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

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2
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3answers
509 views

how to solve $ {\partial u \over \partial t} - k {\partial ^2 u \over \partial x^2} =0$ [duplicate]

How do I solve the following PDE for it's general solution? $$ {\partial u \over \partial t} - k {\partial ^2 u \over \partial x^2} =0$$ How do I determine the general the solution of this equation ...
1
vote
4answers
140 views

Method of characteristics. Small question about initial conditions.

Okay, so we're given a PDE $$x \frac {\partial u} {\partial x} + (x+y) \frac{\partial u} {\partial y} = 1$$ with initial condition: $u(x=1,y)=y$ So $a=x, b=x+y, c=1$ $\Rightarrow$ ...
10
votes
2answers
748 views

Sobolev space is an algebra

How do you prove that the Sobolev space $H^s(\mathbb{R}^n)$ is an algebra if $s>\frac{n}{2}$, i.e. if $u,v$ are in $H^s(\mathbb{R}^n)$, then so is $uv$? Actually I think we should also have ...
3
votes
1answer
281 views

PDE - solution with power series

I am learning this method for solving PDSs by means of power series. Since I am studying it with lecture notes and I can't find any other book that describes this method, I am going to summarize it ...
3
votes
1answer
158 views

Solution of a differentiation in integral form

How will I get the solution in the form of integration $$ \phi (0,t)=\frac{R^{3}}{2}\frac{A}{\sqrt{\pi }}\int_{0}^{\infty }k^2e^{-R^{2}k^{2}/4}\cos (\sqrt{k^2+2} t)\ dk. $$ from the equation, when ...
19
votes
2answers
754 views

Applications of Pseudodifferential Operators

I am very interested in just about anything that has to do with PDE's, and inevitably pseudodifferential operators comes up. Its obvious that such a novel way of looking at PDE's would be important, ...
6
votes
2answers
456 views

Explaining the method of characteristics

I am learning about solving p.d.e.s by the method of characteristics at the moment. I was given an "algorithm" to solve these problems but I want to know also what is going on, how it works and what ...
4
votes
1answer
525 views

Sobolev embedding for $W^{1,\infty}$?

From the Evans'PDE book I learned $W^{1,\infty}(U)$ coincide with Lipschitz continuous $C^{0,1}(U)$ with $U\in\Bbb{R}^n(n\geqslant1)$ is bounded and $\partial U\in C^1$. I wonder the counterpart ...
4
votes
2answers
280 views

Change variables into Fokker-Planck PDE

I have a question regarding a PDE (Fokker-Planck) and change of variables. I have a problem deciding what route to take after I use the chain route. I have an expression $$\frac{\partial u}{\partial ...
2
votes
0answers
233 views

The canonical form of a nonlinear second order PDE

Can anyone help me find the canonical form of $$x^2u_{xy} - yu_{yy} + u_x - 4u = 0?$$ I don't know how to solve it because $a = 0$. I just got that it's hyperbolic since $a=0$ , $b =(x^2)/2$, $c= ...
1
vote
1answer
3k views

Solve Burgers' equation

Solve Burgers' equation $$u_t + uu_x =0,$$ with $u=u(x,t)$ and the side condition $u(x,0)=x$. I am not sure on how to find the solution $u(x,t)$. I have learned the method of characteristics. ...
84
votes
8answers
38k views

Has Prof. Otelbaev shown existence of strong solutions for Navier-Stokes equations? [closed]

Moderator Notice: I am unilaterally closing this question for three reasons. The discussion here has turned too chatty and not suitable for the MSE framework. Given the recent pre-print ...
25
votes
7answers
8k views

Good 1st PDE book for self study

What is a good PDE book suitable for self study? I'm looking for a book that doesn't require much prerequisite knowledge beyond undergraduate-level analysis. My goal is to understand basic ...
3
votes
1answer
562 views

Indication on how to solve the heat equations with nonconstant coefficients

I was just wondering how to start solving a heat equation with non-constant coefficients like $$u_t-(x^2u_x)_x=0, \quad x\in (1,e), \quad t>0$$ $$u(1,t)=u(e,t)=0, \quad t>0$$ $$u(x,0)=u_0 ...
5
votes
2answers
136 views

Sobolev spaces - about smooth aproximation

Consider $\Omega $ a open and bounded set of $R^n$ . Let $u \in H^{1}(\Omega)$ a bounded function. I know that exists a sequence $u_m \in C^{\infty} (\Omega) $ where $u_m \rightarrow u$ in ...
2
votes
0answers
318 views

About finding the general solution of first-order totally nonlinear PDE with two independent variables

Recently I feel very interested about finding the general solution of first-order totally nonlinear PDE with two independent variables. However, most PDE books only discussed little about finding the ...
1
vote
1answer
106 views

Finding the uniquely determined region of a PDE

(a) Solve the equation $yu_x+xu_y=0$ with the condition $u(0,y) = e^{-y^2}$. (b) In which region of the xy plane is the solution uniquely determined? I did the first part but I don't understand ...
27
votes
1answer
4k views

Mathematical precise definition of a PDE being elliptic, parabolic or hyperbolic

what is the general definition for some partial differential equation being called elliptic, parabolic or hyperbolic - in particular, if the PDE is nonlinear and above second-order. So far, I have ...
43
votes
1answer
2k views

What is the solution to Nash's problem presented in “A Beautiful Mind”?

