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

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1answer
61 views

Second order pde in 2 variables [closed]

How I find the general solution of $u_{xx}-u_{yy}+\dfrac {2u_x}x=0$ equation by $w=x^nu$ transform?
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35 views

Solution procedure for poisson equation

Consider the Poisson equation in the rectangle $Q=\{(x,y):0<x<1,0<y<1\}$, $$u_{xx}+u_{yy}=F(x,y)$$ $$u(0,y)=0,\,u(1,y)=0$$ $$u(x,0)=\phi(x),\,\,u(x,1)=\psi(x)$$ My Question: Is ...
3
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1answer
40 views

Regularity of solutions to a transport equation

Currently I am working on a transport equation and have been able to prove the existence and uniqueness of a weak measurable solution to said equation. I am now working in trying to jot down (with ...
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0answers
57 views

solving a 2nd order PDE with constant coefficients

This question is followed up from this question system of non-homogeneous advection equations \begin{equation} \left\{ \begin{array}{lll} u_t+b_1 u_x=(r+l_1)u-l_1v,\\ v_t+b_2 v_x=(r+l_2)v-l_2u,\\ ...
1
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1answer
55 views

Boundedness of $A$ in the operator equation $Au = f$ of $-\Delta u(x)=f(x)$.

We consider the boundary value problem on a bounded, open domain $\Omega \subset \mathbb R^n$ determining $u : \Omega \rightarrow \mathbb R$ such that $$-\Delta u(x)=f(x), \qquad ...
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0answers
28 views

Numerical method for wave equation with nonlinear forcing in 1+1 D

I am looking for numerical method to solve the equation $$ \square \phi = \frac{\partial^2 \phi}{\partial x^2} - \frac{\partial^2 \phi}{\partial t^2} = \lambda \phi^3 \, $$ for real $\phi = ...
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1answer
21 views

Analytical solution to PDE exists?

I have the following PDE: $\delta_\epsilon(S(\epsilon)\phi(x,\epsilon))+\delta_x\phi(x,\epsilon) = -T(\epsilon)\phi(x,\epsilon)$ Deltas represent partial derivatives, for ease of notation. Does it ...
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0answers
66 views

How to do this estimate

If $a,b$ are two vectors in $\mathbb R^n$ satisfy the following relation \begin{equation} \frac{|a|^2}{1+(1-|a|^2)^{\frac{1}{2}}}\geq \frac{|a|^2-2\langle ...
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2answers
42 views

An intuitive definition of the frequency spectrum of a function.

In a PDE book I'm reading, the author introduces the Fourier transform by first introducing the Fourier series, and then the Fourier integral representation of a function. The Fourier integral ...
1
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1answer
29 views

general solution function using Method of Characteristics

Suppose I am given a function $f(x, y, z)$ that is such that $3 f_x + xf_y + 2yf_z = 0$ I want to know how to write down a general representation for functions with such a property. Proceeding ...
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2answers
233 views

Numerical methods (for ODE/PDE) that could take approximate solutions/good initial guesses, and further refine it to an certain accuracy

I am currently playing with an old analog computer, which could solve time-dependent ODE/PDEs pretty fast, without time-stepping; thus there is no convergence issues caused by time-stepping because of ...
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0answers
24 views

Closure for Legendre Polynomials

Suppose I have a PDE which can be solved by an expansion on the Legendre Polynomial basis, for an axisymmetric problem in spherical coordinates. For the derivation for such a problem, see these ...
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0answers
27 views

Monotonic convergence of Newton's method for boundary value problems

I’m interested in solving nonlinear elliptic boundary value problems of the type $$ -a\Delta u + f(u) = 0, $$ $$ u|_\Gamma = u_0 $$ by Newton’s method when its convergence is global and monotonic. ...
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2answers
42 views

Pdes definition of spaces

I am reading Temam's book Navier Stokes Equations and he defines $E(\Omega) = \left\{u \in L^2\left(\Omega\right), \ \operatorname{div}(u) \in L^2\left(\Omega\right)\right\}$. Later he says that if $p ...
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1answer
38 views

Heat equation with sin initial condition

How do i find the analytical solution of the heat equation: $$U_t = U_{xx} + \sin{\pi x}$$ subject to $u(0,t) = u(1,t) = 0$ and $u(x,0) = \sin(\pi x).$ I appreciate its a pretty common/general ...
1
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0answers
57 views

system of non-homogeneous advection equations

I would like to solve this system \begin{equation} \left\{ \begin{array}{lll} u_t+b_1 u_x=(r+l_1)u-l_1v,\\ v_t+b_2 v_x=(r+l_2)v-l_2u,\\ \end{array} \right. \end{equation} First , I would like to ...
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0answers
23 views

System of multi-valued Hyperbolic semi-linear inhomogeneous PDEs

Recently I am working on solving a hydrodynamic problem which is a system of multi-valued Hyperbolic semi-linear inhomogeneous PDEs, \begin{eqnarray} \partial_t U+ \sum_{i=1}^3 ...
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1answer
30 views

Computing weak derivatives on an open square

I am looking at the computation of weak derivative in the blog https://sunlimingbit.wordpress.com/2012/11/25/one-example-related-to-weak-derivative-2/ For equation (3), I have some confusions. Here ...
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1answer
15 views

Does particular+homogeneous capture all solutions for a linear pde?

