Questions tagged [differential-equations]
Ordinary or partial differential equations. Delay differential equations, neutral equations, integro-differential equations. Well-posedness, asymptotic behavior, and related questions.
1,655
questions
4
votes
1
answer
189
views
Nirenberg problem in conformal change
Let $(\mathbb{S}^n,g_0)$ be the standard sphere, $n\geq 3$, consider the Nirenberg problem$$
-k(n) \Delta_{g_0} u+R_0 u=R u^{\frac{n+2}{n-2}}, \quad u>0\,\text{ on }\, \mathbb{S}^n,
$$
where $k(n)=...
- 419
0
votes
0
answers
63
views
How to solve the ODE with variable coefficients?
How to solve the ODE: $L(\varphi)=\ddot \varphi - (n-2) \tanh t \dot \varphi + n\varphi\frac{1}{\cosh^2 t }=0$, where $\sinh t=\frac{e^t-e^{-t}}{2}$, $\cosh t=\frac{e^t+e^{-t}}{2}$, $\tanh t=\frac{\...
- 419
2
votes
1
answer
185
views
Super harmonic function
If $u>0$ in $\mathbb{R}^n\backslash\{0\}$ ($n\geq 2$) and $-\Delta u>0$ in $\mathbb{R}^n\backslash\{0\}$, is it true that $\liminf_{|y|\rightarrow 0}u(y)>0$?
- 419
1
vote
0
answers
75
views
Domain where the fractional Laplacian operator is a closed operator
Consider the fractional Laplacian defined by
$$(-\Delta)^s u(x) = P.V. \int_{\mathbb{R}^N} \frac{u(x) - u(y)}{|x - y|^{N + 2s}}dy, \ s \in (0,1).$$
Also consider that
$$D((-\Delta)^s) = \{u \in H^s(\...
- 111
0
votes
2
answers
149
views
Convergence of solutions to parametrized ODE when no limiting ODE exists
There is plenty of literature on the convergence of the solutions to the real ODE, parametrized by $N \in (0;\infty)$,
\begin{equation}
f_N' (x)
=
a_N (x) \cdot f_N (x)
+ b_N (x)
\end{equation}
to the ...
- 335
9
votes
0
answers
179
views
When is the solution to a linear system of ODEs an algebraic variety?
Question: Are the following observations well known, and in what general context?
Let $A$ be a diagonalizable $n\times n$ matrix over $\mathbb{C}$ and consider the following system of differential ...
- 3,741
-4
votes
1
answer
122
views
Charpit's method and a nonlinear PDE
I have the nonlinear PDE
$$p^2 + 2q = x$$
with the initial condition $u(0, y) = -y^2$, and $y > 0$.
Here's what I have done so far:
I defined the function $F$ to be equal
$$F(x, y, p, q, u) = p^2 + ...
1
vote
0
answers
84
views
is dp/dt = P(1 - 2P^2) a Logistic Differential Equation? [closed]
I currently going through a differential equations course and I am presented with the question:
$$\DeclareMathOperator{\D}{d\!}
\text{is }
\frac{\D p}{\D t} = p(1 - 2p^2)\text{ a logistic DE}?
$$
I ...
1
vote
0
answers
85
views
Wave equation on $[0,1]$ with mixed boundary conditions
Consider the wave equation $u_{xx}-u_{tt}=0$ on the unit interval $x\in[0,1]$. Take mixed boundary conditions ($\alpha_{1,2}^2+\beta_{1,2}^2 \neq 0$)
\begin{align*}
\alpha_1 u(0,t) + \beta_1u_x(0,...
- 439
3
votes
1
answer
160
views
Must solutions to the time-independent Schrodinger equation that have discrete or negative eigenvalues be square-integrable?
This is a very straightforward question (although the answer may not be!) about a ubiquitous textbook physics situation. I assume that the answer is probably well-known, but I asked it on both Math ...
- 1,243
2
votes
0
answers
146
views
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 ...
0
votes
1
answer
89
views
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 ...
- 101
-1
votes
1
answer
75
views
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!
3
votes
0
answers
215
views
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,\...
- 1,121
1
vote
1
answer
126
views
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 ...
- 449
2
votes
0
answers
42
views
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 ...
- 21
1
vote
1
answer
204
views
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}...
- 35
4
votes
3
answers
290
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 ...
- 41
1
vote
0
answers
80
views
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 ...
- 987
4
votes
1
answer
131
views
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 ...
- 677
0
votes
0
answers
165
views
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
.
...
- 21
2
votes
0
answers
75
views
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\...
- 51.6k
4
votes
1
answer
391
views
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
$$u(0,t)=A(t) \qquad u(1,t)=B(t).$$
...
- 41.8k
1
vote
1
answer
108
views
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
$$
\|u\|_{L^{\infty}(\mathbb{R}^{1+2}\...
- 1,133
1
vote
0
answers
62
views
Continuity in the uniform operator topology of a map
I have a question concerning the continuity for $t>0$ in the uniform operator topology $L(X)$ of the following map: $$t\mapsto A^\alpha R(t)$$ where A is the infinitesimal generator of an analytic ...
