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Questions tagged [pde]

Questions on partial (as opposed to ordinary) differential equations.

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17 views

Partial Differential Equation System: Klein-Gordon field: spherical symmetry: analytical solution

The PDE system that follows is the hamiltonian form of the Klein-Gordon equation in units $\hbar=c=m=1$: $$\partial_t \varphi = \varpi\qquad\partial_t\varpi=\frac{1}{r}\partial^2_r(r\varphi)-\...
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0answers
13 views

How to solve first order pde with unkown source term

I am trying to solve a problem in differential geometry. I want to find a change of coordinates that rectify a set of vectors. I know this diffeomorphism exists (because the vectors commute). However ...
1
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0answers
9 views

Existence of solution to Laplace equation

Given condition: Let $\Omega$ is a smooth bounded domain in $\mathbb{R}^2$. Suppose there exist $w(>0)\in H_{0}^1(\Omega)$ satisfying the inequality $-\Delta w\leq C e^w$ in $\Omega$ for some ...
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22 views

Feynman-Kac formula in action.

Briefly speaking, the Feynman-Kac formula gives a way for constructing $C^2$ functions satisfying certain PDEs in the classical sense (at least, it's how it is explained in Oksendal's book that I am ...
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0answers
19 views

Characterization of ellipticity for linear differential operators

Let $P(D)$ be a differential operator of order $m$. I would like to show that $P(D)$ is elliptic if for some open and bounded subset of $\mathbb{R}^n$ and for some real $s$ and $c>0$ and all $\...
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0answers
19 views

Finding a particular solution to a linear PDE

I want to solve the PDE $$\frac{\partial u}{\partial t}+x_1(x_2-x_3) \frac{\partial u}{\partial x_1}+x_2(x_3-x_1) \frac{\partial u}{\partial x_2}+x_3(x_1-x_2) \frac{\partial u}{\partial x_3}=\sum_{i=1}...
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1answer
51 views

Exact vs approximate Riemann solvers

I am trying to understand numerical methods for conservation laws. I am confused with few terminologies. I have the following doubts. When do we say that a numerical scheme for a conservation law ...
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0answers
33 views

Numerical solution to 1D convection-diffusion equation

I am looking to numerically solve the convection-diffusion equation $$ \frac{\partial p}{\partial t}=-\frac{\partial}{\partial x}\left(u(x)p(x,t)-D(x)\frac{\partial p}{\partial x}(x,t)\right) $$ where ...
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0answers
25 views

Energy Dissipation in Euler forward

I am currently learning numerical methods for partial differential equations. While learning ENO methods and Runge-Kutta time-stepping algorithms, we mentioned Euler forward time-stepping. I know ...
2
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0answers
22 views

Definition of weak solution of a PDE that is given in the nondivergent form

Firstly, I would like to introduce two problems. A Riemann problem for a system of conservation laws given in divergent form: $$(1) \hspace{1cm} \begin{cases} u_t+f(u)_x=0 \\[2ex] u(x,0)= \...
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0answers
39 views

Solving a nonlinear reaction-diffusion PDE

I am trying to find a solution to the nonlinear reaction-diffusion PDE: $$u_t = \Delta u + \lambda u - u^3$$ where $(x,t) \in \Omega \times (0,\infty)$ and $\Omega\subset \mathbb{R}^n$ is a bounded ...
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1answer
35 views

Solutions to the heat equation with a ring of coolant

I'm interested in solutions to the heat equation for the following problem $ \dfrac{\partial u}{\partial t} = c^2 \nabla^2 u $ on $ \left\{ (x,y) \vert x^2+y^2 \leq R^2 \right\} $ such ...
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1answer
28 views

Concavity of $\log u(x)$

Let $u>0$ be a solution of the following equation $$\begin{cases}-[|u'(x)|^{p-2}u'(x)]'=\lambda_1\cdot u(x)^{p-1}&,x \in (a,b)\\u(a)=u(b)=0 \end{cases},$$ where $\lambda_1$ is the minimum of ...
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0answers
23 views