Specifically, I'm trying to solve $\Delta f(x) = 1$ inside the unit ball in $\mathbb{R}^3$ with $f(x)=1$ for $\|x\|=1$. I start by finding radially symmetric solutions. The laplacian in spherical ...
2
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0answers
55 views

$\{v_n \} $ is bounded in $W_0^{1,q}$ for $ 1 \leq q < \frac{N}{N-1}$

I have a encountered to a problem in reading article . Can someone look at the page 9 in this article and give a hint that why $\{v_n \} $ is bounded in $W_0^{1,q}$ for $ 1 \leq q < ...
1
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1answer
48 views

Free wave behaviour of $\frac{\partial^2 \phi}{\partial x^2} - \frac{\partial^2 \phi}{\partial t^2} = \lambda |\phi|^2 \phi$.

I am playing with the partial differential equation $\frac{\partial^2 \phi}{\partial x^2} - \frac{\partial^2 \phi}{\partial t^2} = \lambda |\phi|^2 \phi$. $\phi(x,t)$ is complex and the domain is not ...
3
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0answers
42 views

How can we show that $u$ as a weak solution has properties $u \in L^{\infty}(\Omega)$ , $ u>0 $

Let $\Omega$ be an open domain in $\mathbb {R^n}$ and $f \in C^{\infty}(\Omega)$ then how can we prove there is a weak solution $u \in ‎‎ W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega) \cap ...
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1answer
34 views

the equivalence of a absolute value function $|D^2 u|$ in problem 10(b) evans pde chapter 5

Can someone tell me whether $|D^2 u|$ is equivalent of writing $\frac{\nabla u}{|\nabla u|}\, D^2 u$? This relates to the post Integrate by parts to prove this inequality I wasn't sure why ...
4
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2answers
115 views

Is the following PDE boundary value problem well-posed?

My Question Is the following Poisson boundary value problem well-posed, as stated? If so, how could I go about solving it? If not, what would it need to be well-posed? Does it satisfy the ...
2
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1answer
45 views

Unique weak solution of Poisson's equation

Let $\Omega$ be an open set in $\mathbb{R}^n$ and now consider the weak formulation of Poisson's equation $$\int_{\Omega} \langle Du,Dv \rangle = \int_{\Omega}{fv}$$ for $v \in H_0^1$ and $u \in ...
2
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1answer
25 views

Maximum condition for second order differential operators

Let $A$ be a second order differential operator such that $$Af(x) = \sum_{ij} a_{ij}(x) \big(\partial_i \partial_j f(x)\big) + \sum_j b_j(x) \partial_j f(x) $$ Assume that $x \in B(0,r)\Rightarrow ...
2
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0answers
59 views

How to use Fourier transform to solve Fisher-Kolmogorov?

How can I use Fourier Transform to solve Fisher-Kolmogorov Equation in 1D? \begin{equation} u_t(x,t) = u_{xx}(t) + u(1-u) \end{equation} \begin{equation} u(0,x) = \phi(x) \end{equation} with ...
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0answers
23 views

$AX=B$: How to solve for $X$ if elements of matrix A are matrices

Objective: I am trying to solve for $C$ in 2D space (x,y) and time from following PDE. $$\begin{align} \text{PDE: }\frac{\partial C}{\partial t} + \nabla\left(v.C - D\nabla{C} \right)= \alpha.C ...
2
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1answer
61 views

$‎\Delta ‎^2(.)-‎\frac{‎‎\lambda‎‎}{|x|^4}(.): W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega) \to W_0^{-2,2}(\Omega) ‎$ is coercive.

I am reading an article and there, author claim that $$‎L(.)=\Delta ‎^2(.)-‎\frac{‎‎\lambda‎‎}{|x|^4}(.): W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega) \to W_0^{-2,2}(\Omega) ‎$$ is coercive if ‎‎$ ‎0\leq ...
2
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1answer
16 views

Maximum principle, quantitative version

Is it true that $$ \|f\|_{L^\infty(\Omega)}\leq C\|\Delta f\|_{L^\infty(\Omega)}+C\|f\|_{L^\infty(\partial\Omega)} $$ for all $f\in C^2(\Omega)$ and smooth $\Omega$?
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1answer
84 views

a vector calculus problem when coping with problem 9 chapter 5 evans

Hi I was trying to understand the solution given by Ray Yang in the post question 9 - chap 5 evans PDE It gets to the sort of things I am quite bad at... When I get to the point ...
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2answers
34 views

Is the parabolic heat equation with pure neumann conditions well posed?

