Questions tagged [partial-differential-equations]
Questions on partial (as opposed to ordinary) differential equations - equations involving partial derivatives of one or more dependent variables with respect to more than one independent variables.
22,534
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Asymptotic solution to an integro-differential equation
I am trying to obtain an asymptotic solution to the integro-differential equation of the form,
$$
\frac{\partial f(x,t)}{\partial t} = \int_{-\infty}^\infty \Lambda\left(\frac{|x-x'|}{x_0}\right)\frac{...
- 925
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0
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35
views
Solving quasi linear partial differential equation.
I am trying to attempt to solve $$u_t +2ut u_x = tu$$, with the initial condition, $$ u(x,0)=x$$.
I have been trying the method of charecterisitcs and stuck after this step
$$ \frac{dt}{1}= \frac{dx}{...
- 152
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votes
1
answer
34
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Integration by parts of 3 vectorial functions
I am considering the following integral
$$
\int_{\mathbb{R}^d} ( [\nabla a ]\cdot \nabla b ) \Delta c \ dx,
$$
with $a, b$ and $c$ vanishing in the infinity.
By considering integration by parts,...
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38
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How to solve spinorial differential equations.
I've asked this question in the Physics Exchange Forums, but I think this is a more appropaite site to ask this question. How would you solve the following differential equation, and find its ...
- 689
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An estimate using a variant of Holder's inequality in Fourier space
I have a problem of an estimate of $(1.3)$ in the paper https://arxiv.org/abs/2010.10460, which says that
Assume
$$\mathcal{F}\{Q_m[f,g]\}(\xi)=\frac{1}{(2\pi)^{\frac{3}{2}}}\int_{\mathbb{R}^3}m(\xi,\...
- 31
1
vote
0
answers
40
views
Convolution and fractional operator
Let $h\in C_c^\infty(\mathbb R^n)$ and consider the equation
$$(-\Delta)^s u = h,$$
where $(-\Delta)^s u$ denotes the fractional Laplacian of $u$.
Let $\Gamma$ be its fundamental solution, i.e. $(-\...
- 376
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24
views
Green's Function for Parabolic PDE with Time-dependent Coefficients
I'm trying to work through Polyanin's solution in Handbook of Linear Partial Differential Equations for Engineers and Scientists for the following PDE:
$$
\frac{\partial C}{\partial t} = \frac{1}{\...
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1
vote
1
answer
80
views
Is the weak derivative a Radon-Nikodym derivative?
It seems reasonable to me to conjecture that the weak derivative is a special case of Radon-Nikodym derivative between measures. Recall that (missing some technical details):
Weak derivative: given a ...
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Show that $\{p,\{p,\phi\}\}>0$ on $T_{\Sigma}^{*}\mathbb{R}^{n}$
I am reading an article and I found the following situation,
Suppose we have a differential operator $P(x,D)$ of order $m$ in $\mathbb{R}^n$. Let $\Sigma$ be an oriented hypersurface in $\mathbb{R}^n$,...
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Question about convergence in Sobolev spaces
Let $\Omega \subseteq \mathbb{R}^d$ be bounded and open with $\partial\Omega$ Lipschitz, $1<p<\infty$. Is it true that if $(\varphi_n)_{n\geq 1}\subset C^{\infty}_c(\Omega)$ with $\varphi_n\to \...
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Differential equation and surface equation
A surface is drawn satisfying $r+t=0$ and touching $x^2+z^2=1$ along its section $y=0$. Obtain its equation in the form $x^2(x^2+z^2-1)=y^2(x^2+z^2)$.
I have done the CF of the partial differential ...
0
votes
1
answer
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How to prove that $\frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}\in L^1(\mathbb R^{2n})$?
Let $\Omega$ be an open bounded domain of $\mathbb R^n$. Let $s\in(0, 1)$, $p\in (1, \infty)$ and consider the Banach space
$$X^{s, p}(\Omega)=\{u\in W^{s, p}(\mathbb R^n): u=0 \text{ in } \mathbb R^n\...
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Partial derivatives of a complex function
I have a problem considering the partial derivatives of a complex function. I am going to try to sketch the problem as best I can:
I have a module, which takes as an input K and M (stiffness and mass ...
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1
answer
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What the meaning of 'in the trace sense'? [closed]
enter image description here
There are always descriptions about the value of boundary. I want to know if this condition is necessary.
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Discrete Schrodinger Equation - Solution as Bessel functions
Consider the DSE with no nonlinearity term: $$i\partial_{z}(q_n(z)) = q_{n+1}(z) + q_{n-1}(z)$$ I want to solve this exactly and have been told it can be done with Bessel functions, but I do not see ...
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1
answer
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Prove that the solution to a PDE is non-unique
Given the PDE:
$$x u_x - y u_y = 0$$
How do I show that every solution that satisfies $u(1, y^{3}) = y^3$ is not unique?.
If we apply the method of characteristics to this PDE, we get that:
$$u(x, y) =...
