Questions tagged [pde]

Questions on partial (as opposed to ordinary) differential equations which involving partial derivatives of one or more dependent variables with respect to more than one independent variables.

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26 views

Some questions on Alfréd Rényi's paper ON THE THEORY OF ORDER STATISTICS

when I read ON THE THEORY OF ORDER STATISTICS written by Alfréd Rényi, I found this statement, and I can't figure out why this satisfied, could anyone give me some idea? A necessary condition of ...
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25 views

When can one use only the particular solution?

I have the following PDE $$xu_x+u_y-(y+z)u_z=0$$ So I have to solve $$\frac{dx}{dt}=x,\frac{dy}{dt}=1,\frac{dz}{dt}=-(y+z),\frac{du}{dt}=0$$ I got $$x(t,s_1,s_2)=e^t\cdot f_1(s_1,s_2)$$ $$y(t,s_1,...
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19 views

Equality p-Laplacian

Prove that $$\Delta_p u= \text{div}(|\nabla u|^{p-2}\nabla u)=|\nabla u|^{p-2} ({(p-2)\Delta_{\infty}} u+\Delta u)$$ I tried do this from the definitions but it didn't lead me to the any good ...
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27 views

What type of PDE is this ? Are there any known instances of modelling with this equation?

In 1D, are there any known PDE (...or reaction-diffusion) models with a form similar to the equation: $\frac{\partial U(x,t)}{\partial t} = \alpha \big[1 - U(x, t)\big]\Big(U(x,t) + \nabla^2 U(x,t) \...
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1answer
82 views

When does $\int \frac{dx}{x} = \ln|x|$ and when $\int \frac{dx}{x} = \ln(x)$?

Sorry for the provocative question but I am often see a solution where the absolute value is neglect for example: $$ \begin{cases} xuu_x+yuu_y=u^2-1, x>0\\ u(x,x^2)=x^3\\ \end{cases} $$ We ...
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26 views

Is it possible to classify a PDE using the method of characteristics?

I know that the method of characteristics is the technique to solve some PDEs. But I obtained an assignment to classify PDE (hyperbolic, elliptic or parabolic) using the very same method. I was ...
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8 views

Recovery of Classical Solution for the Homogeneous Dirichlet Problem

In section 9.5 of Brezis' text, Brezis goes through the example of the homogeneous Dirichlet problem. My issue is with the final step where he recovers a classical solutions from the weak solutions. ...
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1answer
34 views

Brezis (Variational Formulation for Boundary Value Problems)

Brezis considers the inhomogeneous Dirichlet problem, \begin{align} -\Delta u+u=f\quad &\text{in }\Omega,\\ u=g\quad &\text{on }\partial\Omega, \end{align} where $f$ is given on $\Omega$ and $...
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1answer
20 views

Non Constant Coefficients PDE (part of solving a differential equation using symmetry methods)

How would I solve: $$ y^3 \eta_{xx} - y \eta_x =0 $$ where $\eta(x,y)$ I started by dividing by $y^3$ to get: $$ \eta_{xx} - \frac{1}{y^2}\eta_x = 0$$ I then thought to let $u = \eta_x$ and then ...
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32 views

Uniform $C^k$ bounds from a Laplacian bound.

In a paper I am reading, I am given smooth solutions to a family of Monge-Ampere equations on a smooth, compact subset of a variety. The paper claims that uniform $C^k$ bounds on the family of ...
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3answers
71 views

Solve $\frac{1}{x}z_x+\frac{1}{y}z_y=4$

$$\begin{cases} \frac{1}{x}z_x+\frac{1}{y}z_y=4\\ z(1,y)=y^2-1.\\ \end{cases}$$ So we started with: $$\frac{dx}{dt}=\frac{1}{x}\rightarrow x^2(t,s)=2t+f_1(s)$$ $$\frac{dy}{dt}=\frac{1}{y}\...
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1answer
35 views