The parabolic heat equation is a partial differential equation given by $\frac{du}{dt}=\nabla^2u+f$. If i impose an initial condition u(x,0) and pure homogeneous neumann boundary conditions that ...
2
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1answer
44 views

Weak convergence and integrals

Assume $$u_k\rightharpoonup u,\quad v_k\rightharpoonup v\quad\text{in}\quad L^1(0,T;Y)\tag{1}$$ and $$\int_0^T u_k(t)\varphi'(t)\ dt=-\int_0^T v_k(t)\varphi(t)\ dt\tag{2}$$ for some $\varphi\in ...
1
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1answer
14 views

Maximum principle of p-Laplacian operator

Suppose $\Delta_p u \geq \Delta_p v$ then can it be said that $u \geq v$? The domain considered is a bounded subset of $\mathbb{R}^n$.
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1answer
69 views

For which $1\le p\le\infty$ does $u$ belong to $W^{1,p}$(\Omega)$?

Hi could anyone help with a solution for problem 7 Evans PDE chapter 5? I think it is basically about checking which $p$ allows $$\int_{\Omega} |u|^p dx+\int_{\Omega}|Du|^p dx<\infty$$ ? But I ...
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0answers
31 views

non existence of weak derivative evans pde chapter 5 example 2

Hi Im looking at the this very basic example, in proving the non existence of weak derivative. I am confused in the last line $$\cdots\lim_{m\to\infty}\left(\int_0^2 v\phi_m dx-\int_0^1\phi_m ...
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1answer
23 views

Bound on the size of the solution to $u_{t} - \Delta u - u \leq 0$

Let $B$ denote the open unit ball in $\mathbb{R}^{d}$ and let $u$ be a smooth function such that $u_{t} - \Delta u - u \leq 0$ in $U_{T}:= B \times (0, T]$ and $u = 1$ on the parabolic boundary (that ...
0
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1answer
13 views

Regularity for non-homogenous elliptic PDE

Assume $L$ is an elliptic differential operator (second order, with coercive associated bilinear form) with smooth coefficients, and that $\Omega$ has smooth boundary. Does there exist a result of ...
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0answers
26 views

How to apply Runge-Kutta to an implicit scheme?

I see there are some differences in the solution as I increase the resolution of my grid. I'm using Operator Splitting to solve Diffusion Reaction equation \begin{equation} \frac{\partial ...
1
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1answer
59 views

Quadrature formula on triangle

I am looking for a quadrature formula on the triangle, with points at the vertices and at the mid-edges, so 6 points, and that is exact for polynomials of degree at least 2, with weights strictly ...
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0answers
25 views

non-linear PDE finite difference approach

How to approach this equation using finite difference method ...
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0answers
22 views

Reference For a PDE text that treats non homogenuous boundary conditions Rigorously

I am interested in reading a text or paper where elliptic and parabolic PDE's are discussed on bounded domains with non-homogeneous boundary conditions. I haven't been able to find anything in the ...
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0answers
27 views

Closedness of the range of differential operator first order

The fact that the range of gradient from $H_0^1$ to $L_2$ is closed is well known. In general we can define some kind of weak derivative in the form \begin{equation} Du=\sum_{i,j}a_{ij}\partial_i u_j ...
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1answer
28 views

Which 2D domain with fixed area has the lowest laplacian eigenvalue?

I know that a disc has the lowest laplacian eigenvalue among domains with fixed area. But how do I prove it?
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1answer
25 views

Control of $L^{\infty}$ Norm of 3d Heat Equation Solution for $L^{3}$ Initial Data

Let $w_{t}$ denote the 3-dimensional heat kernel $$w_{t}(x)=(4\pi t)^{-3/2}e^{-\left|y\right|^{2}/(4t)},\qquad y\in\mathbb{R}^{3}, \ t > 0$$ Suppose $f\in L^{3}(\mathbb{R}^{3})$, and let ...
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0answers
28 views

Why L belongs to the dual space $H^{-1}$

I'm studying pde using Evans book. In chapter 6 he introduces second order partial differential operators for example : $L= \sum{D_i(a_{ij}D_j(u))+\sum{b_iD_i(u)+cu}}$ I can't understand why $L \in ...
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0answers
19 views

2D Poisson equation and Bessel Functions

If I were to solve a inhomogeneous 2D Laplace's equation (in polar coordinates), of the form: $$\nabla^2 f= J_2(r)$$ How would I use the Hankel or the Mellin transform to find a solution? I couldn't ...
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0answers
83 views

{$\mathbb u$ $\in W^{2,2}(\Omega)$ , such that $u=0$ , $ \Delta u=0 $ on $\partial \Omega $} $\subseteq$ $W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$

I have a question that it maybe simple but I can not understand why we have : {$\mathbb u$ $\in W^{2,2}(\Omega)$ , such that $u=0$ , $ \Delta u=0 $ on $\partial \Omega $} $\subseteq$ ...
1
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1answer
39 views

Help with method of characteristics question.

I'm trying to solve the partial differential equation, $$ \frac{\partial F}{\partial t} = (z-1)\left(kF - d \frac{\partial F}{\partial z}\right) $$ where $k$ and $d$ are constants greater then $0$, ...
0
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1answer
31 views

Heat equation with fourier transformation

I want to understand a solution from an exercise where we should find a solution of the heat equation: $$\frac{\partial u(x,t)}{\partial t}=\sum_{j=1}^{n}\frac{\partial^2 u(x,t)}{\partial x_j^2} $$ ...