Questions on "Partial Differential Equations", as opposed to "ordinary differential equations".

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265 views

Time Derivative of PDE solution

Consider the PDE $$ \frac{\partial \psi}{\partial t}(t,x)=\psi''(t,x)+2\psi\psi'(t,x)-\tilde V'(x)~ on~ [0,D/2]\times(0,\infty)\\ \psi(0,t)=0, \psi(D/2,t)=-k, \\\psi(\cdot,0)=\psi_0 $$ for some ...
3
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1answer
17 views

Regularity for a parabolic problem with nonsmooth coefficients

I'm looking for references on the regularity of the (weak) solution to the parabolic problem with nonsmooth coefficients. In most literature, like Evans, the coefficients are often assumed to be ...
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1answer
265 views

Numerical techniques for solving systems of first-order semi-linear hyperbolic PDEs in two variables

I need to solve such systems of PDEs numerically and I'm wondering what the "standard" method (if there is one) is. The systems I'm interested in have the form: $\frac{\partial F_i}{\partial t} + ...
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0answers
50 views

Cauchy problem in pde [on hold]

I think that this question is related to Cauchy problem in PDE. Please explain the solution explicitly? And please which books should I study for such type questions? Especially, the book contains ...
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0answers
38 views

How to solve a system of first-order partial differential equations?

I have a system of first-order partial differential equations. \begin{align} & -\frac{a}{a_1} \frac{\partial P_{12}}{\partial a} - \frac{a b}{a_1 b_1} \frac{\partial P_{12}}{\partial b} + ...
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2answers
38 views

3D Cauchy problem for the PDE $ yu_x-xu_y+u_z=0 $

I will answer the question myself but let me know what you think of my correctness. We have the Cauchy Problem $$ yu_x-xu_y+u_z=0 $$ with data $u(x,y,0) = x+y$.
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0answers
16 views

Compute average and maximum value of a field over a streamline

I'm working on a code solving a set of PDEs. I have a vector field, $\vec{v}(x,\theta,z,t)$ (it's a velocity) and a scalar field, $c(x,\theta,z,t)$. I have a $2\pi$-periodicity in $\theta$. The ...
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1answer
40 views

Comparison of the gradients of two harmonic functions near the boundary

Let $\Omega$ an open bounded domain in $R^n$. Let $u,v$ be nonconstant smooth functions in the interior of $\Omega$ and harmonic in $\Omega$. Suppose that $u,v \in C(\overline{\Omega})$ and $u \geq ...
2
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2answers
39 views

First order Cauchy problem

I'm trying to solve the PDE \begin{align*} au_x + (bx+cu-1)u_y = d \\ u(x,0) = 0. \end{align*} So far, I got the solution for the case $cd-ab \neq 0$ as \begin{align*} u(x,y) = \frac{d}{cd-ab}\left( ...
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0answers
39 views

Method of Characteristics for a PDE

I'm working through a problem at the moment, and I've got an answer, but it seems far too complicated... I've been asked to use the method of characteristics to solve the following PDE; $$x^2 u_x ...
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13 views

Reference for p-capacitary functions

Consider $\Omega_1 $ and $\Omega_2$ two open bounded and convex sets in $R^n$with $\Omega_1 \supset \overline{\Omega}_2$. The unique weak solution of the problem $$ \begin{cases} \Delta_p u = 0 ...
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12 views

Determining the semigroup and relating it to the generator via the functional calculus

In the theory of Hille Yoshida, we have a (definitely real) banach space $L$ and on it a contractive, strongly continuous semigroup of operators. ($T_tT_s=T_{t+s}$ and $T_0=I$.) We then define the ...
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1answer
108 views
+400

Non-ellipticity of Yang-Mills equations

Let $D=\text{d}+A$ be a metric connection on a vector bundle $E\rightarrow M$ with curvature $F=F_D$. How does one prove that the Yang-Mills equations $$ \frac{\partial}{\partial ...
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0answers
20 views

PDE and Taylor's formula

I'm looking to a prove that a function that satisfies the following equations is actually $f(x,t)=x^3 \pm tx$ after changing coordinates. Here are the equations: 1) $\frac{\partial^3 f}{\partial ...
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2answers
534 views

Symbol of a (non linear) differential operator

I am interested in knowing whether there is a definition for the symbol of a PDO which is NOT linear. In Wikipedia and in the book I am reading (An Introduction to Partial Differential Equations by ...
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2answers
126 views

Definition of the principal symbol of a differential operator on a real vector bundle.

I'm trying to understand the construction of the dirac operator on a manifold, but actually I guess that doesn't really matter for the question at stake. I'm interested in understanding a definition ...
19
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1answer
523 views

Seeking Fourier series solution on Laplace equation…still looking, am I on track?

