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

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asking a way to prove an inequality

Assume $\Omega$ is a bounded smooth domain in $\mathbb R^N $ with $N \ge 5 $ and $u \in C^2(\Omega)$ . I want to proof $$\int_{\Omega}\frac{|\nabla u|^2}{|x|^2}d{x} \;\ge\; ...
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1answer
30 views

A classical solution of Poisson's equation is also a weak solution

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain and $u\in C^0(\overline{\Omega})\cap C^2(\Omega)$ be a solution of $$\left\{\begin{matrix}-\Delta u&=&f&&\text{in }\Omega\\ ...
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1answer
51 views

Prelim problem $\sup_{\bar\Omega}|\nabla u|^2\leq C(\sup_{\Omega}u^2+\sup_{\Omega}f^2+\sup_{\Omega}|\nabla f|^2)+\sup_{\partial\Omega}|\nabla u|^2$

This is an old prelim problem. Let $\Omega \subset \mathbb{R}^n$ open and bounded with smooth boundary. If $u\in C^3(\bar\Omega)$ solves $$ -\sum_{i,j=1}^n a_{ij}(x)\,u_{x_ix_j}(x)=f(x)\quad ...
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7 views

Spectral collocation

I'm new to spectral methods, and I'm trying to solve an advection-diffusion equation in 1d: $\frac{\partial u}{\partial t}=A \frac{\partial u}{\partial x} + B \frac{\partial ^2u}{\partial x^2} $ I ...
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1answer
17 views

How to solve a simple differential equation in the way of weak solutions?

I want to prove that the solution equation $y'=y$ is $y=Ce^x$, where $C$ is a constant. Here $y$ belongs to the space of linear operators on $C_0^\infty(\mathbb{R})$, and $y'$ is its weak ...
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11 views

Poisson partial differential equation under Neumann boundary conditions

I'm trying to find solutions for the Poisson equation under Neumann conditions, and have a couple of questions. More specifically, I'm interested in the gradient of the function $\phi(x)$ in a space ...
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34 views

Find a function that plays the same role as distance function?

If $E$ is a manifold , then the distance is smooth outside $E$. For a compact set whose capacity is zero, can we find a function that plays the same role as the distance function? I.e., can we find ...
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1answer
27 views

Help understanding variable substitutions

The concrete problem is this Let $$u(t,x)=v(\frac{x^2}t)$$ on $\mathbb R^+\times\mathbb R$. Show that $$u_t=u_{xx} \Leftrightarrow 4zv''(z)+(z+2)v'(z)=0$$ Now, while I would like to know the ...
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38 views

Solving higher order pdes

So here's my problem, while you solve the Euler Bernoulli beam Equation by separation of variables, how do I have to prove the separated function of space are orthogonal? If so, are hyperbolic sines ...
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1answer
18 views

non constant harmonic function

If $u$ is harmonic function on disk with radius $R$ around the origion, and non constant in it. why is it true that $u$ cannot be constant in any sub-Disk (i.e disk with radius less than $R$) thanks ...
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2answers
36 views

What is the purpose of the weight function $w(x)$ in a Finite Element Method?

I have just started looking into finite element methods. Suppose we have an equation for the strong $$L(u) = s$$ Then the integral form of the equation is given by $$\int_0^1 L(u)w(x) dx = ...
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22 views

The one dimensional wave equation.

Find the deflection $y(x,t)$ of the vibrating string of length $\pi$ and ends fixed, corresponding to zero initial velocity and initial deflection $f(x)=k*(\sin x - \sin 2*x)$, given $c^2 = 1$. I ...
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1answer
23 views

Is function $u$ nice when all $\Delta^k u$ are nice?

Let $\Omega \subset\mathbb{R}^d$ has smooth boundary and $$\Delta^k u \in W^{2,2}(\Omega)\cap W^{1,2}_0(\Omega) \qquad k\geq0$$ Show that $u\in W^{n,2}(\Omega)$ for all $n\in \mathbb{N}$. This ...
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13 views

an estimate about the Besove space?

Recently, I study a book about the Besov space and nonlinear partial differential equations, http://link.springer.com/book/10.1007%2F978-3-642-16830-7, whose authors are Hajer Bahouri, Jean-Yves ...
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2answers
27 views

A solution of a differential equation of first order in the large-variable limit

The differential equation reads: $ \dfrac{\partial R (t)}{\partial t} = \dfrac{c_2}{R^2} + \dfrac{c_3}{R^3} + O(R^{-4})$, Where $c2 > 0$ and $c3 > 0$, how to get the solution of the ...
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1answer
19 views

Differential spherical wave equation, why is the result the same for real and imaginary parts?