Asymptotic behaviour of heat equation solution

In our lecture we discussed the solutions to \begin{cases} u_t - u_{xx} = 0, & x \in (0,1) \\ u_x(0,t) = u_x(1,t) = 0, & t > 0 \\ u_x(x,0) = u_0(x) \in L^2(0,1) \end{cases} and found $$ u(...
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1answer
46 views

Solving Euler Homogenous of Degree One Equation in 2D

is there some analytical solution to the following problem: Finding all $f:\mathbb{R}^2 \to \mathbb{R}$, $f\in C^1$ such that: $$ x \cdot \nabla f(x) = f(x)$$ This Euler Relation means that $f$ is ...
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1answer
28 views

Lax-Milgram Lemma, Alternative Proof in Finite Dimensional Case

I am asking myself, if we would weaken the assumptions of the Lax-Milgram Lemma to the finite dimensional case Lax Milgram Lemma Let ($V$, $(\cdot, \cdot$, $\Vert \cdot \Vert$) be a (real) Hilbert ...
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3answers
34 views

Solving Exponential PDE [on hold]

Can anyone help me solve this? I am having a hard time figuring this out. $\frac{dy}{dx}=e^x+y$
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26 views

Geometry of Envelope form definition

I had read about the envelope of the family of the curve. It is defined as a curve which is tangent to each member of the family at a single point and it is union of all such points. To find envelope ...
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0answers
21 views

Mean field games and approximate Nash equilibria

I am learning mean field games (MFG) through the notes by Cardaliaguet: https://www.ceremade.dauphine.fr/~cardaliaguet/MFG20130420.pdf. I have a question about a step in theorem 3.8 on page 17. Let ...
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1answer
57 views

Solving PDE $xu_x+(x+t)u_t=1$ with $u(1,t)=t$ [duplicate]

Solve $xu_x+(x+t)u_t=1$ such that $u(1,t)=t$. Is the solution defined everywhere? I had known that this specific problem is related to a Heat Equation problem. I tried solving for its ...
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1answer
77 views

Finding explicit solution of PDE $xu_x+tu_t=pu$

Let $p\in \mathbb{R}$ and consider $xu_x+tu_t=pu$. Find a) Characteristic curves for the equation. b) Find an explicit solution for $p=4$, where $u=1$ on the unit circle. What I tried: a) ...
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0answers
14 views

Showing the value functions for an N-player differential game solve a coupled system of parabolic PDE

I'm interested in deriving this system: $$\begin{cases}-\partial_t v^{N,i} - \sum_{j} \Delta_{x_j}v^{N,j} + \sum_{j \neq i} D_{x_j}v^{N,j} \cdot D_{x_j}v^{N,i} + \frac12 |D_{x_i}v^{N,i}|^2 = F^i(\...
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1answer
42 views

How does one show that $\biggl(\dfrac{\partial}{\partial{t}}-\dfrac{\partial^{2}}{\partial x^{2}}\biggr)$ is a linear operator?

How does one show that $\dfrac{\partial{u}}{\partial{t}}-\dfrac{\partial^{2}{u}}{\partial{x^{2}}}$ is a linear operator? At first, I was thinking that there might be some distributive property that I'...
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1answer
23 views

Boundedness of travelling front norm

For a reaction diffusion equation $u_t=u_{xx}+f(u)$, assume that $\mu_1<\mu_2$ are equilibria and $V(\xi), \xi=x-ct$ with $c>0$ is a traveling front, where $\lim_{\xi\to -\infty}V(\xi)=\mu_2$ ...
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2answers
40 views

How do we get the function $f(y')=f(bx-ay)$?

