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

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70 views

Differential equation with the solution of $(1+ax/2)\exp(-ax)$

Is there any linear differential equation which has following solution $$y=(1+ax/2)\exp(-ax)$$ $a$ is constant.
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2answers
109 views

On the abstract bootstrap principle in the book “Nonlinear Dispersive Equations” by Terence Tao

In Terence Tao's book "Nonlinear Dispersive Equations", he gives the following "Abstract bootstrap priciple": "Let $I$ be a time interval, and for each $t \in I$ suppose we have two statements, a ...
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1answer
54 views

Exists $C$ constant: $||f||_{L^{\infty}(\mathbb{R})}\leq\{ ||f||_{L^2(\mathbb{R}}+||f'||_{L^2(\mathbb{R})} \}, \forall f\in S(\mathbb{R})$

Show that there exists $C$ constatant such that $||f||_{L^{\infty}(\mathbb{R})}\leq\{ ||f||_{L^2(\mathbb{R}}+||f'||_{L^2(\mathbb{R})} \}, \forall f\in S(\mathbb{R})$. This is a question in my ...
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1answer
67 views

show a PDE has no distribution solution in $\mathbb{R^2}$\{0}

" Consider the following equation in the plane $x\frac{\partial u}{\partial y}-y\frac{\partial u}{\partial x}=f(x^2+y^2)$ where $f(t)$ is a $C^\infty$ function of the real variable $t$ such that ...
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2answers
47 views

PDE course question

What courses do I need for a course in partial differential equations? My university has a prereq of Multivariate Calculus and Ordinary Differential Equations. However, I opened up a book on pdes in ...
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2answers
878 views

Show that the Poisson kernel is harmonic as a function in x over $B_1(0)\setminus\left\{0\right\}$

Show that the Poisson-kernel $$ P(x,\xi):=\frac{1-\lVert x\rVert^2}{\lVert x-\xi\rVert^n}\text{ for }x\in B_1(0)\subset\mathbb{R}^n, \xi\in S_1(0) $$ is harmonic as a function in $x$ on ...
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1answer
100 views

Regularity of ultra-weak solutions

Suppose to have in the sense of distributions $$ -\Delta u = f \in L^q (B_1(0)) $$ where $B_1(0)\subset R^2$ and $q>1.$ Can I infere that $u\in L^{\infty}(B_1(0))$? Which results could I use? ...
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2answers
96 views

Book searching in Elliptic Equation

I am learning a course with the subject of Elliptic Equations. If you know about it, please recommend me a book on Elliptic Equations. And if that's possible, someone post these books/author/...that ...
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1answer
51 views

Divergenceless proving

We can define a topological current, \begin{equation} J_{top}^u = \frac{1}{2v} \epsilon^{\mu \nu} \partial_\nu \phi \end{equation} How to prove it divergenceless? where the the condition is ...
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1answer
145 views

Analytical Solution for Elastic Bar under applied end velocity

Say, a thin long rod is occupying the space $[0,L]$. It's isotropic, linear elastic, homogeneous. The partial differential equations for stress $\sigma(x,t)$ and displacement $u(x,t)$ are as follows ...
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2answers
75 views

Dirichlet problem on $[0,1] \times [0, \pi]$

Let $\Omega := [0, 1] \times [0,\pi]$. We are searching for a function $u$ on $\Omega$ s.t. $$ \Delta u =0 $$ $$ u(x,0) = f_0(x), \quad u(x,1) = f_1(x), \quad u(0,y) = u(\pi,y) = 0 $$ with $$ f_0(x) ...
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1answer
182 views

Vector analysis: $(\vec v \cdot \vec \nabla) \vec v=(\vec \nabla \cdot \vec v) \vec v$?

If I know that $\vec \nabla \cdot \vec v=0$, can I say that: $$( \vec v \cdot \vec \nabla )\vec v=\underbrace{(\vec \nabla \cdot \vec v)}_{=0} \vec v=0 $$ ? Note: this is a question I asked in ...
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2answers
94 views

About smooth approximation in a Sobolev space

I want to prove the following fact : Consider $\Omega \subset R^n$ a bounded and open set. Let $v \in H^{1}_{0}(\Omega)$ a nonnegative function. Then exists a sequence $v_m$ in ...
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1answer
401 views

Constants in Laplace's equation for a cube

I'm working a Laplace's equation $\Delta F=0$ for a cube in Cartesian coordinates $((0,0,0),(a,a,a))$ and after separation I have $$\frac{X''(x)}{X(x)} +\frac{Y''(y)}{Y(y)}+\frac{Z''(z)}{Z(z)} = ...
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1answer
318 views

How to determine if the sums and products of harmonic functions is also harmonic?

Suppose I know that $u + iv$ is non-constant and analytic on a domain $D$, then I know that $u$ and $v$ are harmonic on $D$ and not both constant; but how do I then determine whether $3u^2v - v^3 + ...
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2answers
188 views

Navier-Stokes system

I have to study this system which name is Navier-Stokes. Can you explain please what means that $p$, $u$ and $(u \cdot \nabla)u$. What represents in reality ? Tell me please, how shoul I read the ...
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2answers
140 views

A Poisson Equation: how to solve it?