I wasn't very sure whether to ask this in the physics forum or here, but the question regards mathematics much more than it does physics. The following wave function is given (spherical wave): ...
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15 views

Prequisites for a PDE course (Strauss)

This question is quite general. In four days I will enroll in a PDE course which will use Strauss as the textbook. However, today when I checked the course description I found that 'multivariable ...
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7 views

Parabolicity of high order PDEs

I know that the traditional classification of PDEs into parabolic, elliptic, and hyperbolic is applicable for the second order equations. However, I often see remarks about parabolicity of higher ...
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1answer
15 views

Hamiltonian Constant on integral curves

Let $H \in C^{2}(\mathbb{R}^2)$ and let $(x(t),y(t))$ be a solution to the equations $$\frac{dx}{dt} = \frac{\partial}{\partial y} H(x(t),y(t))$$ $$\frac{dy}{dt} = -\frac{\partial}{\partial x} ...
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1answer
39 views

Issues with solving PDE

It's been a while since I've had to solve the heat equation, and so I am having a slight issue. The question is as follows: A long, hollow, rigid tube, of length $L$ and constant cross section is ...
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23 views

All possible flat conformal metrics of dimension greater than 2

Combining List of formulas in Riemannian geometry and Conformal symmetry, is there a proof which states $$ x^\mu \to \frac{x^\mu-a^\mu x^2}{1 - 2a\cdot x + a^2 x^2} $$ represents all possible ...
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1answer
13 views

How can I conclude this “gluing property” for these Sobolev functions?

Let $u \in W^{1,2}(\Omega) \cap C(\Omega)$, where $\Omega$ is an open bounded domain in $R^n$ with smooth boundary.Let $B(x,R) \subset \overline{B(x,R)} \subset \Omega $ a ball. Consider $u^{\star} ...
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1answer
24 views

solving the Laplace equation

I want to know that: What is the general solution of the (2 and 3 dimentional)Laplace equation $f_{xx}+ f_{yy}=0$ and $f_{xx}+ f_{yy} +f_{zz}=0$? How can we solve the Laplace equation? With many ...
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1answer
28 views

Fourier transform of PDE on finite and infinite bound simultaneously.

Consider $$u_{xx} + u_{yy} = 0 $$ on the bounds: $$o < x < L$$ and $$-\infty<y<\infty$$ The initial condition is: $$u(0,y) = f(y)$$ and $$u(L,y)=g(y)$$ I've tried performing fourier ...
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1answer
26 views

mean value theorem for heat equation

Hi I am looking at the proof of mean value property for heat equations (evans chapter 2 theorem 3) Again, I got through every step except the very last line of the proof $$\phi(r)=\lim_{t\to ...
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27 views

Bounding a sum on lattice points in an annulus

How do we bound $\Sigma_{\beta<|\vec{k}|<\alpha}|\vec{k}|^{-2}$ by using an integral comparison type method ? How about $\Sigma_{\beta<|\vec{k}|<\alpha}|k|^{-4}$ Here ...
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2answers
43 views

Significance of Sobolev spaces for numerical analysis & PDEs?

I never had an option to take a Functional Analysis module. I am tied up with other work for the next two months so I won't get a chance to self-study it until September. So one thing I was wondering ...
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14 views

Hadamard's method of descent

I would like to apply the Hadamard's method of descent to calculate the solution of the homogeneous wave equation in one dimension descending from the solution in two dimensions. This is the ivp in ...
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1answer
43 views

Wave equation- solution extincts in time

I am currently dealing with the following task: Let $u$ be a solution to the wave equation on $\mathbb{R}^3 \times (0,\infty)$ with initial conditions $u(x,0)=g, u_t(x,0)=h$ where $g,h \in ...
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Conformal transformation of a region bounded by a curve $y=x^a, a \in \mathbb{R}$

I would like to solve the 2D Laplace equation $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} =0$ on the positive upper half plane: $0 <x<\infty$ and $0 < y < ...
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1answer
40 views

How to solve 2D Laplace Equation over an infinite rectangular strip (bounded on two edges), with Dirichlet boundary conditions

Is it possible to solve Laplace equation $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}=0$, over an infinite rectangular strip defined by $0 < x < \infty$ and $0 < y ...
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1answer
32 views

Applying Fourier transform to heat equation with source

I haven't had any experience with applying of FT to heat equation with source. But this popped up in an exercise. Any help in the right direction would be great. Consider: $$\frac{\partial ...
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Please help: My MATLAB code for solving a 2D Schrödinger equation keep giving me weird output.

I've been trying to solve the following Schrödinger equation numerically, \begin{equation} -(\frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2})\Psi + \frac{\sinh^2(y) + ...
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1answer
40 views

Are all IVP's and BVP´s essentially eigenvalue/eigenfunctions problems?

