Questions tagged [differential-equations]

Ordinary or partial differential equations. Delay differential equations, neutral equations, integro-differential equations. Well-posedness, asymptotic behavior, and related questions.

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Regularity of elliptic partial differential equation with mixed Dirichlet-Robin boundary condition, to prove $u\in H^{2}(\Omega)$

I have posted this problem on Math Stackexchange but got no reply. When I deal with the wave equation with dynamical boundary condition, I am confused by the regularity of the following elliptic ...
monotone operator's user avatar
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1 answer
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Finding minimal $\gamma$ that satisfies the integral equation

I have a function $f(z) \in [0,\infty]$ that satisfies $-1 < f(z) < 0$. I would like to find the minimal $\gamma$ that satisfies: $$ \int_0^{\gamma} f(z)dz = \log(1+f(0)).$$ Clearly, I cannot ...
nir's user avatar
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Cauchy problem for convolution operators

I don't know how to solve the following Cauchy problem: $$f'(x)=-x f\ast g(x) \qquad \text{ and }\quad f(0)=1. $$ Could you please help me with this. Thank you in advance!
yassine yassine's user avatar
3 votes
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An attempt to extend polynomial rings

Suppose $\mathbb{K}$ is a field of characteristic $0$. Let $S_n=\mathbb{K}[[x_1,\dots,x_n]]$ be the ring of formal power series in $n$ variables, $L_n$ be its fraction field and $W_n=\mathbb{K}[x_1,\...
Zerox's user avatar
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1 answer
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Integral inequality implies majorization by solution of ODE

Let $f:[0, \infty)\to [0, \infty)$ be non-increasing (and not necessarily differentiable nor continuous) and satisfy $$f(t)\leq f(0)-C\int_{0}^{t}f(s)^{1/2}ds,$$ where $C>0$. How can one show that ...
Shaq155's user avatar
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Symplectic (or alike) integrator for system with Coulomb singularity and time-dependent potentials

I am trying to calculate classical trajectories for a single a ion and a single electron inside an RF trap. Therefore, I am dealing with a two-body system that possesses: Coulomb potential with a ...
michalt's user avatar
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1 answer
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What's a good approximation for the first derivative at the endpoints of given datapoints for a cubic spline interpolation?

I'm using a cubic spline interpolation for given data points. The boundary condition for the spline is that $f'(a)$ and $f'(b)$ are given (I'm using a finite difference formula $\frac{y_1-y_0}{x_1-x_0}...
Gogoman96 X's user avatar
4 votes
3 answers
286 views

Coupled Riccati equations

Is there a general solution (in terms of simple known functions) for the following system of coupled non-linear EDOs ? $$x'(t) = -a_1x^2 -bxy$$ $$y'(t) = -a_2y^2 -bxy,$$ where $a_1$, $a_2$ and $b$ are ...
silmar's user avatar
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Poisson equations for tensors on compact Riemannian manifold

Let $({M},g)$ be a compact Riemannian manifold with Levi-Civita connection $\nabla$. It is well known that the Poisson equation $$\Delta f=S$$ where $\Delta:C^{\infty}({M})\to C^{\infty}({M})$ denotes ...
B.Hueber's user avatar
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Properties of the displacement field, assuming only smooth charge distribution and Gauss's theorem

In physics, the displacement field satisfies Gauss's theorem: $$ \int_{\partial \Omega} {\bf D}\ {\bf n}\operatorname{d\!}S = \int_{\Omega} \rho\operatorname{d\!}V, $$ where $\Omega$ is a bounded ...
MikeTeX's user avatar
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Numerical approaches to functional equations

I'm interested in finding numerical approaches to solving functional equations such as f(xy)=f(x)+f(y), where the equations had no derivatives or integrals, and contains arguments involving x and y . ...
Doug Brunson's user avatar
2 votes
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Differential inequality with convex constraint

The following problem looks to be classical, but I fail to find a reference for it. If you know it, please help me. Given a bounded measurable function $h:{\mathbb R}_+\to\mathbb R$ and a number $a\...
Denis Serre's user avatar
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Periodicity and Burger's equation

Consider the 1-dimensional Burger's equation on a finite interval $I=(0,1)$, $$u_t+uu_x=u_{xx}$$ with initial condition $$u(x,0)=f(x)$$ and boundary conditions $$u(0,t)=A(t) \qquad u(1,t)=B(t).$$ ...
T. Amdeberhan's user avatar
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1 answer
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How to prove $ \|u\|_{L^{\infty}}\leq C\|\partial_1\square u\|_{L^1} $ for any $ u\in C_0^{\infty}(\mathbb{R}^{1+2}) $?