The following is from Partial Differential Equations by Strauss: Let us solve $au_{x} + bu_{y} = 0$, where $a$ and $b$ are constants not both zero. Coordinate Method Change variables (or “...
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1answer
22 views

Questions about $au_{x} + bu_{y}=0$, for $u(x,y)$ and $v=ai +bj$

I have a couple of questions in regards to the following passage from Partial Differential Equations by Strauss: When we say that the solution is constant on each such line, are we restating that $bx-...
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27 views

A series involving spectrum of $- \Delta_{\mathbb{S}^2}$

Infinite series involving eigenvalues of the Beltrami-Laplace operator on Riemannian manifolds as well as $L^p$-estimates of eigenfunctions arise in the study of the nonlinear Schrödinger equation (...
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40 views

Solving differential equations with derivatives to non-integer powers

In particular, for an equation of the form, $$x' = a (y')^b$$ where $b$ is a real number (not necessarily an integer), and $x'$ and $y'$ are each derivatives with respect to a different variable (e.g. ...
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17 views

Streamlines of conserved current rarely end up in a “node”?

Intro for motivation: Suppose $(\rho,\mathbf{J})$ are a certain density- and current profile. Suppose $\rho\geq 0$, $\rho(x,t)=0\Rightarrow \mathbf{J}(x,t)=0$ and finally $\partial_t \rho + \nabla \...
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2answers
48 views

Solving a pde on the unit square

I'm having trouble solving for $u(x,y)$ in the following pde on the unit square: $$ - \nabla^2u = x(1-x) ,\;\; \text{for } 0 < x < 1, 0<y<1, \\ u(x,0) = 1 \text{ and } u(x, 1) = 2,\;\; \...
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0answers
22 views

Definition of Hölder Space on Manifold

Can anyone point me to a reference for the definition of Hölder spaces for manifolds (with boundary)? Every paper I have looked at says these are "defined in the ordinary way" and no one says what ...
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0answers
25 views

Laplace Equation on Rectangle with Inhomogeneous Neumann BCs and PBCs

I will preface this question by referring to a similar question here which did not give a full solution. I found this question on the January 2017 Applied Mathematics Qualifying Exam from the ...
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1answer
61 views

Shallow water equations in differential form and cylindrical coordinates

I am ignoring bed height or any other extra terms. The version of SWE when written in cartesian is: $ \begin{pmatrix} h \\ hu \\ hv \\ \end{pmatrix}_t + \begin{pmatrix} hu \\ hu^2 + \frac{g}{2}...
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27 views

HJB with discontinuity at boundary

I have an optimization problem whose value function I denote by $U(x,t)$ where $x\in [0,1]$ is a state variable and $t\in [0,1]$ is time. The HJB equation for my optimization problem when $x<1$ is ...
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20 views

Boundary condition for a bulk-surface and bulk-bulk diffusion reaction system

Consider this simple example below and the corresponding geometries. I simplified these equations from the real system. Geometry 1 The first geometry is a sphere. Inside this sphere a species $b(t,...
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35 views

Can Navier Stokes be solved by assuming there is some recursive function that calls upon itself?

Let $$y=e^y = e^{e^y} = e^{e^{e^y}}=e^{e^{e^{e^{...}}}}$$ The derivative is $$\frac{\partial}{\partial x} e^y = e^y\frac{\partial y}{\partial x} = yy_x$$ Compare this with the quasilinear homogenous ...
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2answers
80 views

Einstein field equation,pde and differential geometry

I'm a math undergraduate student with some interest in mathematical physics with basic knowledge of partial differential equation. When I was reading a wikipedia article about einstein field equation,...
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2answers
60 views

wave equation problem

hello I have a homework and i think this problem is wrong: $$ \left\{ \begin{array}{ll} u_{tt} &= u_{xx}\\ u_{x}(0, t) &= 0 \\ u_x (1, t) &= t \\ u(x,0) &= 0 \\ u_t(x,0) &= \cos{...
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1answer
34 views

Asymptotic behavior of $u_t= u_{xx}+au$

Consider the following one-dimensional reaction-diffusion equation: $$u_t= u_{xx}+au$$ on $\Omega=(0,1)$ with Dirichlet boundary conditions with $a>0$ and a nonnegative initial condition $u_0$. If $...
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11 views

