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
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votes
0
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29
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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 ...
- 906
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\...
- 309
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 ...
- 309
39
votes
1
answer
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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 ...
- 491
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 ...
- 309
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
$$
\begin{cases}...
2
votes
0
answers
29
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finding weak form of nonlinear differential equation for FEM simulation
The following is the well-known nonlinear differential equation for director's distribution at static equilibrium in liquid crystal displays(LCD). I want to obtain weak form of the given differential ...
- 31
0
votes
0
answers
85
views
Sobolev estimates on domain with boundary
Could someone point me to a reference for the proof of the following Sobolev estimate
$$
\|u\|_{L^{2 d /(d-2)}(\Omega)} \leqslant C(\|f\|_{L^{2 d /(d+2)}(\Omega)} + \|g\|_{(\partial\Omega)})
$$
for ...
- 11
1
vote
0
answers
30
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Boundedness of solution for a differential inclusion with normal cone
Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a positive polynomial function and
$$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))$$ has a ...
- 309
1
vote
0
answers
32
views
The existence of an optimal distributed control problem
Consider $\Omega$ as a bounded interval of $\mathbb{R}$, and let $y\in L^{\infty}(\Omega \times (0,T))$ be a mild solution of the following parabolic partial differential equation:
\begin{equation}\...
- 21
1
vote
0
answers
48
views
Closed convex hull of a sequence of distributions
I encounter the following problem when trying to apply the theory of fractional derivatives. On the natural numbers $\mathbb{N}$, for an $\alpha>1$, define a distribution $p_{\alpha}$ on $\mathbb{N}...
- 131
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answers
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starting methods for general linear methods
I'm self studying general linear methods and are looking for more detailed worked examples of creating starting methods following section 533 of Butchers Numerical Methods for Ordinary DiffEquations.
...
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0
answers
48
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Upper-bound on energy of nonlinear boundary-value problem
The problem:
Consider the following boundary-value problem for the function $\rho : \mathbb{R}^{+} \to \mathbb{R}$ with boundary conditions $\lim_{x\to \infty}\rho(x) \to 1$ and $\lim_{x\to 0}\rho(x)...
- 105
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votes
0
answers
50
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Internal symmetries of partial differential relation via the nonholonomic jet bundle
On a smooth n-dimensional Riemannian manifold $M$, suppose I have the kth order partial differential relation (PDR) written in the form:
$$\mathcal{R}=\mathcal{R}\left(x^{i},u^{i},u_{j}^{i},u_{jk}^{i}\...
- 160
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 ...
- 309
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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
\...
- 309
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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 $u^{\frac{4}{n-2}}g_\text{flat}$,
$$
A^u = -\frac{2}{n-2}u^{-\frac{n+2}{n-2}}\...
- 309
2
votes
1
answer
83
views
The linearization problem of fully nonlinear equation on standard sphere
For a metric $g$ on $\mathbb{S}^{n}$ $(n\geq 3)$, the $\sigma_k$-curvature of $g$ is defined as follows. Let $Ric_{g}$, $R_{g}$ and $A_{g}$ denote respectively the Ricci curvature, the scalar ...
- 309
4
votes
1
answer
165
views
Nirenberg problem in conformal change
Let $(\mathbb{S}^n,g_0)$ be the standard sphere, $n\geq 3$, consider the Nirenberg problem$$
-k(n) \Delta_{g_0} u+R_0 u=R u^{\frac{n+2}{n-2}}, \quad u>0\,\text{ on }\, \mathbb{S}^n,
$$
where $k(n)=...
- 309
0
votes
0
answers
52
views
How to solve the ODE with variable coefficients?
How to solve the ODE: $L(\varphi)=\ddot \varphi - (n-2) \tanh t \dot \varphi + n\varphi\frac{1}{\cosh^2 t }=0$, where $\sinh t=\frac{e^t-e^{-t}}{2}$, $\cosh t=\frac{e^t+e^{-t}}{2}$, $\tanh t=\frac{\...
- 309
2
votes
1
answer
142
views
Super harmonic function
If $u>0$ in $\mathbb{R}^n\backslash\{0\}$ ($n\geq 2$) and $-\Delta u>0$ in $\mathbb{R}^n\backslash\{0\}$, is it true that $\liminf_{|y|\rightarrow 0}u(y)>0$?