- 31
1
vote
1
answer
52
views
How to find the maximum value of the following difference equation without using iterative method？
$E(i+1)=(I-AT)E(i)+1/2(AT)^2$
How to find the maximum value of $E$ in this expression without using the iterative method? An approximate estimation is also acceptable. Only the $E$ vector is unknown, ...
- 13
0
votes
0
answers
156
views
Solving a nonlinear differential equation
I need to solve the following equation:
$$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)}$$
with $\eta>1$.
Undoubtedly, the differential ...
- 43
1
vote
1
answer
93
views
Are there PDEs in which Hessian appears in the weak formulation
Before stating the question, I would like to first use an example for the type of formulation that I'm interested in.
Suppose we consider the continuity equation $\partial_t \rho + \mathrm{div}( \rho ...
- 768
1
vote
0
answers
63
views
Finding all polynomials that become zero when certain differential operators act on them
Consider some differential operators that do not have $x^n$ type of coefficients, i.e., only as powers of $\partial_x$, $\partial_y$ or sum of a few such terms with constant coefficients. For example, ...
- 11
2
votes
1
answer
300
views
Property so that $f(t)\equiv 0$ for all $t\geq T$ for some finite $T>0$?
Let $f:[0, \infty)\to [0, \infty)$ be non-decreasing and satisfy for all $t>t_{0}$, $$f(t)+C\int_{t_{0}}^{t}f^{\gamma}(s)ds\leq \frac{1}{t-t_{0}}\int_{t_{0}}^{t}f(s)ds,$$ where $0<\gamma<1$ ...
- 449
2
votes
0
answers
67
views
Methods for holonomic recurrences
I wanted to ask if anyone knows of good texts/resources on methods for solving holonomic recurrence relations (if there are any general analytical approaches):
$$p_1(n)a(n)+p_2(n)a(n-1)+\dotsb+p_k(n)a(...
- 21
2
votes
0
answers
100
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 $$L u(x):=\sum_{i,...
- 81
2
votes
2
answers
236
views
Domain of Schrödinger operators
Let $S$ be a Schrödinger operator on $\mathbb{R}$, $Su=-u''+Vu$ with $V\geq1$ continuous and going to $+\infty$ at infinity (you can think of it as $x^2+1$). I wondering which assumptions do I have to ...
- 41
2
votes
0
answers
60
views
Higher order energy method for nonlinear damping wave equation(reference request)
When I deal with Energy decay rate estimates of the wave equation$$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_{...
0
votes
1
answer
197
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:
$$...
- 109
2
votes
2
answers
116
views
Uniqueness of a second order linear ode
I am currently confronted with the following equation $$
0=w''(t)(t^2-t)+w'(t)((2n-1)t^2-n)+w(t)(n-1)^2t
$$ for $t\in(-1,1)$. So $w:(-1,1)\rightarrow\mathbb{R}$. The following assumption is also in ...
- 21
1
vote
1
answer
195
views
Solution of nonlinear differential equation $g = c_1 f^2 + c_2 (f')^2$ for function $f$
I would like to find an analytic solution (if possible) of the differential equation:
$g = c_1 f^2 + c_2 (f')^2$
Where $c_1$ and $c_2$ are constants, $g$ is a known function of $x$, $f$ is another ...
- 73
1
vote
1
answer
59
views
ODE with conditions within the interval
Can anyone please recommend some publications related to ODEs with non-initial, non-boundary conditions, but conditions for points inside the interval, on which the ODE is defined?
3
votes
1
answer
115
views
Analyticity of central stable manifolds
Let $X$ be a real analytic vector field defined on $\mathbb{R}^n$. Assume the origin $0 \in \mathbb{R}^n$ is a zero of $X$. Assume, furthermore, that we know that the center-stable manifold (in the ...
- 1,379
1
vote
1
answer
186
views
PDE involving curl
Let $G:\mathbb{R}^3\rightarrow\mathbb{R}^3$ be smooth vector field over $\mathbb{R}^3$. For which vector fields $F:\mathbb{R}^3\rightarrow\mathbb{R}^3$ does the PDE
$$\dfrac{\partial}{\partial t}\...
- 205
3
votes
1
answer
197
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 ...
- 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 ...
- 151
2
votes
1
answer
107
views
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.
0
votes
0
answers
99
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{...
- 291
3
votes
1
answer
341
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 $i=1,\ldots, r$, ...
1
vote
0
answers
118
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
$$
\mathbf{u} (\mathbf{x}; t) = \...
3
votes
1
answer
301
views
Find $f:\mathbb R^3 \to \mathbb R$ s.t. $(f - 1)\Delta f + f^2 = 0$?
I wish to solve the following nonlinear PDE that I derived in statistical physics. (I was curious if I could include higher order terms into a model for heat transfer described in a homework problem.) ...
- 547
4
votes
1
answer
174
views
Euler operator as part of a cochain complex
I am studying chapter 4 of Olver's "Applications of Lie groups to differential equations", about symmetries in differential equations coming from a variational principle.
The Euler operator ...
3
votes
1
answer
124
views
A type of singular limit for systems of differential equations
Suppose I have a system of differential equations for the unknowns $(x_1,v_1,\ldots,x_N,v_N)$ (interpreted as the positions & velocities of $N$ labeled particles),
$$\begin{cases}\dot{x}_{i,\...
- 863
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 $M^n\backslash\Sigma$ ($u$ ...
- 63