I wish to solve the Poisson Equation:$$u_{xx}+u_{yy}=1,$$with $u(x,y)$vanishing on $r=a$. I know that the final solution should be the sum of the homogenous part and the non-homogeneous part, which ...
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2answers
152 views

Find the general solution of the PDE $ 2u_x-3u_y = x$

Find the general solution of the PDE $2u_x-3u_y=x$ , where $u=u(x,y)$ . We have to use the method of characteristics. I know that the slope is $-\dfrac{3}{2}$ and the characteristic lines are ...
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2answers
711 views

PDE problem: heat equation with periodic BC

I need help with this exercise: Given the PDE $$u_t=-10u_{xx}\tag{1},$$ with periodic boundary conditions in $[-1,1]$: $$u(-1,t)=u(1,t), \qquad u_x(-1,t)=u_x(1,t).$$ A) Obtain the solution $u(x,t)$ ...
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2answers
1k views

Fundamental solution of Heat equation

For every bounded function $u_0\in C(\mathbb R)$ the function $$u(t,x)=\frac{1}{\sqrt{4\pi t}}\int_{-\infty}^{\infty}e^{-\frac{-(x-y)^2}{4t}}u_0(y)dy$$ is a solution of $\dfrac{\partial v}{\partial ...
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2answers
83 views

Mid-step in solving a pde

A mid-step in solving a pde requires integrating the function $$f'(-x) = -2x + e^x$$ with respect to $x$. What do you do differently when integrating a function of $-x$ rather than of $x$? Am I ...
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1answer
345 views

Partial integro-differential equation

I don't know if there is a method to solve this following integro - differential equation: $$\partial_t{u(x,t)}\int_{0}^{x}u(\zeta,t)d\zeta=\partial_{xx}u(x,t)$$ Can someone give me some hint? ...
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2answers
738 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
190 views

Laplace's equation

I am very confused. So I have Laplace's equation $\nabla^2\phi(x,y)=0$ and B.C.'s $\phi(x,0)=f(x); \,\,\,\,\,\, \phi(x,1)\equiv0$ where I have to solve it by Fourier transform. So I take the ...
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2answers
59 views

Solving Simple Partial Differential Equation

I can't solve this partial differential equation. $$x\frac{\partial \phi}{\partial x}+y\frac{\partial \phi}{\partial y}+ (\alpha+1-x)\phi =0$$ The short answer in the book which i read from it , ...
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1answer
20 views

An easy computation of an integral

I have a question about an integral. I don't know how to quick compute $$\int_{\mathbb{R}^n}\phi(x,1)|x|^2dx$$where$$\phi(x,1)=(4\pi)^{-\frac{n}{2}}e^{-|x|^2/4}$$ which is the fundamental solution of ...
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1answer
45 views

Mass conservation for heat equation with Neumann conditions

I'm trying to implement Neumann boundary conditions to solve the heat equation with an explicit scheme. I checked mass conservation, but it doesn't seem to hold. Could someone check if my boundary ...
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2answers
21 views

Quasilinear PDE:$(x^2+1)u_x +2xu_y = 0, u(0,y)=\phi(y)$

I am trying to solve the PDE $$(x^2+1)u_x +2xu_y = 0, u(0,y)=\phi(y)$$ The solution is given here at the end of the third page and begining of the fourth and the author concluded at a step that ...
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2answers
42 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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2answers
42 views

Characterization of $H^k$ by Fourier transform

Let $u\in H^k(\mathbb{R}^n)$ a function with complex-valued. Question 1: If $u\in H^k(\mathbb{R}^n)$ and $D^\alpha u\in L_2(\mathbb{R}^n)$, by definition for each $|\alpha|\le k,$ then we have ...
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2answers
20 views

A general solution of a partial differential equation with $f(x,y)$

I need to find a general solution to such a PDE: $$u_x-u_y=f(x,y)$$ I am able to find a solution if $f(x,y)=0$ or $f(x,y)=u$. But I have no idea how to get the general solution. Has anybody got any ...
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1answer
30 views

Gradient of a solution of an elliptic equation on the boundary

Let $u\in C^2(B_1(0))\cap C^1(\overline{B_1(0)})$, which satisfies $$\triangle u(x)=f(x)\qquad x\in B_1(0)$$ $$u(x)=0\qquad x\in\partial B_1(0)$$ Prove $$\left|\frac{\partial u}{\partial ...
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1answer
57 views

explicit solution to linear PDE — boundary value problem

I am looking for an explicit, closed form expression for the solution to a boundary value problem given by the linear PDE: $$ u_{xx}+au_y-bu+c=0 $$ with the BCs $ u_x(-g,y)=ku $, $u_x(g,y)=-ku$, $ ...
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2answers
50 views

Uniqueness and Existence of $u^{\prime\prime}(x) + u^{\prime}(x) = f (x)$ with conditions.