Equivalent definition of subharmonicity on a Riemann surface

Let $X$ be a Riemann surface. This is the definition of subharmonicity I have known (and have been trying to work with) for a while. Here is my definition of subharmonic: Let $A$ denote the set of ...
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36 views

Solving a boundary value problem in terms of eigenfunctions

I am having trouble solving the second part of this question. The first part stated: Let $r, \theta$ be the polar coordinates in $\mathbb{R}^2$ and $\Omega$ the unit disk $r < 1$. Solve the ...
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22 views

Dirichlet problem for the wave equation on the half-line

The following material comes from the textbook written by Walter A. Strauss. Please look at the part circled in red ink. If $t<0$, the reduction of $\phi_{\text{odd}}$ to $\phi$ seems to go wrong. ...
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1answer
40 views

On the locality of the gradient of a Sobolev function

Let $\Omega$ be an open subset of $\mathbb{R}^n$. Suppose that $u,v\in W^{1,1}(\Omega)$ and that $u = v$ on a Borel subset $E\subseteq \Omega$. Question: is it true that $\nabla u= \nabla v$ a.e. in $...
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1answer
69 views

Steady states of $u_t= u_{xx}+\pi^2u$

I just put the following one-dimensional reaction-diffusion equation in Mathematica: $$u_t= u_{xx}+au$$ with $\Omega=(0,1)$ with Dirichlet boundary conditions. When $a<9$, no matter the initial ...
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1answer
22 views

Long time stability in nested Bochner spaces

Let $V\subset H$ be densely, compactly and continuously embedded, which, among others, means that $$\|x\|_H \leq \|x\|_V\quad (*)$$ for all $x\in V $. I have the following stability estimates for a ...
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1answer
34 views

Heat equation and Fourier series

I am trying to solve the following boundary value problem involving the heat equation: $$ \frac{\partial{u}}{\partial{t}} - \frac{1}{4} \frac{\partial^2{u}}{\partial{x}^2} = 0,\;\; t > 0 \text{ and ...
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0answers
77 views

System of 16 coupled Integro-differential equations

I have a question: How can i solve numerically a system of 16 coupled integro-differential equations? $ - a{{s'}_{11}} + b{{s'}_{33}} + c{{s'}_{44}} + Ng_1^2(\int_0^z {({s_{14}}(t) + {s_{23}}(t))} dt)...
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1answer
29 views

Heuristic argument to prove integrable harmonic function is null

Referring to this question on harmonic function in which is proved the sequent Let $u(x)$ a harmonic function in $\mathbb{R}^n$ such as: \begin{equation} \int_{\mathbb{R}^n}|u(x)|dx =K< \...
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20 views

3D heat diffusion equation in terms of convolution

The solution to 1-D heat equation can be expressed via 1-D convolution. More formally, let $\frac{\partial u}{\partial t} = -k \frac{\partial^{2}}{\partial x^{2}}$, then we can find a solution $u = \...
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26 views

a system of 16 coupled Integro-differential equations [closed]

I have a question: How can i solve a system of 16 coupled first-order integro-differential equations? $ - a{{s'}_{11}} + b{{s'}_{33}} + c{{s'}_{44}} + Ng_1^2(\int_0^z {({s_{14}}(t) + {s_{23}}(t))} dt)...
2
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0answers
23 views

Set of Differential Equations - Partition of Unity

I am trying to find the solution for the following set of differential equations: Find $q(u) = [q_1(u), \ldots, q_n(u)]^\top$ where $u \in \mathbb{R}^n$ and $n \in \mathbb{N}$ such that $$ \sum_{k=1}...
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22 views

Source of extract: Partial Differential Equations

Can anyone maybe identity the source of this extract from a textbook for Partial Differential Equations. I would like to purchase the full version.