Solution to Laplace equation with Dirichlet boundary conditions

I'm having trouble finding the solution to the problem: $u_{xx}+u_{yy}=0$ with the boundary conditions $u(x,1)=-x,u(x,-1)=x,u(1,y)=-y,u(-1,y)=y$. I've been suggested to try separation of variables, ...
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0answers
27 views

Complete integral for the first order PDE [on hold]

On the domain $K = \{(x,t)|x\in \mathbb R^n \text{ and } t\in \mathbb > R^+ \}$ let $h:K\rightarrow \mathbb R$ obey the partial differential equation $$h_t+\frac{1}{2}|Dh|^m=0,$$ where $D$ is ...
1
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1answer
18 views

Is there exists a convex function in exterior domain and have zero limit at infinity?

Recently I came into a problem and wondering if there exists such a function $f:\mathbb{R}^n\setminus\overline{B_1}\rightarrow\mathbb{R} $is convex and having zero limit at infinit. Meaning that I ...
7
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3answers
101 views

Boundary and initial conditions in quasi linear first order pde

I cannot understand what we are looking to find in such a problem... For example consider the pde $u_t+u_x=u$, with $x,t>0$ (1) and initial and boundary conditions: $u(x,0)=1$, for $x\ge0$ (2)...
2
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1answer
26 views

Subsolution of Laplace equation

Let $\Omega$ be a bounded open subset of $\mathbb{R}^2$ and $w(>0)\in H_{0}^1(\Omega)$ satisfies the equation $$ -\Delta w\leq e^w\text{ in }\Omega. $$ Let $v(>0)\in H_{0}^1(\Omega)$ satisfies ...
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0answers
15 views

Pazy Theorem 6.1.4 proof (existence of mild solution)

In Semigroups of Linear Operators and Applications to Partial Differential Equations by Pazy, Theorem 6.1.4 establishes the unique existence of the mild solution when $f$ is "locally Lipschitz" in $u$,...
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1answer
48 views

Solution of $u_t=\mathcal{F}u$

I tried to solve the equation $u_t=\mathcal{F}u$, where $\mathcal{F}$ denotes the Fourier transform, with initial data $u(x,0)=u_0(x)$. The solution should be given by $$ u(x,t)=e^{\mathcal{F}t}u_0=\...
1
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1answer
51 views

General solution vs General Integral

I have been reading some notes about Quasi Linear PDE and I stuck to the following issue: Consider the PDE: $xz_x+yz_y=xe^{-z}$ then my notes read as: "the general solution is: $G(x/y, e^z-x)...
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1answer
30 views

Why does $\log|(x,y) - (ξ,η)| $ satisfy the laplace equation for all $(x,y) \ne (ξ,η)$

Why does $G = \log|(x,y) - (ξ,η)| $ satisfy the laplace equation for all $(x,y) \ne (ξ,η)$ Could someone explain why this is true? I thought $\frac{\partial G}{\partial x} = \frac{1}{(x,y) - (ξ,η)}$ ...
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0answers
13 views

Systems of Partial differential equations: initial-boundary value problems: in search of numerical code test.

I am in search of constrained evolution problems defined in some region $$(t,r)\in[0,\infty)\times[0,R]$$ concerning PDE's systems and boundary conditions of the general form $$ x_1' = f_1(\vec{x},\...
0
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1answer
19 views

Solve $u_x + 4xu_y = 1 + u^2$ for $u(0,y)=y$

I got a weird result so I'm not sure I did this right Let the initial condition be $u(0, y_0) = y_0 $ for some $y_0$ By the method of characteristics let $$\frac{dx}{ds} = 1 \to x = s + A$$ $$x(s=0)...
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0answers
18 views

Plot the solutions of the following PDEs at the specified times

I am studying for my PDEs final and this is one of the practice questions from Strauss and I don't understand how to solve it. Plot $u$ vs $x$ for the following problem on the half-line $$u_{tt} = c^...
5
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0answers
39 views

An inequality about linear PDE

I am trying to solve the following problem: Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^n, \ n\geq 2$. Let $u\in C^2(\overline{\Omega})$ be a solution of $$\left\{\begin{array}{ll}u_t-...
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0answers
26 views