Okay, I've been working at this a couple of days now, I will try to give relevant details but will omit some intermediate steps. The problem as given says: Consider the BVP for $u=u(x,y)$: ...
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1answer
27 views

Lp bounds of the Heat Kernel

These days, I am struggling with a problem which seems very straightforward (and I'm pretty sure it is straightforward) but it resists to my attempts to prove it. Here it is: Let $\mathcal H_t$ be ...
3
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2answers
210 views

Heat equation and semigroup theory.

Theorem: Let $X$ be a Banach space, $\{T(t)\}_{t\geq 0}$ a $C_0$-semigroup on $X$ and $U_0\in X$. If $A:D(A)\subset X\to X$ is the infinitesimal generator of $\{T(t)\}_{t\geq0}$, then the function ...
2
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1answer
670 views

FFT for solving Poisson equation in 3 dimensions with periodic boundary conditions

In electrostatics, the laplacian of the electrostatic potential $\Delta V(\mathbf{r})$ arising from a charge distribution $\rho(\mathbf{r})$ is $$ \Delta ...
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0answers
14 views

Green's function of a Markov Chain, and maybe of a Feller Process?

How are the Green's functions of a markov chain related to the notion from PDE theory? For instance, if the markov chain (i.e. discrete state space) is continuous time, then the Green's function I'm ...
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0answers
25 views

About the Hopf lemma for the p-laplacean

My question is about this article :http://arxiv.org/pdf/1204.6578v1.pdf . Consider $U$ an open bounded domain in $R^n$ and $u \in W^{1,p}(U)$ a p-harmonic function in $U$(the definition is on the ...
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0answers
24 views

existence and uniqueness of volterra integral equation of the first kind

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To know the existence and uniquness of solution of volterra integral equation(VIE) of the first kind, we differentiate it and convert to the VIE of the second kind. ...
2
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3answers
224 views

Really nice functional equation with second partial deratives.

Show that $f:\mathbb{R}^2\to\mathbb{R}$, $f \in C^{2}$ satisfies the equation $$\frac{\partial^2 f}{\partial x^2} - \frac{\partial^2 f}{\partial y^2} = 0$$ for all points $(x,y) \in \mathbb{R}^2$ if ...
3
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2answers
86 views

The relationship between two definitions of star-shaped domain

There are two definitions of star-shaped domain. One is given in wikipedia as follows. Def1: A set $S$ in the Euclidean space $\mathbb{R}^n$ is called a star domain (or star-convex set, star-shaped ...
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0answers
25 views

Di Perna-Lions theory

I'm reading the paper of Di Perna and Lions "Ordinary differential equations, transport theory and Sobolev spaces". I'm not understanding the proof of corollary II.2; in particular I don't understand ...
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1answer
461 views

An inequality in Evans' PDE

In Section $9.2$ Theorem $5$ of Lawrence Evans' Partial Differential Equations, First Edition the author proves that for a large enough $\lambda$, the equation $$\begin{array}-\Delta u+b(\nabla ...
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0answers
39 views

Is my function singular at these two points?

My function $S(x,y,t)$ satisfies the following PDE $$\frac{\partial S(x,y,t)}{\partial t}=-H(x,y)$$ where the known function $H$ is singular when $x=\alpha$ and/or $x=\beta$, hence my question: ...
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0answers
40 views

Elliptical Coordinates PDE, wave equation and separation of variables

I need some help with this problem. I know how to use the method of separation of variables and that the constant lambda should give you trig functions with solutions at some interval of pi, which ...
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1answer
30 views

Riesz measure associated with a subharmonic function

In page 101, corollary 4.3.3., from Armitage and Gardiner's book on potential theory, the authors prove that any subharmonic function, can be identified with a positive measure (Riesz measure). In ...
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1answer
44 views

Hardy's inequality

Hardy's inequality (for integrals, I think) presented in Evans' PDE book (pages 296-297) contains a formula whose notation is substantially different than the conventional estimate presentation of ...
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1answer
21 views

Problem on energy of a Discrete Galerkin Method

I'm reading an article from this website: article question is in page 3,about a wave equation,and use the Galerkin method to discrete the space. (1) page4 why the author use the fraction ...
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0answers
41 views

Using the method of characteristics to find a general solution to PDE

I want to find the general solution to $3U_x-4U_y = x^2$ using the method of characteristics. I'm given the answer which is $U(x,y)=\frac{x^3}{9}+F(3y+4x)$ but I'm having trouble getting to this ...
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1answer
41 views

Eigenvalues and Eigenfunctions of Integral Equation

Compute the eigenvalues and eigenfunctions of $$ \varphi(x) - \lambda \int_0^1 e^{x+s} \varphi(s) ds = f(x) $$ Are there functions $f$ such that the inhomogenous equation has for every real $\lambda$ ...
22
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2answers
933 views

