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

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Property of a tranform equation

Let $\Omega\subset\mathbb{R}^N, N=2$ or $N=3$ be a bounded smooth domain, $T>0$ and $y=y(x,t)\in L^{\infty}(\Omega\times (0,T))^N$ is a given vector function such that $\nabla\cdot y=0$. Consider ...
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Find the Eigenfunctions and Eigenvalues for $u_{xx} + \frac{2}{x}u_x + u_{yy} + \theta u = 0$

From an old qual exam: Find the eigenfunctions and eigenvalues associated with the BVP \begin{align} &u_{xx} + \frac{2}{x}u_x + u_{yy} + \theta u = 0,\quad 0 < x < a, \quad 0 < y < ...
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Parallel Programming the 1-D dam breakage problem

I am to write a parallel program to simulate the 1D dam break problem by using the Galerkin Equations with WENO limiter. The equations are on domain [0,2000]. At the beginning a dam divides the ...
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Existence of first order PDEs?

It seems the follow theorem is classical, but I don't know how to proof it: For $x\in\Omega\subset R^n$, where $\Omega$ is a domain with smooth boundary, consider the system of PDEs: ...
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24 views

Do there exist nontrivial global solutions of the PDE $ u_x - 2xy^2 u_y = 0 $?

Consider the following PDE, $$ u_x - 2xy^2 u_y = 0 $$ Does there exist a non-trivial solution $u\in \mathcal{C}^1(\mathbb{R}^2,\mathbb{R})$? It is clear that all solutions for $u\in \mathcal{C}^1( ...
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Reasons for convergence

I am interested to know if anyone can see the reasoning behind the convergence $$\int_{\Omega}c(u_{k},\nabla u_{k})(u_{k}-u)dx \rightarrow 0$$ in equation (2.82), page 50 in the following book ...
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Is this really a solution to the PDE?

Suppose we have a function $C(x,\nu,\tau)$ and the PDE $$-C_{\tau}+\frac{1}{2}\nu C_{xx}-\frac{1}{2}\nu C_x+\frac{1}{2}\eta^2\nu C_{\nu\nu}+\rho \eta \nu C_{x\nu}-\lambda(\nu-m)C_{\nu}=0\tag{1}$$ ...
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37 views

A question on deriving d'Alembert's formula from change of variables

Let's assume $x \in \mathbb{R}$ and $t \ge 0$. I am asked to find d'Alembert's formula to the wave equation $$\begin{cases}u_{tt}-u_{xx}=0 & \text{in } \mathbb{R} \times (0,\infty) \\ u = g, u_t = ...
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Equipartition of energy

Let $u$ solve the initial-value problem or the wave equation in one dimension: $$\begin{cases}u_{tt}-u_{xx}=0 & \text{in } \mathbb{R} \times (0,\infty) \\ u = g, u_t = h & \text{on } ...
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+50

Examples of applications of mononotone and pseudomonotone operators

Hi I am aware that the following question is quite broad, but I would appreciate any feedback even if it is in the form a reference. I am interested in some standard examples in engineering (or any ...
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1answer
16 views

Stokes' Rule for an initial-value problem

Assume $u$ solves the initial-value problem $$\begin{cases}u_{tt}-\Delta u = 0 & \text{in } \mathbb{R}^n \times (0,\infty) \\ u = 0, u_t = h & \text{on }\mathbb{R}^n \times \{t=0\}. ...
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1answer
87 views

The well-posedness of Laplace equation on half-space

In $2$ dimension, take $-\Delta u=0$ on $\{(x,y\},y\geq 0\}$ with $u(x,y=0)=f(x)$, $u_y(x,y=0)=g(x)$ where $f$ and $g$ are smooth function. I want to justify whether this problem is well posed. My ...
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1answer
45 views

The Gauss-Green theorem for unbounded domain

This question comes to me when I deal with the following PDE problem. Suppose we have \begin{cases} -\Delta u=0 & x\in \mathbb R^N\setminus B(0,1)\\ u=0 & x\in\partial B(0,1)\\ u\to 0 & ...
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Discretization of the Anisotropic Diffusion Operator for Finite Difference Method

I have to derive and apply a Finite Difference scheme to solve a steady state, anisotropic, diffusion equation. So I have to find a discretization of the following equation $$\nabla \cdot ( ...
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The Neumann Problem on a Half-space when dimension is $2$

Take $\Omega:=\{x=(x_1,x_2):\,-\infty<x_1<\infty,\,x_2>0\}$, i.e., the half-space, and I am interested in the Neumann problem \begin{cases} \Delta u=0&x\in \Omega\\ ...
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Functions over a finite domain that cannot be represented by Fourier series

Given a double Fourier series for some $f:[0,L]\times [0,R]\to \mathbb{R}$ of the following form $$\sum_{k,l=0}^{\infty}a_{kl}\cos\left(\frac{2k\pi ...
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Poisson equation with a implicit nonlinearity [on hold]