Are all IVP's and BVP´s essentially eigenvalue/eigenfunctions problems, or does there exist some IVP or BVP which is not of this "nature"? (By "essentiell" I mean "can be regarded as")
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About the standard textbook example for Laplace Equation and Separation of Variables

Here is a standard example typically cited to illustrate the technique of separation of variables. Suppose we have a semi-infinite rectangular strip formed by boundaries $x=0, y=0, y=a$ and we are to ...
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28 views

Solution to a pde

I have a PDE system that I am trying to solve at steady state. When I make the appropriate substitutions, I get an equation of the following form: $$\frac{1+M}{M}\frac{d^2 M}{d x^2}=1$$ Is there a ...
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1answer
32 views

ODE's & PDE's: Homogenous piecing vs Eigenexpansion vs Green functions

I don't know if i'm within rules of the forum to ask this question. If i'm not please comment before downvoting. If you know of a source that answers these questions, please suggest. It would be ...
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1answer
18 views

How to show that the Poisson equation has its maximum on the boundary?

$$\Delta f=h, x\in\Omega$$ $$f=F, x\in\partial \Omega$$ where $h$ is $C^1$ function s.t. $0\le h$, $0\le h'$ and the domain is in 3D. then, I want to show that $f$ has its max on boundary. please ...
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25 views

General solution of laplace equation

I want to know that: What is the general solution of the (2 and 3 dimentional)Laplace equation $f_{xx}+ f_{yy}=0$ and $f_{xx}+ f_{yy} +f_{zz}=0$? With many thanks for your help.
0
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1answer
14 views

Simplification of this fourier transform signum function

Given this equation: $$\frac{-1}{4c}[\int_{ -\infty}^{\infty}g(\varpi)Sgn(x - ct - \varpi).d\varpi -\int_{-\infty}^{\infty}g(\varpi)Sgn(x+ct - \varpi ).d\varpi ]$$ Where sgn is the signum function, ...
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9 views

lattice variation, cylindrical discretisation of PDE

Given an energy functional $ E=\int_{0}^{\infty} \,dr.r\left[\frac{1}{2}\left(\frac{d \phi}{dr}\right)^2 - S.\phi\right] $, I am told that discretizing on a lattice $ r_{i}=ih $ (h=lattice size, i is ...
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6 views

mean value formula for weak solution of Laplace equation

Let $u \in H^{1}(\Omega)\cap C(\Omega)$ ($\Omega$ an open and boundad domain in $R^n$ with smooth boundary). Consider a ball $B(x,R) \subset \overline{B(x,R)} \subset \Omega$. Let $u^{\star} \in ...
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37 views

Prove using Green's theorem that the boundary value problem has at most one solution

Prove using Green's theorem that the boundary value problem $$\frac{\partial}{\partial{x}}\left ( (1+x^2)\frac{\partial{u}}{\partial{x}}\right )+\frac{\partial}{\partial{y}}\left ( ...
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1answer
30 views

Partial differential equation help

I have a linear pde $ 3U_{xx} -2U_{xt} -U_{tt} = 0 , -\infty < x > \infty$ I have solved this and found the general solution which is $ U(x,t) = F(x-3t) + G(x+t) $ G ,F are ...
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84 views

Why is my solution “not enough”?

The question is as follows Let $G(x,y)$ be the Green's function for a bounded domain $\Omega$. Prove that $G(x,y)<0$. My solution is in the following picture, where the professor marks "This ...
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1answer
32 views

Counterexample to Rellich-Kondrachov Compactness Theorem, case $q=p^*$

I was searching some counterexample for I was searching some counterexample for Rellich-Kondrachov Compactness Theorem (You can see: PDE, Evans, chapter 5), for the case $q=p^*$ and I found this ...
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Initial and boundary value problem

I have to find the solution of the initial and boundary value problem $$u_{tt}(x,t)-u_{xx}(x,t)=1+\sin x, x \in (0,1), t>0 \\ u(x,0)=\sin x, x \in (0,1) \\ u_t(x,0)=0, x \in (0,1) \\ ...
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41 views

The Dirichlet problem for the Laplace equation: classical solutions versus weak solution

Let $B_R$ a ball in $\mathbb{R}^n$. Consider $u^{\star} \in H^{1}(B_R) $ and $f \in H^{1}(B_R) \cap C(\overline{B_R})$. Suppose that $u^{\star}$ minimizes $$\int_{B_R} |\nabla u|^2, u \in \{ v \in ...
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1answer
22 views

partial differential equations , first order help

Consider the linear first order non-homogeneous partial differential equation $U_x+yU_y-y=ye^{-x}$ By using the method of characteristics show that its general solution is given by ...
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23 views

How to apply Cauchy-Kowalevsky Theorem.

The Cauchy-Kowalevsky theorem is stated in my notes as: For the Cauchy problem: $$ \begin{cases} u_{y}=F(x,y,u,u_{x}) \\ u(x,0)=h(x) \end{cases} $$ If $h$ is analytic in a neighborhood of ...