in a Banach Space: $$ A=\begin{pmatrix} 5 & -2 \\ 1 & -1 \\ \end{pmatrix} ?$$ I have tried ...
0
votes
1answer
63 views

Theorem with an example

I have this theorem In the paper they give an example: But here $H_1$ is not satisfied ! How to correct it please? http://mathoverflow.net/questions/163788/theorem-with-an-example
4
votes
1answer
79 views

First eigenvalue of Laplacian and Poincaré inequality

Any idea on how to solve: $\int_{\Omega} |\nabla u|^2 d^n x=\lambda_1\int_{\Omega}u^2 d^n x$, with $u\in H^1_0(\Omega)$ and $\Omega\subset\mathbb{R}^n$, and $\lambda_1$ the first eigenvalue of the ...
0
votes
1answer
19 views

The index of some perturbation about elliptic operator with Robin boundary condition

Let $I$ be an closed interval $[0, 1]$. $C^{2}(\bar{I})$ is the space of all $C^{2}$ functions on $(0, 1)$ with continuity at boundary and usual maximal norm. $C(\bar{I})$ is the space of all ...
1
vote
0answers
28 views

The uniqueness of a solution of a system of differential equations

Suppose I have a system of delay differential equations $\frac{d G_1(\phi_1(s))}{ds}= F_1(\phi_1(s),\phi_2(1-s),1-s)$ $\frac{d G_2(\phi_2(1-s))}{ds}= F_2(\phi_1(s),\phi_2(1-s),s)$ where $G_1, G_2, ...
0
votes
0answers
26 views

proof of existence theorem

Let the equation $$-\mathrm{div}(A(x,u)\nabla u) + a_0(x)u=f(x,u,\nabla u)$$ Let $\Omega$ an open bounded of $\mathbb{R}^n$, and $A(x,)$ an patrix defined by $$\forall u \quad \mbox{fixed} \quad, ...
1
vote
1answer
55 views

differential system

We consider the differential system $$ \begin{cases} & y'(t)=a y(t)^3 + b z(t)\\ & z'(t)=c z(t)^5 - b y(t) \end{cases} $$ with $t>0$ $y(0)=y_0, z(0)=z_0,\quad a<0,\quad c<0,\quad ...
0
votes
0answers
62 views

Help with Gronwall's Inequality

I'm trying to solve an extra credit hw problem that has to do with Gronwall's Inequality. I understand (or think I do) how to find an upper bound when $p=0$, but I'm unsure of how to handle the extra ...
0
votes
1answer
31 views

Similar outputs from different transfer functions

I have two rational transfer functions with the same denominator: $$ H_{0}(s) = N_{0}(s)/D(s),\,\,H_{1}(s) = N_{1}(s)/D(s)$$ I would like for the two outputs from the system, ...
0
votes
1answer
27 views

Weak convergence of the 4-th degree of a weak convergent sequence

Good day! We solve an optimal control problem $$ J(u) = \|y - y_d\|^2 \to \inf $$ where $y$ is a solution of the PDE $$ \frac{dy}{dt} + Ay = Bu. $$ $A$ is a nonlinear operator, $(Bu, v) = ...
4
votes
1answer
89 views

Zeros/poles at Laplace and at Fourier Transform

I recently started "relearning" the Laplace transform, and I noticed something. It seems to me that the intuitive idea of poles and zeros is different between these two transforms! For example, in ...
1
vote
1answer
22 views

Differential equations and surjectivity of some linear operators

Let $a_0,a_1,...a_{n-1}$ be some continuous functions $[0,1]\longrightarrow \mathbb{R}$. Consider a linear operator $D:C^n[0,1]\longrightarrow C[0,1]$ which maps each $y\in C^n[0,1]$ to ...
1
vote
1answer
36 views

Does $C([0,T];H^{s+1}) \cap C^1([0,T];H^{s})=C^1([0,T];H^{s+1})$?

I have asked this question with other problems, but about this part nobody answers. So I want to ask again, and put some details in it. My question is whether the following equality is correct. If it ...
0
votes
1answer
25 views

Are there methods to determine the stability of functional differential equations?

For nonlinear differential equations there are methods to determine the stability of fixed points and limit cycles using a phase plane analysis. If I have a functional differential equation with two ...
0
votes
2answers
39 views

Finding function given its Jacobian and the initial condition

Consider continuously differentiable function $f:\mathbb{R}^k\mapsto \mathbb{R}^k$. We know that $f(x_0)=y_0$ and the Jacobian matrix is given for all $x$. I'd like to know the explicit for of the ...
0
votes
0answers
51 views

Studiy of a differential operator

Let $V=W^{1,p}_0(\Omega)$ and this dual space $V'=W^{-1,p'}(\Omega)$ with $p'$ the conjugate of $p$. Let $A(u)=-\mathrm{div}(|\nabla u|^{p-2} \cdot \nabla u)$. How we can prove that $A$ is monotone? ...