- 309
1
vote
0
answers
47
views
Domain where the fractional Laplacian operator is a closed operator
Consider the fractional Laplacian defined by
$$(-\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(\...
- 111
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 ...
- 325
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votes
0
answers
75
views
Solutions to fourth order PDE
I'm interested in finding analytic solutions to this PDE:
$$-\frac{s^3}{xyz} \frac{\partial^4}{\partial s^4} H(s,x,y,z) -\frac{3s^2}{xyz} \frac{\partial^3}{\partial s^3} H(s,x,y,z) -\frac{s}{xyz} \...
9
votes
0
answers
173
views
When is the solution to a linear system of ODEs an algebraic variety?
Question: Are the following observations well known, and in what general context?
Let $A$ be a diagonalizable $n\times n$ matrix over $\mathbb{C}$ and consider the following system of differential ...
- 3,661
-3
votes
1
answer
62
views
Charpit's method and a nonlinear PDE
I have the nonlinear PDE
$$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,...
- 243
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 ...
- 1,213
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 ...
- 101
-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,\...
- 992
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 ...
- 439
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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:
$$...
2
votes
0
answers
27
views
Symplectic (or alike) integrator for system with Coulomb singularity and time-dependent potentials
I am trying to calculate classical trajectories for a single a ion and a single electron inside an RF trap. Therefore, I am dealing with a two-body system that possesses:
Coulomb potential with a ...
- 21
1
vote
1
answer
82
views
What's a good approximation for the first derivative at the endpoints of given datapoints for a cubic spline interpolation?
I'm using a cubic spline interpolation for given data points. The boundary condition for the spline is that $f'(a)$ and $f'(b)$ are given (I'm using a finite difference formula $\frac{y_1-y_0}{x_1-x_0}...
- 35
4
votes
3
answers
244
views
Coupled Riccati equations
Is there a general solution (in terms of simple known functions) for the following system of coupled non-linear EDOs ?
$$x'(t) = -a_1x^2 -bxy$$
$$y'(t) = -a_2y^2 -bxy,$$
where $a_1$, $a_2$ and $b$ are ...
- 41
0
votes
0
answers
121
views
Gauss's theorem under the convolution product
Assume that $\rho$ is a smooth scalar field in $\mathbf R^3$ and that $D$ is a measurable vector field in $\mathbf R^3$, such that, for every bounded domain $\Omega$ with smooth boundary $\partial \...
- 677
1
vote
0
answers
56
views
Poisson equations for tensors on compact Riemannian manifold
Let $({M},g)$ be a compact Riemannian manifold with Levi-Civita connection $\nabla$. It is well known that the Poisson equation
$$\Delta f=S$$
where $\Delta:C^{\infty}({M})\to C^{\infty}({M})$ denotes ...
- 701
4
votes
1
answer
117
views
Properties of the displacement field, assuming only smooth charge distribution and Gauss's theorem
In physics, the displacement field satisfies Gauss's theorem:
$$
\int_{\partial \Omega} {\bf D}\ {\bf n}\operatorname{d\!}S = \int_{\Omega} \rho\operatorname{d\!}V,
$$ where
$\Omega$ is a bounded ...
- 677
0
votes
0
answers
76
views
Numerical approaches to functional equations
I'm interested in finding numerical approaches to solving functional equations such as
f(xy)=f(x)+f(y),
where the equations had no derivatives or integrals, and contains arguments involving x
and y
.
...
- 21
2
votes
0
answers
70
views
Differential inequality with convex constraint
The following problem looks to be classical, but I fail to find a reference for it. If you know it, please help me.
Given a bounded measurable function $h:{\mathbb R}_+\to\mathbb R$ and a number $a\...
- 49.5k
4
votes
1
answer
355
views
Periodicity and Burger's equation
Consider the 1-dimensional Burger's equation on a finite interval $I=(0,1)$,
$$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).$$
...
- 40.2k
1
vote
1
answer
98
views
How to prove $ \|u\|_{L^{\infty}}\leq C\|\partial_1\square u\|_{L^1} $ for any $ u\in C_0^{\infty}(\mathbb{R}^{1+2}) $?
It comes from estimates for wave equations.
For any $ u=u(t,x)\in C_{0}^{\infty}(\mathbb{R}^{1+2}) $, which is a smooth compactly supported function, prove that
$$
\|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 ...
- 31
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, ...
- 13
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 ...
- 43
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
$$
\...
- 1