It comes from estimates for wave equations. For any $ u=u(t,x)\in C_{0}^{\infty}(\mathbb{R}^{1+2}) $, which is a smooth compactly supported function, prove that $$ \|u\|_{L^{\infty}(\mathbb{R}^{1+2}\...
Luis Yanka Annalisc's user avatar
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Continuity in the uniform operator topology of a map

I have a question concerning the continuity for $t>0$ in the uniform operator topology $L(X)$ of the following map: $$t\mapsto A^\alpha R(t)$$ where A is the infinitesimal generator of an analytic ...
Jaouad's user avatar
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1 answer
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How to find the maximum value of the following difference equation without using iterative method?

$E(i+1)=(I-AT)E(i)+1/2(AT)^2$ How to find the maximum value of $E$ in this expression without using the iterative method? An approximate estimation is also acceptable. Only the $E$ vector is unknown, ...
chen chen's user avatar
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0 answers
156 views

Solving a nonlinear differential equation

I need to solve the following equation: $$y'(t)+2[\cos y(t)+\Omega(t)]=0,$$ where $$\Omega(t)=-2\eta +\frac{2(\eta^2-1)}{\eta-\cos(4\sqrt{\eta^2-1}t)}$$ with $\eta>1$. Undoubtedly, the differential ...
Young Q's user avatar
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1 answer
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Are there PDEs in which Hessian appears in the weak formulation

Before stating the question, I would like to first use an example for the type of formulation that I'm interested in. Suppose we consider the continuity equation $\partial_t \rho + \mathrm{div}( \rho ...
Kacper Kurowski's user avatar
1 vote
0 answers
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Finding all polynomials that become zero when certain differential operators act on them

Consider some differential operators that do not have $x^n$ type of coefficients, i.e., only as powers of $\partial_x$, $\partial_y$ or sum of a few such terms with constant coefficients. For example, ...
Han Yan's user avatar
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2 votes
1 answer
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Property so that $f(t)\equiv 0$ for all $t\geq T$ for some finite $T>0$?

Let $f:[0, \infty)\to [0, \infty)$ be non-decreasing and satisfy for all $t>t_{0}$, $$f(t)+C\int_{t_{0}}^{t}f^{\gamma}(s)ds\leq \frac{1}{t-t_{0}}\int_{t_{0}}^{t}f(s)ds,$$ where $0<\gamma<1$ ...
Shaq155's user avatar
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Methods for holonomic recurrences

I wanted to ask if anyone knows of good texts/resources on methods for solving holonomic recurrence relations (if there are any general analytical approaches): $$p_1(n)a(n)+p_2(n)a(n-1)+\dotsb+p_k(n)a(...
Doug Brunson's user avatar
2 votes
0 answers
98 views

Strong differentiability and Sobolev function

Assume that $u: B\subset\Bbb R^n\to R$ is two times differentiable almost everywhere in the unit ball $B=B^n$ that is continuous on boundary and that solves the PDE almost everywhere $$L u(x):=\sum_{i,...
Dejv's user avatar
  • 81
2 votes
2 answers
233 views

Domain of Schrödinger operators

Let $S$ be a Schrödinger operator on $\mathbb{R}$, $Su=-u''+Vu$ with $V\geq1$ continuous and going to $+\infty$ at infinity (you can think of it as $x^2+1$). I wondering which assumptions do I have to ...
BlueCharlie's user avatar
2 votes
0 answers
59 views

Higher order energy method for nonlinear damping wave equation(reference request)

When I deal with Energy decay rate estimates of the wave equation$$u_{tt}-\Delta u=0\ in\ \Omega$$ with acoustic boundary conditions$$z_{tt}+\varphi(z_{t})+z-g*z+u_{t}=0\ on\ \Gamma_{1},$$ $$\partial_{...
monotone operator's user avatar
0 votes
1 answer
194 views

Numerical reconstruction of Einstein's field equations

A few analytic solutions are known to the Einstein field equations: $$ R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R - kT_{\mu\nu} = 0$$ Taking a preexisting analytic solution such as Schwarzchild's solution: $$...
James's user avatar
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2 answers
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Uniqueness of a second order linear ode

I am currently confronted with the following equation $$ 0=w''(t)(t^2-t)+w'(t)((2n-1)t^2-n)+w(t)(n-1)^2t $$ for $t\in(-1,1)$. So $w:(-1,1)\rightarrow\mathbb{R}$. The following assumption is also in ...
mhmmm1997's user avatar
1 vote
1 answer
192 views

