# Questions tagged [differential-equations]

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

1,536
questions

0
votes

0
answers

29
views

### Determinant of 2D non-positive second order partial differential operator

If I have an ordinary second order differential operator the Gelfand-Yaglom method is often useful to calculate its (regularized) determinant. The great advantage is that one doesn't have to calculate ...

1
vote

0
answers

27
views

### Fractional Laplacian in higher order case

Let $n\geq 2$ and $\sigma \in (0,\frac{n}{2})$. Denote the critical Sobolev exponent $2_{\sigma}^*:=\frac{2n}{n-2\sigma}$, consider Sobolev space $E$ which is the space of real-valued functions $u\...

2
votes

0
answers

28
views

### Maximum principle for poly-harmonic equations

If $u_1\geq 0$ and $u_1\neq 0$, and satisfies
$$-\Delta u_1=|u_1|^{\frac{4}{n-2}} u_1\quad \text { on }\, \mathbb{R}^n,\quad n\geq 3,$$
it follows from maximum principle that $u_1>0$. My question ...

39
votes

1
answer

2k
views

### What is the shape of the perfect coffee cup for heat retention assuming coffee is being drunk at a constant rate?

Note: I asked this on Mathematics SE and even though @TheSimpliFire offered a bounty on it, no-one had a good answer
Find the optimal shape of a coffee cup for heat retention. Assuming
A constant ...

4
votes

1
answer

130
views

### A system of linear PDEs with boundary conditions

I'm researching on discrete/semi-discrete/smooth differential geometry. Recently, I could simplify one of my geometric problems (in the smooth scenario) into the solutions of a system of linear PDEs ...

0
votes

0
answers

43
views

### The equivalence of fully nonlinear Yamabe equation

On a Riemannian manifold $(M, g)$ of dimension $n \geq 3$, consider the Schouten tensor
$$
A_g=\frac{1}{n-2}\left(\operatorname{Ric}_g-\frac{1}{2(n-1)} R_g g\right),
$$
where $\mathrm{Ric}_g$ denotes ...

6
votes

2
answers

217
views

### Existence of solutions to the heat equation on nonsmooth domains

Let $\Omega \subset \mathbb{R}^n$ be a compact domain and for given functions $g: \partial \Omega \times [0,T] \to \mathbb{R}$ and $h: \Omega \to \mathbb{R}$ consider the heat equation
$$
\begin{cases}...

2
votes

0
answers

29
views

### finding weak form of nonlinear differential equation for FEM simulation

The following is the well-known nonlinear differential equation for director's distribution at static equilibrium in liquid crystal displays(LCD). I want to obtain weak form of the given differential ...

0
votes

0
answers

85
views

### Sobolev estimates on domain with boundary

Could someone point me to a reference for the proof of the following Sobolev estimate
$$
\|u\|_{L^{2 d /(d-2)}(\Omega)} \leqslant C(\|f\|_{L^{2 d /(d+2)}(\Omega)} + \|g\|_{(\partial\Omega)})
$$
for ...

1
vote

0
answers

30
views

### Boundedness of solution for a differential inclusion with normal cone

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a positive polynomial function and
$$C=\mathbb{R}^n_+ = \{x\in\mathbb{R}^n:x_i\geq 0,i=1,\ldots,n\}.$$ If we know
$$\dot{x}(t)=-\nabla f(x(t))$$ has a ...

1
vote

0
answers

32
views

### The existence of an optimal distributed control problem

Consider $\Omega$ as a bounded interval of $\mathbb{R}$, and let $y\in L^{\infty}(\Omega \times (0,T))$ be a mild solution of the following parabolic partial differential equation:
\begin{equation}\...

1
vote

0
answers

48
views

### Closed convex hull of a sequence of distributions

I encounter the following problem when trying to apply the theory of fractional derivatives. On the natural numbers $\mathbb{N}$, for an $\alpha>1$, define a distribution $p_{\alpha}$ on $\mathbb{N}...

-2
votes

0
answers

32
views

### starting methods for general linear methods

I'm self studying general linear methods and are looking for more detailed worked examples of creating starting methods following section 533 of Butchers Numerical Methods for Ordinary DiffEquations.
...

0
votes

0
answers

48
views

### Upper-bound on energy of nonlinear boundary-value problem

The problem:
Consider the following boundary-value problem for the function $\rho : \mathbb{R}^{+} \to \mathbb{R}$ with boundary conditions $\lim_{x\to \infty}\rho(x) \to 1$ and $\lim_{x\to 0}\rho(x)...