Non-linear partial differential equation

I would like to find out if there is any specific method -apart from numerics- for finding solutions of a non-linear PDE of the form $$\nabla \times \mathbf{A} = \pm\lambda\mathbf{A} \tag{1}$$ under ...
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2answers
91 views

Partial Differential equations and applications- Reference request

I will be taking up a PDEs course next semester and would like to find some good references. The topics covered in the syllabus is given below. Partial differential equations: Conservation laws, ...
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2answers
41 views

Show Uniqueness of Solution for Boundary Value Problem

Let $G \subseteq R^n$ be a simple, connected and bounded region with smooth boundary and let $f : \overline G \to \mathbb R$, $g : \partial G \to \mathbb R$ be continuous. Show that the following ...
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1answer
23 views

Blow up of a solution of the p-Laplace equation at a boundary point

My doubt is about a proof from the lemma 2.8 of the article 18 of this homepage: http://www.math.kth.se/~henriksh/Henriks_page/publications.html The result that I said is: Lemma: Let $D_1$ and $D_2$ ...
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1answer
49 views

Solve the given differential equation by using Green's function method

I am really struggling with the concept and handling of the Green's function. I have to solve the given differential equation using Green's function method $$\frac{d^{2}y}{dx^{2}}+k^{2}y=\delta ...
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0answers
14 views

Missing explanation in this paper of Masmoudi.

In this paper, on page 4, beginning in the line above 3.8, the authors begin a discussion of a given variational problem. I follow their argument until they begin the line of reasoning that begins ...
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0answers
27 views

Showing $u = \phi_\epsilon * u,$ ie Function Equals Convolution with Function

My question is the following: Starting with the mean value property for harmonic $u \in C^2(\Bbb R^3)$, ie $$u(x) = {1 \over {4\pi R^2}} \int_{\partial B_R(x)}u(y) dy,$$ deduce that if $\phi \in ...
0
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1answer
37 views

Definition of elliptic pde

http://en.wikipedia.org/wiki/Elliptic_partial_differential_equation Very confusing. So are there non-linear elliptic pdes? It seems to be the case. And what about $$A u_{xx} + C u_{yy} + F u = 0$$ ...
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25 views

Show that different eigenfunctions of integral kernel are orthogonal

Consider the integral operator $K \varphi := \int_0^1 k(x,s) \varphi(s) ds$ with a continuous and symmetric kernel $k : [0,1]^2 \to \mathbb R$ which has at least two different eigenvalues $\lambda_1$ ...
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2answers
26 views

Fourier Transform of Poisson Equation

While trying to solve the Poisson Equation by using Green's Function I have to Fourier transform the equation i.e $$-\nabla^{2}\phi(r)=\rho(r).$$ In the book after Fourier transform, the solution ...
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0answers
29 views

Non stationary solutions of the PDE $u_t + u_x = u_{xx}$

Problem. Consider the PDE $$ u_t + u_x = u_{xx}, \qquad (t,x) \in (0,+\infty) \times (0,1). $$ (i) Write the unique solution $\overline{u}=\overline{u}(x)$ which does not depend on time and ...
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1answer
24 views

Show that Fourier series arising in solution of differential eqn. converges uniformly

Let $f \in L_2(0,\pi)$ have the Fourier expansion $f(x) = \sum_{n=2}^{\infty} f_n\sin(nx)$. Compute (formally) the boundardy value problem $$ u''(x) + u(x) = f(x) \qquad \mbox{ for } 0 < x < ...
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2answers
82 views

In three dimensions, the Laplacian of $1/r$ is $0$ outside the origin

Why does the following hold? $$\Delta_{3}\frac{1}{r}\Bigg\vert_{\mathbb{R}^3 \setminus \left\lbrace 0 \right\rbrace}=0$$
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0answers
9 views

Converting from mixed boundary condition to dirichlet boundary condition

I have found a finite element method code in MATLAB that I want to use however the boundary conditions are dirichlet -- more specifically the function is 0 at the boundary. However, the PDE I'm ...
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17 views

Accuracy of a finite-difference method for numerically solve a PDE or BVP

When solving the Poisson Equation $$-u''(x)=f(x)$$ with Dirichlet-Neuman boundary conditions $$u(0)=0, u'(1)=0$$ using a finite difference 2-order centered scheme and a 2-order upwind ...
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0answers
24 views

Easy solution to Yamabe problem for surfaces

The Yamabe problem asks if, given a Riemannian manifold $(M,g_0)$, it is possible to find a conformal metric $g$ on $M$ with constant scalar curvature. I would like to know if there is some "easy" ...