Solution of nonlinear differential equation $g = c_1 f^2 + c_2 (f')^2$ for function $f$

I would like to find an analytic solution (if possible) of the differential equation: $g = c_1 f^2 + c_2 (f')^2$ Where $c_1$ and $c_2$ are constants, $g$ is a known function of $x$, $f$ is another ...
Alex's user avatar
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1 vote
1 answer
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ODE with conditions within the interval

Can anyone please recommend some publications related to ODEs with non-initial, non-boundary conditions, but conditions for points inside the interval, on which the ODE is defined?
Ivan Matychyn's user avatar
3 votes
1 answer
114 views

Analyticity of central stable manifolds

Let $X$ be a real analytic vector field defined on $\mathbb{R}^n$. Assume the origin $0 \in \mathbb{R}^n$ is a zero of $X$. Assume, furthermore, that we know that the center-stable manifold (in the ...
Paul's user avatar
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1 vote
1 answer
182 views

PDE involving curl

Let $G:\mathbb{R}^3\rightarrow\mathbb{R}^3$ be smooth vector field over $\mathbb{R}^3$. For which vector fields $F:\mathbb{R}^3\rightarrow\mathbb{R}^3$ does the PDE $$\dfrac{\partial}{\partial t}\...
MrPie 's user avatar
  • 185
3 votes
1 answer
190 views

Stability analysis of equilibrium point of non-linear ODE system with Jacobian going to infinity

Let's say I have a nonlinear system of ODEs, where one of equations looks like: $$ \frac{dX_i}{dt} = a_1X_0+\dotsb+a_j\frac{X_j^{0.5}}{X_j^{0.5}+b_j^{0.5}}+\dotsb. $$ And equilibrium point is 0. I ...
Omega's user avatar
  • 31
2 votes
1 answer
107 views

Can I characterize functions (in 2D), which will have compactly supported/support contained Poisson solution?

I have the problem of solving Poisson equation in 2D $$ \Delta u = f $$ Let's say for a moment I want to solve it on $\mathbb{R}^2$, for $f(x,y), x\in \mathbb{R}, y\in \mathbb{R}$. I know however that ...
VojtaK's user avatar
  • 151
2 votes
1 answer
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References for group of invariance of the Painlevé property

I'm searching for some references about groups of invariance of the Painlevé property other than the book of Robert Conte or more generally birational transformation of Riemann surfaces.
Redouane Khaled's user avatar
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0 answers
97 views

Polynomial / quadratic autonomous system of ODEs – proving monotonicity / convexity

Problem: Consider the autonomous ODE system \begin{align*} \dot{x} &= (1-x) (z-xy)\\ \dot{y} &= \tfrac 1 2 y^2 - (a+xy)(1-y) \\ \dot{z} &= \tfrac 1 2 z^2 - \tfrac 1 2 y^2 + (a+xy)z \end{...
Pavel Kocourek's user avatar
3 votes
1 answer
333 views

Existence of solution to linear inhomogeneous first order PDEs systems

Maybe I am asking a triviality. If that is the case, please let me know and I will close the question. I have searched a lot but I didn't find an adequate and forceful response. For $i=1,\ldots, r$, ...
A. J. Pan-Collantes's user avatar
1 vote
0 answers
116 views

The norm of Sobolev space involving the time

Question. Is the following way of writing the norm of a Sobolev space involving the time correct? I would be grateful for any help. Let's assume we have a function $$ \mathbf{u} (\mathbf{x}; t) = \...
Abdulhameed Qahtan Abbood Alta's user avatar
3 votes
1 answer
298 views

Find $f:\mathbb R^3 \to \mathbb R$ s.t. $(f - 1)\Delta f + f^2 = 0$?

I wish to solve the following nonlinear PDE that I derived in statistical physics. (I was curious if I could include higher order terms into a model for heat transfer described in a homework problem.) ...
Talmsmen's user avatar
  • 577
4 votes
1 answer
172 views

Euler operator as part of a cochain complex

I am studying chapter 4 of Olver's "Applications of Lie groups to differential equations", about symmetries in differential equations coming from a variational principle. The Euler operator ...
A. J. Pan-Collantes's user avatar
3 votes
1 answer
123 views

A type of singular limit for systems of differential equations

Suppose I have a system of differential equations for the unknowns $(x_1,v_1,\ldots,x_N,v_N)$ (interpreted as the positions & velocities of $N$ labeled particles), $$\begin{cases}\dot{x}_{i,\...
Matt Rosenzweig's user avatar
5 votes
1 answer
194 views

How to extend this PDE?