Here is my question: Given $u^{\prime\prime}(x) + u^{\prime}(x) = f (x)$ with the conditions $u^\prime(0) = u(0) = {1\over 2}[u^\prime(l) + u(l)]$ Where $f(x)$ is a given function, is the solution ...
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3answers
79 views

Symmetry of PDEs

I am trying to solve the following problem: Show that $$(u,t,x) \rightarrow (u, \frac{t}{c^2t^2 - x^2}, \frac{x}{c^2t^2 - x^2})$$ is a symmetry of the Wave equation $u_{tt} - c^2u_{xx} = 0$. Some ...
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2answers
42 views

First order partial differential equation

How to find L if the form is: $$\left(\frac{\partial L}{\partial x}\right)^2-\left(\frac{\partial L}{\partial y}\right)^2=-1$$ The author wrote, $$L=y+ax^2+...$$ but I didn't get how? Edit: where L ...
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2answers
72 views

Finding a solution for $x\frac{\partial u}{\partial x}+2y\frac{\partial u}{\partial y}=x^2$

To find a solution for $x\frac{\partial u}{\partial x}+2y\frac{\partial u}{\partial y}=x^2$ knowing that $u(x,y)=1$ if $xy=1.$ I thought it may be useful to do the change $v=\log x, w=\frac{1}{2}\log ...
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1answer
50 views

Laplace equation in polar coordinates

Solve the Laplace equation in polar coordinates $u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta\theta}=0$ within the domain $0<\theta<\pi, 1<r<2$ subject to boundary conditions ...
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1answer
144 views

Weak derivative as an $L^2$ limit of the difference quotient

Let $u \in H^1(\mathbb{R})$. Show that $$\left\| \frac{u(x+h)-u(x)}{h} - u' \right\|_2 \to 0\quad \text{ as } h \to 0, $$ where $u' \in L^2(\mathbb{R})$ is the weak derivative of $u$. In other words, ...
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3answers
116 views

PDE - Energy - Wave Equation

I dont know how solve a) and b), I'm read the book of Walter Strauss, but I have a lot of doubts, for the c) first, I tried estimate $(k(t)e^{-2at})'$ and integrate the inequal, but not unwind... :( ...
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1answer
56 views

Analytic solution for a type of PDE systems

Peace be upon you, I have the following system of partial differential equations \begin{align*} \begin{cases} \frac{\partial}{\partial a}S(a,b,c,d)=f_1(a)\\ \frac{\partial}{\partial ...
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1answer
90 views

Boundary conditions for the time-independent Schrödinger equation on the sphere

if you have a free Schrödinger operator on a sphere $-\Delta \psi(\theta,\phi) = E\psi(\theta,\phi),$ then the substitution $\psi(\theta,\phi) = f(\theta)e^{i n \phi}$ leaves you with the ...
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1answer
45 views

Partial differential equation $3u_y+u_{xy}=0$

I am only starting my PDE course and I have problems solving this easy equation. $$3 \frac{\partial u}{\partial y} + \frac{\partial ^2 u}{\partial x \partial y} = 0$$ Here's what I've tried: ...
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1answer
39 views

Counterexample for existence of a minimiser in a variational problem

I'm trying to find an example of a minimisation problem of the form $$ \inf \{ J(u) := \int_{\Omega} f(x)|u(x)| + |\nabla u(x)|^2:\, u \in H^1, \, \int u = 1\}$$ with $\Omega$ an open and bounded ...
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1answer
185 views

Waves on the Half Line - difference between Dirichlet and Neumann problems?

I'm having trouble finding the solution of the one-dimensional wave equation for $x \in (0,\infty) $ and $ t > 0 $ with initial conditions: $$u(x,0) = f(x), \space u_t(x,0) = g(x), \space x > 0 ...
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2answers
54 views

Find a general solution of the PDE for $u=u(x,y)$ by using ODE techniques.

Find a general solution of the PDE for $u=u(x,y)$ by using ODE techniques. This is a simple one, but I got stuck towards the end. I wanted to use separation of variables because it was the easiest ...
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1answer
112 views

What method can be used for solving this fokker Planck equation and how?

Let's have this equation: \begin{equation} \frac{\partial p(x,t)}{\partial t} = - a \frac{\partial p(x,t)}{\partial x} + \frac{1}{2} b \frac{\partial^2 p(x,t)}{\partial x^2} \end{equation} a and b ...
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1answer
113 views

Mean value proof in Evans PDE

Here is the proof I don't really understand about the part beginning using Green's formula. How can Du(y) become du/dv . Is is using the directional derivative formula ? Aslo how can you get/pull ...
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1answer
72 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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1answer
37 views

Sturm Liouville form

How do you put $u'' +c u' +d =0$ into regular SL-form? Can not see how it's an eigenvalue problem without a first order term. But the theorem states EVERY second order operator can be put into SL ...