Description of the PDE $u_{\xi\xi}=a^2u_{yy}-u_{\xi}$

I have the following PDE: $$\frac{\partial^2u}{\partial\xi^2}=a^2\frac{\partial^2u}{\partial y^2}-\frac{\partial u}{\partial\xi}$$ First of all, this is a PDE, right? And after that is nonlinear and ...
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0answers
12 views

Calderon-Zygmund inequality on $W^{k,p}$

Let $\Omega$ be a bounded region of $\mathbb R^n$, we have the Calderon-Zygmund inequality, i.e. $\|\nabla^2u\|_{L^p}\leq C\|\Delta u\|_{L^p}$. Q For $k\geq1$, can we have $$\|\nabla^2u\|_{W^{k,p}}\...
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0answers
17 views

Solving the wave equation with initial conditions $u_x=0$ at $x=0,\pi$ [duplicate]

So we have, $u_{tt}-c^2u_{xx}=0$, with initial conditions $u_x=0$ at $x=0,\pi$ Without loss of generality, let's take $c=1$, and suppose our solution is in the form $u(x,t)=X(x)T(t)$. Then we can ...
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0answers
43 views
+50

An integral estimate in conformal geometry

Let $g_0$ be a Riemannian metric of positive scalar curvature $R_0$ on a closed 4-manifold, and for some fixed constant $C$, define the set \begin{equation} \mathcal{S} = \{u\in C^\infty(M): ||u||_{W^{...
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0answers
34 views

Show that the following linear PDE has a solution $f(x)$ satisfying $\int_D f(x)dx=0$ [on hold]

$$-\Delta f(x)=v(x),$$ where the function $v(x)$ satisfying, $\int_Dv(x)dx=0$ with $\nabla v\cdot \vec{n}=0$. Here $\vec{n}$ is a unit normal vector, and $D\subset \mathbb{R}^d$. I don't understand ...
1
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1answer
28 views

Sound wave equation: Neumann boundary conditions

In this paper it's described the solution of the damped wave equation in cylindrical coordinates $$ \nabla^2\left(c^2\rho_1+\nu\frac{\partial\rho_1}{\partial t}\right)-\frac{\partial^2\rho_1}{\...
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0answers
21 views

Globally hypoelliptic operator

Are the eigenfunctions of a globally hypoelliptic operator in the Schwarz space $S$.
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3answers
150 views

Heat Equation + Uniform Convergence in time -> Harmonic Limit

Assume we have $u \in C^3(\mathbb{R}^n \times (0,\infty))$ satisfying the heat equation $$ \Delta u(x,t) = \partial_t u(x,t)$$ and a function $u_0:\mathbb{R}^n \to \mathbb{R}$ with unknown regularity (...
0
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1answer
12 views

Decay of Dirichlet exterior problem

Assuming $\Omega \subset \mathbb R^n, (n\ge 3)$ is a domain. Consider the Dirichlet exterior problem \begin{align} &\Delta u = 0 ~~~~x\in \mathbb R^n/\Omega\\ &u|_{\partial \Omega} =1 \tag{1}...
1
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1answer
47 views

Laplace of the PDE $ x \frac{\partial(w)}{\partial(x)} + \frac{\partial(w)}{\partial(t)}=xt$.

Question is find the Laplace transform of this equation: $$ x \frac{\partial(w)}{\partial(x)} + \frac{\partial(w)}{\partial(t)}=xt$$ Boundary conditions : $$ w(x,0) = 0 \qquad x \geq0 $$ $$w(0,t)=0 \...
1
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0answers
27 views

Showing that the Riemann invariant $\frac{1}{2} (u^2+v^2) + \int \frac{c(p)^2}{p} dp $ is conserved along the characteristic $dy/dx = v/u$

I need to show that the Riemann invariant $R = \frac{1}{2} (u^2+v^2) + \int \frac{c(p)^2}{p} dp $ is conserved along the characteristic $dy/dx = v/u$. My system of equations are: \begin{aligned} (pu)...
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2answers
37 views

Good book on solitons focusing mainly on analysis of PDE

I have a good background in PDEs and functional analysis and I would like to learn more about solitons. The most popular references that I see pop up are Drazin and Kasman, but the first is a little ...
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2answers
48 views

Are both answers for $xu_x + yu_y = 0$ valid?