I was watching the said movie the other night, and I started thinking about the equation posed by Nash in the movie. More specifically, the one he said would take some students a lifetime to solve ...
12
votes
3answers
792 views

Do discontinuous harmonic functions exist?

A function, $u$, on $\mathbb R^n$ is normally said to be harmonic if $\Delta u=0$, where $\Delta$ is the Laplacian operator $\Delta=\sum_{i=1}^n\frac{\partial^2}{\partial x_i^2}$. So obviously, ...
5
votes
1answer
1k views

Variational formulation of Robin boundary value problem for Poisson equation in finite element methods

So I am confused about some details of obtaining a variational formulation specifically for Poisson's equation. I am in a Scientific Computing class and we just started discussing FEM for Poisson's ...
4
votes
2answers
1k views

Find the Green's Function and solution of a heat equation on the half line

Consider the heat equation on the half line $$u_t = ku_{xx},\quad x > 0,\, t > 0,\\ u(x,0) = 0, \,x \in\mathbf R,\\ u(0,t) = \alpha(t),\, t > 0. $$ Find Green's function and the solution. ...
2
votes
2answers
424 views

Boundaries in heat equation

I have this heat equation: $u_t = 9u_{xx} - 7u + f(x,t),$ $f(x,t) = 1; 0 < x < l; 0 < t < T $ $u(x,0) = 6x^2 - 5x +2$ $u(0,t) = 3t + 2$ $u(l,t) = t + 3$ $l = 1$ My problem is ...
3
votes
3answers
142 views

If $\frac{\partial \varphi}{\partial x}=f(x,y),\frac{\partial\varphi}{\partial y}=g(x,y)$, what is $\varphi$?

Suppose we have a real-valued function $\varphi(x,y)$ such that $$ \frac{\partial \varphi}{\partial x}=f(x,y)\quad\text{and}\quad\frac{\partial\varphi}{\partial y}=g(x,y) $$ for some functions $f$ and ...
2
votes
1answer
109 views

Differentiating the determinant of the Jacobian of a diffeomorphism (don't understand a proof)

For each $t$, let $A_t:\Omega_0 \to \Omega_t$ be a bi-Lipschitz map between open sets in $\mathbb{R}^n$. The map is also invertible. It satisfies $$\frac{d}{dt}A_t(y) = w(A_t(y),t)$$ where $w$ is a ...
5
votes
2answers
269 views

Regularity of an infinite series arising with the heat equation

Let $(t,y)\in(0,\infty)\times\mathbf{R}$, and $\displaystyle f(t,y) \equiv \sum_{k=-\infty}^{\infty}\frac{\exp(-(y-2\pi k)^2/2t)}{\sqrt{2\pi t}}$. This infinite series arises if one attempts to solve ...
4
votes
3answers
366 views

Convergence of a sequence of periodic functions

Motivated by the homogenization theory which studies the effects of high-frequency oscillations in the coefficients upon solutions of PDE, I am thinking about the following question. Let the ...
2
votes
2answers
213 views

variational problem-exercice

Let $\Omega$ an open bounded connexe and regular, and let $f \in L^2(\Omega)$ We consider the variational problem: find $u \in H^1(\Omega)$ such: $$\displaystyle\int_{\Omega} A \nabla u \cdot \nabla v ...
2
votes
1answer
231 views

Finding an analytical solution to the wave equation using method of characteristics

Okay so I am super confused on what the method of characteristics is and what it means geometrically. So my first question is if anyone could kindly explain what characteristic lines are, why its ...
1
vote
1answer
193 views

The issue of treating an inverse Fourier transform in terms of a tempered distribution.

Consider the wave equation $$ u_{tt}=\Delta{u} \quad u(x,0)=f(x) \quad u_t(x,0)=g(x) \tag{*} $$ A solution to this equation is given by $$ u(.,t)=f*\partial_t\Phi_t+g*\Phi_t \tag{**} $$ where ...
1
vote
2answers
149 views

How to model multi-step cell differentiation

Can I better explain cell lineages using PDEs or stochastic?
0
votes
2answers
219 views

Simplifying PDE

I want to ask you a question. For example I have an equation: $$u_{tt}-7u_{xx}-u_{x}=0 $$ To solve it I must first simplify it, right? I mean I must remove $u_x$. I suppose, that I must use next ...
41
votes
5answers
10k views

Is the problem that Prof Otelbaev proved exactly the one stated by Clay Mathematics Institute?

Recently, mathematician Mukhtarbay Otelbaev published a paper Existence of a strong solution of the Navier-Stokes equations, in which he claim that he solved one of the Millennium Problems: existence ...
19
votes
2answers
776 views

Intuitive explanation of the difference between waves in odd and even dimensions

Motivation: In odd dimensions, solutions to the wave equation: $u_{tt}(x,t)=\nabla u(x,t)$, $u_t(x,0)=0$, $u(x,0)=f(x)$, ($t\geq 0, x\in \mathbb{R}^n$) have the nice property that the value of ...
22
votes
2answers
438 views

Why is the Laplacian important in Riemannian geometry?