Let $(M^n,g)$ and $(N^m,h)$ be Riemann manifolds without boundary of dimension $n$ and $m$ respectively and $u:(M^n,g)\to (N^m,h)$ be a map satisfying the following PDE on $M^n\backslash\Sigma$ ($u$ ...
Tears's user avatar
  • 63
3 votes
2 answers
180 views

Floquet coefficients under time change

Let's consider two ODEs $\tag{1}\label{1}\frac{du}{dt}=\gamma(u(t))\ F(u(t))$ and $\tag{2}\label{2}\frac{dv}{d\tau}=F(v(\tau))$ where $f\in C^\infty(\mathbb R^n,\mathbb R^n)$ and $\gamma\in C^\infty(\...
herve's user avatar
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0 votes
0 answers
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How do we solve this rather simple ODE (Loewner equation with driving function $\sqrt t$)?

Remember the following result for the Loewner equation: If $\lambda:[0,\infty)\to\mathbb R$ is continuous, then for all $z\in\mathbb C\setminus\{\lambda(0)\}$ there is a uniqe $\zeta(z)\in(0,\infty]$ ...
0xbadf00d's user avatar
  • 161
6 votes
1 answer
296 views

If an initial value problem has a solution on $[0,a)$ for all $a>0$, will it have a solution on the whole $[0,\infty)$?

Consider the initial value problem on $[0,\infty)$: $$x'(t)=f(t, x(t)) \qquad x(0)=0,\label{1}\tag{$*$}$$ where $f:(0,\infty)\times\mathbb R\to\mathbb R$ is a continuous function. Assume that for ...
Feng's user avatar
  • 517
3 votes
0 answers
93 views

Mathematical formulation of beam: get stress/strain from forces and momentum

I'm working with static beams with Euler–Bernoulli model which ODE is $$ \dfrac{d^2}{dx^2} \left(EI \cdot \dfrac{d^2w}{dx^2}\right) = q(x). $$ With a beam along the $x$ axis, the solution consists of ...
Carlos Adir's user avatar
1 vote
0 answers
69 views

About the liouville equation $\Delta u = - \lambda e ^{u}$ on compact manifold with dimension $>2 $

I want to ask about the liouville equation $\Delta u = - \lambda e ^{u}$ on compact manifold with dimension $>2 $. There are many studies on this equation on Riemannian surface (dimension = 2), for ...
Elio Li's user avatar
  • 719
1 vote
0 answers
35 views

Is there an explicit solution to the reaction diffusion system in the following special form?

Suppose $\Omega \subset R^N$ is a smooth bounded domain. Is there an explicit solution to the reaction diffusion equations (RDE) in the following special form? \begin{equation} \left\{\...
Young22's user avatar
  • 11
1 vote
1 answer
169 views

How to rigorously prove that this sequence of stochastic processes converges to a deterministic process?

Assume that for each $n\in\mathbb{N}$, there's a stochastic function $f_n$ of type $\mathbb{R}^{m}\to\Delta\mathbb{R}^{m}$, and for each $x\in\mathbb{R}^{m}$, the distributions $\frac{f_n(x)-x}{\frac{...
Alex Appel's user avatar
0 votes
0 answers
166 views

How does one make sense of singular solutions to constant mean curvature equation?

Background: Consider the following ODE: $$\left(\frac{r^2 \dot{f}}{\sqrt{1+r^2(\dot{f})^2}}\right)' = c r$$ where $c$ is some positive constant (Lagrange multiplier), $f:[0,\infty)\to [0,\pi]$ is the ...
Student's user avatar
  • 653
1 vote
0 answers
29 views

Does elimination imply projection of symmetry?

Let suppose we have system of ODE (linear for simplicity) for two unknown functions $x(t), y(t)$ $$ x''(t) = a(t) x'(t) + b(t) y'(t) + c(t) x(t) + d(t) y(t), $$ and $$ y''(t) = e(t) x'(t) + g(t) y'(t) ...
Dragomir's user avatar
1 vote
0 answers
94 views

Is there an analytic solution of this Burger's type equation?

I came across the following PDE: $$\frac{\partial f(x,t)}{\partial t} -f(x,t)\bigg{(}\frac{\partial f(x,t)}{\partial x}\bigg{)}= 0$$ for $t > 0$ subject to the initial condition $f(x,0) \equiv f_{0}...
InMathweTrust's user avatar

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