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0
answers
41
views
Question about sovolev spaces
Let $\Omega=\{(x, y), 0<|x|<1,0<y<1\} \subset \mathbb{R}^{2}$. Define the function
$$
u(x, y)= \begin{cases}1 & \text { if } x>0 \\ 0 & \text { if } x<0\end{cases}
$$
(a) ...
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1
answer
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Which sobolev embedding gives $W_p^1(\Omega) \hookrightarrow W_{n-\varepsilon}^1(\Omega)$
I am trying to understand a corollary of the Rellich Kondrachov theorem:
Corollary Let $\Omega \in C^1$ be bounded. Then, for $1 \leq p<\infty$,
$$
W_p^1(\Omega) \stackrel{c}{\hookrightarrow} L_p(\...
- 4,207
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0
answers
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Integral sections of higher order jet fields
Let us consider a bundle $(E,\pi, M)$ and let $k\in \mathbb N$. I am going to adopt the notations and conventions by Saunders.
Preliminaries
A first-order jet field on $\pi$ is a section of the bundle ...
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1
answer
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Deriving Lagrangian for common a class of PDEs
I am interested in constructing a Lagrangian for a PDE of type
$$
u_t(t,x) - F(u,u_t,u_x)=0
$$
such that for some functional $\quad I[u] = \int D[u]\mathcal{L}[u,u_t,u_x]$
its associated Euler ...
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answers
43
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Evans' PDE book: definition of (integrals over) smooth surfaces on $\mathbb{R}^n$?
I just started reading Evans' PDE book and am reviewing the appendix. On Appendix A.3 (v), he defines the integral of $f$ over a smooth $(n-1)$-dimensional surface $\Sigma$ in $\mathbb{R}^n$. I was ...
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1
answer
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views
Find the solution for given PDE along unit circle
The problem assigned to me is:
For the equation
$$u_x^2 + u_y^2 = u^2$$
Find the solution when $u|_\Gamma$ = 1, where $\Gamma$ is unit circle at origin.
I was taught about method of characteristics ...
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votes
1
answer
33
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Confusion about Separation of Variable with Periodic Boundary Conditions
I'm terribly confused about the separation of variable method for PDE (heat eq particularly). The given conditions are:
$u_t(x,t)=u_{xx}(x,t), u(0,t)=u(L,t), u_x(0,t)=u_x(L,t), u(x,0)=f(x)$
Since I ...
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votes
1
answer
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views
Choose topic and direction for PhD (NN as universal approximator PDE Equations)
The problem arose of choosing a direction for studying and working within the framework of this topic: neural networks as universal approximators for solutions of partial differential equations.
At ...
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General solution to linear PDE with mixed derivatives
Edit: I've reformulated the problem in a way that makes it easier to express the boundary and initial conditions. This involved expressing it in terms of a different function $g(x,t)$ (which was ...
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answers
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Relation between Poisson equation and Wilson lattice gauge invariance theory
I've recently started writing a library of numerical solvers for elliptic partial differential equations, with particular focus on the Poisson equation. If one considers typical Poisson equation in ...
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0
answers
29
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Bound for $\left\|\nabla^2 u\right\|_{L^2(\Omega)}$
Consider the elliptic equation
$$
\begin{aligned}
-\nabla \cdot A \nabla u=f, & \text { in } \Omega, \\
u=0, & \text { on } \partial \Omega.
\end{aligned}
$$
where $\Omega$ is a bounded domain....
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answers
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views
What is the special name for $u_{t}=u_{xxx}$?
Is there a special name for the following type of PDE?
$$u_{t} = -u_{xxx}, \:\:\: u(x, 0) = f(x), \:\:\: t > 0$$
I found a paper mentioned that it is "linear Airy equation", but I have ...
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answers
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Partial Differential Equation Task [closed]
u_{y} + u ^ 2 * u_{x} = 0 −∞ < x < ∞, y> 0; u(x, 0) = p(x); p(x) is a bounded function. Find the solution in the implicit form. Find its existence region.
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Linear kinetic PDE: Characteristics of the transport operator are given by the flow a Hamiltonian
I am trying to read and understand the article "Hypocoercivity for linear kinetic equations conserving mass." by Dolbeault, Mouhot, Schmeiser. doi: 10.1090/s0002-9947-2015-06012-7 (https://...
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Monotonicity of average in the border for sub/superharmonic functions, for non-euclidean balls
It is a well known fact that a harmonic function in $\mathbb{R}^n$ has the mean value property, namely: the average value of a harmonic function at the border of any (euclidean) ball is equal to the ...
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0
answers
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The question about flows and divergence
On a recent differential equations exam I had a task to prove some result involving closed flows and curls, something I have never seen before so I have no idea how to solve it and I hoped I would ...
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votes
0
answers
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views
$H^{2}_{0}(\Omega)$ is embedded in $L^{\infty}(\Omega)$?
In a research paper https://hal.science/hal-02891557/, the authors used the embedding of $H^{2}_{0}(\Omega)$ in $L^{\infty}(\Omega)$ (see page 16-line 1) to obtain some estimates in their research ...