2
votes

0
answers

50
views

### Internal symmetries of partial differential relation via the nonholonomic jet bundle

On a smooth n-dimensional Riemannian manifold $M$, suppose I have the kth order partial differential relation (PDR) written in the form:
$$\mathcal{R}=\mathcal{R}\left(x^{i},u^{i},u_{j}^{i},u_{jk}^{i}\...

2
votes

0
answers

98
views

### Representation formula for solutions to fully nonlinear equations

Let $n\geq 3$, for a metric $g$ on $\mathbb{S}^n$, the $\sigma_k$-curvature of $g$ is defined as follows. Let $Ric_{g}$, $R_{g}$ and $A_{g}$ denote respectively the Ricci curvature, the scalar ...

0
votes

0
answers

45
views

### Degree equation in Scalar curvature problem

For $1<p<(n+2) /(n-2)$, $n\geq 3$, $K \in C^{\alpha}(\mathbb{S}^n)$, $(0<\alpha<1),$ $(\mathbb{S}^n,g_0)$ is standard sphere, $K>0$, consider the prescribing scalar curvature problem
\...

0
votes

1
answer

76
views

### Kelvin transformation in fully nonlinear equaion

Let $g_\text{flat}$ denote the Euclidean metric on $\mathbb{R}^n$ and $A^u$ denote the $(1,1)$-Schouten tensor of $u^{\frac{4}{n-2}}g_\text{flat}$,
$$
A^u = -\frac{2}{n-2}u^{-\frac{n+2}{n-2}}\...

2
votes

1
answer

83
views

### The linearization problem of fully nonlinear equation on standard sphere

For a metric $g$ on $\mathbb{S}^{n}$ $(n\geq 3)$, the $\sigma_k$-curvature of $g$ is defined as follows. Let $Ric_{g}$, $R_{g}$ and $A_{g}$ denote respectively the Ricci curvature, the scalar ...

4
votes

1
answer

165
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)=...

0
votes

0
answers

52
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{\...

2
votes

1
answer

142
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 $\liminf_{|y|\rightarrow 0}u(y)>0$?

1
vote

0
answers

47
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(\...

0
votes

2
answers

120
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 ...

0
votes

0
answers

75
views

### Solutions to fourth order PDE

I'm interested in finding analytic solutions to this PDE:
$$-\frac{s^3}{xyz} \frac{\partial^4}{\partial s^4} H(s,x,y,z) -\frac{3s^2}{xyz} \frac{\partial^3}{\partial s^3} H(s,x,y,z) -\frac{s}{xyz} \...

9
votes

0
answers

173
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
votes

1
answer

62
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

69
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

57
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,...

3
votes

1
answer

131
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 ...

2
votes

0
answers

101
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

68
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 ...

-1
votes

1
answer

59
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

209
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
vote

1
answer

104
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 ...

0
votes

0
answers

34
views

### Elimination of unknowns in systems of linear differential equations

Let $a_0$, $a_1$, $b_0$, $b_1$, $c_0$, $c_1$, $d_0$, $d_1$ be functions-coefficients, and $f_1$, $f_2$ functions-unknowns.
Let us consider a system of two linear differential equations of order 1:
$$...

2
votes

0
answers

27
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 ...

1
vote

1
answer

82
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}...

4
votes

3
answers

244
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 ...

0
votes

0
answers

121
views

### Gauss's theorem under the convolution product

Assume that $\rho$ is a smooth scalar field in $\mathbf R^3$ and that $D$ is a measurable vector field in $\mathbf R^3$, such that, for every bounded domain $\Omega$ with smooth boundary $\partial \...

1
vote

0
answers

56
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 ...

4
votes

1
answer

117
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 ...

0
votes

0
answers

76
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
.
...

2
votes

0
answers

70
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\...

4
votes

1
answer

355
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
$$u(0,t)=A(t) \qquad u(1,t)=B(t).$$
...

1
vote

1
answer

98
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
vote

0
answers

46
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 ...

1
vote

1
answer

44
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, ...

0
votes

0
answers

153
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 ...

0
votes

0
answers

66
views

### Solving a Differential equation from intersection theory via series expansion

I have the following differential equation $\nabla_\omega \psi=\varphi$ where $\nabla_\omega(\psi)=d(\psi)+\omega(z)\wedge\psi$.
With the local coordinates of $y=z-x_i$ the series expansions is
$$
\...