Consider the nonlinear Poisson equation for $u=u(x,y)$ as follows: $u_{xx}+ u_{yy} = f(u)$ in $\{x,y \mid x^2+y^2<1\}$ $x u_x+y u_y=0$ on $\{x,y \mid x^2+y^2=1\}$ Here $f(u)$ is a function which ...
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1answer
23 views

Neumann problem for Laplace equation on Balls by using Green function

It is well known that for Dirichlet problem for Laplace equation on balls or half-space, we could use green function to construct a solution based on the boundary data. For instance, one could find a ...
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1answer
18 views

Finite difference Coupled PDE

Can someone help me verify if I'm differentiating this correctly? $$\frac{\partial^2 u}{\partial x \, \partial y} = L_x L_y u$$ Choose to define $L_x$ as central difference $$L_x = ...
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Derive $u(x,t)$ as a solution to the initial/boundary-value problem.

Given $g : [0,\infty) \to \mathbb{R}$, with $g(0)=0$, derive the formula $$u(x,t)=\frac{x}{\sqrt{4\pi}}\int_0^t \frac 1{(t-s)^{3/2}}e^{-\frac{x^2}{4(t-s)}}g(s)\,ds$$ for a solution of the ...
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Solutions of the heat equation of the form $u(x,t)=v(x/\sqrt{t})$

Assume $n=1$ and $u(x,t)=v(\frac{x}{\sqrt{t}})$. (a) Show that $u_t = u_{xx}$ if and only if $$v''+\frac z2 v' = 0. \tag{$*$}$$ Show that the general solution of $(*)$ is $$v(z)=c \int_0^z ...
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A nonlinear Poisson equation problem. (Related to Laplace equation)

Let $\Omega\subset R^N$ be open bounded. Define \begin{cases} \Delta u = f(u) &x\in\Omega\\ u=1 & x\in\partial \Omega \end{cases} Q1: Suppose $f(u)=u^m$ where $m$ is odd. Prove that if there ...
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Rewriting the heat diffusion equation with temperature dependent diffusion coefficient to include joule heating.

I am modelling heat flow in a solid round copper conductor with a set area. I plan to discretize and solve numerically in Python. However, I only have a curve fit for thermal conductivity and specific ...
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1answer
21 views

If $u$ is harmonic, prove that $|Du|^2$ is subharmonic.

We say $v \in C^2(\bar{U})$ is subharmonic if $-\Delta v \le 0$ in $U \subset \mathbb{R^n}$. Prove that $v := |Du|^2$ is subharmonic, whenever $u$ is harmonic. This is Exercise 5, part d, ...
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1answer
28 views

Fourier Integral for signum function.

Define the signum function, $\text{sgn}(x)$, by $$\text{sgn}(x)=\begin{cases} 1, & x>0\\0, & x=0\\-1, & x<0 \end{cases}$$ Establish the identity $$\dfrac{2}{\pi}\int_0^ \infty ...
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33 views

Finding the unique weak solution of Non-linear boundary problem

We are given the equation \begin{cases} -\Delta u=0&x\in \Omega\\ \partial_\nu u+\beta(u)=0&x\in\partial\Omega \end{cases} where $\Omega$ is bounded bounded smooth boundary and $0<a\leq ...
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Growth conditions for partial differential equations

Hi I am interested in what the exact purpose is of growth conditions associated with solving partial differential equations. For example the following pde: $$\text{div}(a(x,u,\nabla u)) + c(c,u,\nabla ...
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Mean-value formulas

From PDE Evans, 2nd edition, pages 25-26. THEOREM 2 (Mean Value Formulas for Laplace's equation). If $u \in C^2(U)$ is harmonic, then $$u(x)=\def\avint{\mathop{\,\rlap{-}\!\!\int}\nolimits} ...
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1answer
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only solution to wave equation under certain restriction?

Suppose $u_1, u_2: \mathbb{R}^2 \rightarrow \mathbb{R}$ are two solutions to the wave equation \begin{equation} \partial_t^2u_i = \partial_x^2u_i \end{equation} obeying the restriction $\partial_x ...
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Need help solving non-linear PDE in 2D using method of characteristics

I've been asked to solve $$\frac{\partial u}{\partial y}\frac{\partial u}{\partial x}=xy$$ given $u=u(x,y)$, and the IC: $u=x$ for $y=0$. What I've done so far is defined $p=u_x$ and $q=u_y$. This ...
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Numerical scheme to 1D advection equation

I am trying to numerically solve a system of equations which model the early universe in 1D. The equations I am stuck on are; $$ (1)\quad \partial\rho/\partial t + \partial(\rho v)/\partial x = 0 ...
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Switch integral and sum for Bessel function.