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views
Nonlinear second order partial differential equation Integration
I have the following nonlinear second order partial differential equation:
$\dfrac{\partial^2}{\partial t^2} \log(1 + u) = \nabla^2 u.$
My question is how can i use a finite differences scheme to ...
0
votes
0
answers
32
views
Magnetohydrodynamic equations
Is there any literature related to Slip boundary conditions of friction type for magnetohydrodynamic equations? As there is some results for Navier-Stokes equations with slip boundary conditions of ...
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votes
1
answer
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Frobenius Theorem, a Proof
I would find the proof of Frobenius Theorem (Differential Topology).
Statement: Let $\mathbb{S}$ an overdetermined homogenuous first order PDEs system. $\mathbb{S}$ admits unique maximal solution if ...
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Check how to solve the Fourier series, pde and what I solved [closed]
For $x = \pi/2$, find the value of Fourier series $\sum_{n=1}^{\infty}\frac{1}{n^4}[\sin(n\pi/2)]^2[\cos(n\pi)]^2$.
$$
u_{tt} = u_{xx},\\
u_t(x,0) = 0,\\
u(x,0) = \begin{cases}
x& (0 \leq x \leq ...
- 1
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vote
0
answers
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Well-posedness of linear parabolic PDE
Is there any reference that explicitely comments on the well-posedness (existence of unique solution) for a PDE of the form?
$$\begin{cases}
\frac{\partial}{\partial s}v(s,x)+\varepsilon u\frac{\...
3
votes
0
answers
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views
What is the motivation for Besov spaces?
I am trying to understand the definition of Besov spaces. With such a complicated definition I wonder what is the motivation behind them and why are they so often used in PDE? What advantage do they ...
- 4,287
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1
answer
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General Morrey's Inequality on $\mathbb{R^2}$
Evans shows a general embedding of Sobolev Spaces of bounded open sets with regular boundary onto holder continuous functions.
My question is from a generalization of this result to $\mathbb{R}^n$, ...
- 553
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+50
Evans Chapter 4 Problem 16, initial value problem to Schrödinger equation, convergence
Problem 16 states to discuss the sense in which $u(\cdot,t) \rightarrow g $ as $t\rightarrow 0^+$ defined by $$ u(x,t) = \frac{1}{(4\pi i t)^{n/2}} \int_{\mathbb{R}^n} e^{\frac{i |x-y|^2}{4t}}g(y)dy \...
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Uniqueness of solution of microscopic PDE inhomogeneous neumann boundary condition
The problem is as follows:
$u_k: \Omega_k \rightarrow \mathbb{R}$ such that
$$-div(A(x,\frac{x}{k})\nabla(u_k)) = f(u_k)$$ in $\Omega_k$
$$-A(x,\frac{x}{k})\nabla(u_k)\nu = -kg$$ on $\Gamma_k$
$$u_k=0$...
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votes
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answer
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weak version of maximum principle for not-quite-subharmonic functions
For smooth functions $f(x,y)$ in a disk, if $f$ is subharmonic then it satisfies the maximum principle. What happens if we relax the subharmonic condition, by requiring only certain bounds on $\Delta ...
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Non linear elliptic PDE with gradient term [closed]
I would like to know if there is some book with theory to solve a non linear PDE of the form
$$
-\Delta u(x) + \overrightarrow{g(x)} \cdot \nabla u(x) = f(u).
$$
- 843
2
votes
1
answer
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views
Existence of weak solutions of Poisson equations in $\mathbb{R}^{n}$
We consider the Poisson equation
$$-\Delta u=f \quad \text{in} \quad\mathbb{R}^{n},$$
where $f\in L^{2}(\mathbb{R}^{n})$. We say $ u\in W_{loc}^{1,2}(\mathbb{R}^{n}) $ is a weak solution of the ...
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1
vote
0
answers
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views
Why is this inhomogeneous Sobolev embedding saying anything?
I'm reading Nonlinear dispersive equations: local and global analysis by Terrance Tao and on page 335 he states the following theorem:
if $1 < p < q < \infty$ and $1/p < 1/q + s/d$ then
$$\...
- 1,423
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0
answers
41
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Suggestion for the numerical solution of a nonlinear pde system.
I am faced with the following system of coupled nonlinear partial differential equations
$$
\begin{array}{lcccl}
\varphi_{tt} &-& a_1\varphi_{xx} &+& a_2\varphi_t &=& a_3\sin{\...
- 183
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votes
1
answer
29
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Two scale problems books [closed]
This is not technically a question but more of a help , I need some books to help me go into the theory of two scales and their applications in Partial differential equations.
1
vote
0
answers
41
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Is there a closed form solution to this second order PDE that arose in financial risk modeling?
I am doing mathematical finance and have run across a former colleague's assumption about the convexity of the price of financial securities. The assumption is
$$
\frac{\partial^2 P(x,y)}{\partial x \...
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0
answers
38
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Using Group Symmetries to Translate a Solution Along Itself
Suppose that $G$ be the Lie symmetry group for a given set of differential equations (DEs) $\mathcal{D}$. Let $\mathcal{N}$ be the manifold representing $\mathcal{D}$ in jet space $\mathcal{P}$; i.e. $...
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