As I've learned more Riemannian geometry, many of my teachers have said that studying the Laplacian (and its eigenvalues) is very important. But I must admit, I've never fully understood why. ...
8
votes
2answers
327 views

Mean Value Property of Harmonic Function on a Square

A friend of mine presented me the following problem a couple days ago: Let $S$ in $\mathbb{R}^2$ be a square and $u$ a continuous harmonic function on the closure of $S$. Show that the average of ...
7
votes
2answers
480 views

when does a separate-variable series solution exist for a PDE

I am wondering if there are some conclusions as to when a series solution using the separate variable method to a PDE exists; i.e. for what requirements on the PDE, what requirements on the initial ...
14
votes
2answers
825 views

Why are harmonic functions called harmonic functions?

Are they related to harmonic series in any way? Or something else? Wikipedia didn't help.
7
votes
2answers
1k views

Euler-Lagrange, Gradient Descent, Heat Equation and Image Denoising

For an image denoising problem, the author has a functional $E$ defined $$E(u) = \iint_\Omega F \;\mathrm d\Omega$$ which he wants to minimize. $F$ is defined as $$F = \|\nabla u \|^2 = u_x^2 + ...
6
votes
1answer
360 views

Inverse Laplace Transform of $\bar p_D = \frac{K_0(\sqrt[]s r_D)}{sK_0(\sqrt[]s)}$

I solved the following partial differential equation using Laplace Transform: $\LARGE \frac{1}{r_D}\frac{\partial}{\partial r_D}(r_D\frac{\partial p_D}{\partial r_D})=\frac{\partial p_D}{\partial ...
6
votes
1answer
191 views

continuity and $C^2$ solution of a series

For $\alpha$ with $|\alpha|=2$ let $P$ be a homogenous harmonic Polynom of degree $2$ with $D^\alpha P\ne0$ (e.g. take $P=2x_1x_2$). Choose $\eta\in C^\infty_0(\{x:|x|<2\})$ with $\eta=1$ when ...
5
votes
1answer
793 views

Weakly Harmonic Functions (Weak Solutions to Laplace's Equation $\Delta u=0$) and Logic of Test Function Techniques.

In analysis we often use test functions $\phi\in C_{0}^{\infty}(U)$ in order to make some kind of deduction about another function $u:U\mapsto\mathbb{C}$. For example, if one can obtain the ...
4
votes
1answer
157 views

heat equation with inhomogenous BC and IC

I'm Zekeriya Özkan from Turkey, I'm a master student in Turkey Can you solve the heat equation with conditions $$\frac{\partial^2u}{\partial x^2}=\frac{\partial u}{\partial t}$$ IC: $u(0,t)=1$ BC : ...
4
votes
2answers
372 views

Why solving $\dfrac{\partial u}{\partial x}=\dfrac{\partial^2u}{\partial y^2}$ like this is wrong?

Try let $v=x+y$ , $w=x-y$ , Then $\dfrac{\partial u}{\partial x}=\dfrac{\partial u}{\partial w}\dfrac{\partial w}{\partial x}=\dfrac{\partial u}{\partial w}$ $\dfrac{\partial u}{\partial ...
4
votes
3answers
480 views

Solution to 2nd order PDE

What is the general solution to the differential equation: $$\frac{\partial^2 u}{\partial y^2}=\frac{\partial u}{\partial x}$$ I'm a little stuck because all the techniques I know are unable to ...
10
votes
1answer
491 views

Understanding a theorem concerning Sobolev spaces

I have two doubts in the proof of the theorem below. If you want the detaIls can be found here. Theorem A Let $q$ a given real number $q>p$. Let $u \in W^{1,p}$ be a solution to (E2). Assume ...
7
votes
1answer
159 views

The proof of the Helmholtz decomposition theorem through Neumann boundary value problem

As you know, the Helmholtz decomposition theorem is as follows. Let $\Omega$ be open and simply connected, and bounded region in $\mathbb R^3$. Then the smooth vector field $E : \Omega \to ...
7
votes
2answers
343 views

Counterexample for the solvability of $-\Delta u = f$ for $f\in C^2$

I'm learning for an exam and I'm surprised by the following statement that is given without proof or example: Let $\Omega\subset\mathbb{R}^n$ be open, bounded and connected and let $f\in ...
5
votes
1answer
509 views

reference for Navier-Stokes equation

For understanding the Navier-Stokes equations, are there any references which may include one or more of the followings: mathematical rigorousness motivation preliminaries introduction etc.
4
votes
3answers
118 views

Solution of $\frac{d^2y}{dx^2} - \frac{H(x) y}{b} = H(-x)$

Does the equation $$\frac{d^2y}{dx^2} - \frac{H(x)}{b} y = c H(x)$$ have a solution where $H(x)$ is the Heaviside step function and $b$ and $c$ are constant? Update: What about the second step ...