I haven't real knowledge in Bessel's function and I'd like to know how to switch integral and sum in these two equations. I've already tried a lot of ideas but nothing really works. The first one is : ...
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maximum principle for $p$-Laplace equation

Consider $\Omega \subset R^n$ a bounded domain. Let $\varphi \in W^{1,p}(\Omega) \cap L^{\infty}(\Omega)$. Let $u \in W^{1,p}(\Omega)$ with $\Delta_p u = 0$ in $\Omega$ with $u - \varphi\in ...
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1answer
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singularity and degeneracy of an ODE

I have trouble distinguishing the difference of singularity and degeneracy in the context of ODE theory. Could anyone give me a couple of examples in illustrating the difference of singular point and ...
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1answer
24 views

The tangent hyperplane to the graph of harmonic function

This is an interesting question I found online about Laplace equation. We define a function $u$ :$R^N\to R$'s graph to be the set $\{x,u(x):\,x\in R^N\}\subset R^{N+1}$. Then I want to prove that ...
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How to find the wave equation for a given dispersion relation?

Let's assume I have general wave function written like this: $$ f(x,t) = \displaystyle\int_{-\infty}^{\infty}A(k)e^{i(kx-\omega(k) t)}dk $$ So it's composited from lots of plane waves and each of ...
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$L^2$ regularity of a convolution with newtonian potential.

I am reading Bertozzi, Majda Vorticity and incompressible flow and in page 71 72, we are concerned with recovering the velocity field of a flow from its vorticity. At some point we need to have the ...
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34 views

Show that $\max_{\bar{U}} |u|\le C(\max_{\partial U} |g|+\max_{\bar{U}} |f|)$

Let $U$ be a bounded, open subset of $\mathbb{R}^n$. Prove that there exists a constant $C$, depending on only $U$, such that $$\max_{\bar{U}} |u|\le C(\max_{\partial U} |g|+\max_{\bar{U}} |f|)$$ ...
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59 views

Solving cauchy hyperbolic second order pde

I'm currently taking a course in partial differential equations. I'm trying to solve the following problem (which is, as far as I can tell, a bit above the level of the course): $$\begin{align} ...
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Solving a one dimensional wave equation

Consider the partial differential equation $$\frac{\partial^{2}u}{\partial t^{2}}=c^{2}\frac{\partial^{2}u}{\partial x^{2}}$$ For the region $0<x<\pi$ where $t>0$. With the boundary ...
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can someone help me with method of characteristics please [closed]

Can someone help me with this question please? Please open this photo to see question. Thanks. Please need explain how you solve it. Thanks for looking. pg
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+100

What physical information does the mean value property of heat equation convey?

I'm reading through the Evans' book on PDE, the chapter on heat equation. The definitions are the same as here. I see that mean value property of heat equation is useful for proving maximum principle ...
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Trace map $\gamma:H^1(\Omega) \to H^{\frac 12}(\partial\Omega)$ invertible on $\{ u \in H^1(\Omega) \mid \Delta u =0 \}.$

It is apparently true that the trace map $\gamma:H^1(\Omega) \to H^{\frac 12}(\partial\Omega)$, is invertible when restricted to the domain $$\{ u \in H^1(\Omega) \mid \Delta u =0 \}.$$ I've been ...
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Restriction estimates

What is the defining property of what someone in the harmonic analysis community would call a "restriction estimate?" I see sobolev norms, Fourier transforms, and inequalities relating these. The ...
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Duhamel's principle in constructing heat kernel

I want to construct the heat kernel $k_t(x,y)$, which is the fundamental solution to the heat equation $(\partial_t + \Delta_x)u(t,x) = 0$. There is something that I don't understand about using ...
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Basis of $H^1(\Omega)$ which is orthonormal wrt. $L^2(\partial\Omega)$ inner product?

Let $\Omega$ be a domain with $\partial\Omega$ bounded. Is it possible to find a smooth basis of $H^1(\Omega)$ and $L^2(\Omega)$ which is orthonormal wrt. the $L^2(\partial\Omega)$ inner product? ...
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How to solve $4x^2\cos y\sin y\partial{y}-3x\sin y\partial{x}+8\sin^2y\partial{y}=0$?

$$4x^2\cos y\sin ydy-3x\sin ydx+8\sin^2ydy=0$$ find the solution of this Bernoulli equation. How can I start?
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66 views

The rate of convergence for finite difference methods for Poisson's equation with piecewise constant data

I am solving the following PDE; $$ \nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = \rho, $$ where $\rho(0.5,0.5) = 2$ (zero elsewhere), $0\leq x,y\leq1$ and the ...
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1answer
24 views

Finding the fundamental mode of wave equation in a rectangle

Consider the two-dimensional wave equation: $$\frac{\partial^{2}u}{\partial t^{2}}=c^{2}\left[\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}\right]$$ for some $c>0$. I ...
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1answer
20 views

Solving 2D Laplace eigenfunction equation

I want to solve the equation $$\nabla^{2}\,{\rm P}(x, y) = \frac{k}{c^{2}}{\rm P}(x, y)$$ or $$\frac{\partial^{2}{\rm P}}{\partial x^{2}}+\frac{\partial^{2}{\rm P}}{\partial y^{2}}=\frac{k}{c^{2}}{\rm ...