Solving this problem by the method of characteristic curves we have to solve the ODE $$\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{y}{x}$$ which gives us $$C = \ln(y/x)$$ where $C$ is constant. ...
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1answer
37 views

Mass transport equation Cartesian to polar coordinates

Can someone please advise on how to transform the following equation to polar coordinates? $$\frac{\partial \rho(x,t)}{\partial t}=v\frac{\partial \left(\rho(x,t) L(x)\right)}{\partial x}+D\frac{\...
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1answer
24 views

Characteristic coordinates $ξ(x, y)$ and $η(x, y)$ for $xu_{xx} + u_{yy} = 0$ when $x<0$

How would I determine the characteristic coordinates for $xu_{xx} + u_{yy} = 0$? This PDE reads $au_{xx} + 2b u_{xy} + cu_{yy} = 0$ with $a=x, b=0, c=1$. The polynomial equation $a\lambda^2 -2b\...
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0answers
33 views

On possible closures of the derivative operator

Let $\frac{\partial^2}{\partial x^2}: L^2([0,1]) \to L^2([0,1])$ be the one-dimensional Laplacian, considered as an unbounded, densley defined operator with domain $\mathcal D(\frac{\partial^2}{\...
0
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1answer
28 views

Why the coefficients of G(t) depends on “n” in solving PDE wave equation by separation of variables

I don't understand why the coefficients of the sin and cos terms in G(t) (in the red box in picture 1) depends on "n" Why don't we simply choose them to be equal to 1? picture 1 picture 2 kreyszig'...
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0answers
20 views

Confused of two panels of “vorticity”

According to vorticity ($ \vec{\zeta}=\nabla\times\vec{V}$), flow can be defined as rotational or irrotational flow. I read the material about it in Page.3 (http://web.mit.edu/16.unified/www/FALL/...
2
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1answer
28 views

laplace operator on a laplace

What is the result when Laplace operator is applied on Laplace operator, is it $$\nabla^2(\nabla^2 r)= \nabla^4r = \frac{\partial^4 r}{\partial x^4} + \frac{\partial^4 r}{\partial y^4} + \frac{\...
3
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1answer
62 views
+50

Understanding iterated covariant derivatives to define Sobolev spaces on manifolds

I'm having big troubles understanding the definition of Sobolev spaces on manifolds. Ok, so we have a Riemannian manifold $(M, g)$, and then we can define a natural riemannian measure (which I will ...
0
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1answer
42 views

Symmetric Boundary Conditions/Eigenvalues (PDEs)

Consider the following eigenvalue problem for the Laplacian $-\Delta u = \lambda u$ in $U$ $u + a \left(\frac{\partial u}{\partial v}\right)$ on $\partial U$ where $v$ is the outward unit normal to ...
1
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2answers
30 views

If $u(x)$ is harmonic and equal to $\phi(|x|)$, is $\phi$ continuously differentiable?

I was trying to show that radial harmonic functions on the unit ball (in $\mathbb{R}^n$) are constant. To this end, I suppose that $u$ is a radial harmonic function on the unit ball and write $$ u(x) =...
1
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1answer
30 views

Numerical solution of non-linear first order partial differential equation (HJB)

I am trying to solve a simple optimal control problem using the Hamilton-Jacobi-Bellman equation, numerically in Python. This is proving to be rather difficult as I end up having to solve the ...
0
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2answers
37 views

Quasilinear PDE using method of characteristics

The equation is: $yu_x+uu_y=-xy$ with initial conditions $u=y$ on $x=0$ I first find that $\frac{dx}{y}=\frac{dy}{u}=-\frac{du}{xy}$ Solving $\frac{dx}{y}=\frac{dy}{u}$ we get, $ux=\frac{1}{2}y^2+...