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

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4
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3answers
677 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 ...
17
votes
2answers
2k 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
4answers
408 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 $a=x, b=x+y, c=1$ $\Rightarrow$ ...
49
votes
8answers
20k 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 ...
17
votes
4answers
1k views

The Biharmonic Eigenvalue Problem with Dirichlet Boundary Conditions on a Rectangle

I am interested in solving the following biharmonic eigenvalue problem. $$\begin{array}{cccc} & \Delta ^2 \Psi (x,y) = \lambda \Psi (x,y), & - a \le x \le a & - b \le y \le b \\ ...
7
votes
3answers
2k views

A typical $L^p$ function does not have a well-defined trace on the boundary

This question is from PDE by Evans, 1st edition, Chapter 5, Problem 14. It has been posted here previously, however, I cannot quite put all the information together from the responses there. Hopefully ...
7
votes
2answers
427 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 ...
4
votes
1answer
669 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 ...
9
votes
4answers
1k views

Why separation of variables works in PDEs?

The method of separation of variables is used in many occasions in the upper level physics courses such as QM and EM. But when it is used there is no clear reason why using it is permitted it except ...
6
votes
3answers
3k views

Composition of a harmonic function.

I noticed this question while reading several pdfs of lecture notes, and I'm having trouble figuring it out. Can anyone help? If $f$ is a harmonic function in a domain $D \subset \mathbb{C}$, and $g$ ...
6
votes
1answer
6k 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. ...
5
votes
1answer
182 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 ...
92
votes
8answers
47k 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 ...
23
votes
2answers
1k 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, ...
10
votes
1answer
4k 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 ...
10
votes
2answers
884 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 ...
2
votes
1answer
3k views

question 9 - chap 5 evans PDE

The question is : Integrate by parts to prove : $$\int_{U} |Du|^p \ dx \leq C \left(\int_{U} |u|^p \ dx\right)^{1/2} \left(\int_{U} |D^2 u|^p \ dx\right)^{1/2}$$ for $ 2 \leq p < \infty$ ...
11
votes
2answers
676 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 ...
10
votes
1answer
1k 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 ...
5
votes
1answer
992 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 ...
6
votes
2answers
2k 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. ...
5
votes
1answer
464 views

Dirichlet problem in the disk: behavior of conjugate function, and the effect of discontinuities

Dirichlet's problem in the unit disk is to construct the harmonic function from the given continuous function on the boundary circle. It is solved by the convolution with the Poisson kernel, and we ...
4
votes
0answers
260 views

existence of solution of volterra integral equation of the first kind

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To know the existence and uniquness of solution of volterra integral equation(VIE) of the first kind, we differentiate it and convert to the VIE of the second kind. ...
4
votes
0answers
305 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= ...
43
votes
1answer
6k 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 ...
25
votes
2answers
2k 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 ...
15
votes
3answers
1k 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
2answers
230 views

Sobolev spaces - about smooth aproximation

Consider $\Omega $ a open and bounded set of $\mathbb R^n$ . Let $u \in H^{1}(\Omega)$ a bounded function. I know that there exists a sequence $u_m \in C^{\infty} (\Omega) $ where $u_m \rightarrow ...
4
votes
1answer
383 views

Solve $u_{xx}-3u_{xt}-4u_{tt}=0$ where $u(x,0)=x^2$ and $u_t(x,0)=e^x$

Solve $$u_{xx}-3u_{xt}-4u_{tt}=0$$ where $u(x,0)=x^2$ and $u_t(x,0)=e^x$. My workings so far: I have factored the differential equation in the following way: ...
4
votes
3answers
285 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 ...
4
votes
0answers
572 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
227 views

Uniform convergence in a proof of a property of mollifiers in Evans's Partial Differential Equations [duplicate]

Here are some definitions that was taken of PDE Evans book: Here is a proof of a property of mollifiers: My (elementary) question is: Why is the convergence uniform on $V$? Thanks.
1
vote
2answers
290 views

Wave equation with initial and boundary conditions - is this function right?

If $y(x,t)$ satisfies the 1-dimensional wave equation $$\frac{\partial^2y}{\partial t^2}=c^2\frac{\partial^2y}{\partial x^2}\quad\text{for }0\leq x \leq l$$ with boundary conditions ...
2
votes
1answer
676 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 ...
2
votes
2answers
470 views

Uniform Convergence of Difference Quotients to Partial Derivative

I'm currently reading Evans' PDE book. In it he claims that for $f \in C^2_c(\mathbb{R}^n)$ $$\frac{f(x + he_i) - f(x)}{h} \to \frac{\partial}{\partial x_i}f(x)$$ and $$\frac{\frac{\partial}{\partial ...
51
votes
1answer
5k 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 ...
20
votes
2answers
312 views

Why are $L^p$-spaces so ubiquitous?

It always baffled me why $L^p$-spaces are the spaces of choice in almost any area (sometimes with some added regularity (Sobolev/Besov/...)). I understand that the exponent allows for convenient ...
17
votes
4answers
855 views

What is spectrum for Laplacian in $\mathbb{R}^n$?

I know very well that Laplacian in bounded domain has a discrete spectrum. How about Laplacian in $\mathbb{R}^n$?(not in some fancy-shaped unbounded domain, but the whole domain) Where can I find ...
2
votes
1answer
505 views

Show $\nabla^2g=-f$ almost everywhere

let a continuous function $f(x,y,z)$ be absolutely continuous over every bounded region and let it be in $L^1$ that is $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} ...
6
votes
1answer
898 views

Neumann's problem necessary and sufficient condition (Evans PDE)

Let $U$ be connected. A function $u \in H^1(U)$ is a weak solution of Neumann's problem \begin{equation} (*)\qquad\left\{ \begin{array}{rl} -\Delta = f & \text{in } U \\ \frac{\partial ...
11
votes
1answer
828 views

Why no trace operator in $L^2$?

We have trace operator which allows us to define boundary values of an $H^1$ function. This is because of the fact that $C^\infty$ is dense in $H^1$ under the $H^1$ norm, I believe. I'm sure either ...
3
votes
4answers
2k views

Poincaré inequality for functions with large zero set

I have trouble proving the following problem (Evans PDE textbook 5.10. #15). Could anyone kindly help me solving the problem? I know that I should somehow use Poincaré inequality but I still cannot ...
8
votes
1answer
472 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 ...
3
votes
2answers
698 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 ...
6
votes
2answers
4k views

Finding integrating factor when IF will be a function of x and y

I'm not finding any resource or description or systemic methodology to find integrating factors when the integrating factor will be a function of both x and y. I'm on this problem, $$ ( y - xy^2 ) ...
3
votes
1answer
3k views

Heat equation with time dependent boundary conditions?

Suppose $u(x,t)$ solves \begin{align} u_t&=u_{xx}, \qquad 0 < x < \pi/2,\ t>0,\\ u(0,t)&=e^{-t},\\ u(\pi/2, t)&=t,\\ u(x,0)&=\cos(3x). \end{align} I was following a method ...
1
vote
1answer
731 views

A problem from Evans' PDEs book: find a Lagrangian for a given Euler-Lagrange equation

Find $L=L(p,z,x)$ so that the PDE: $-\Delta u +D\varphi \cdot Du =f $ is the Euler-Lagrange equation corresponding to the functional $I[w]:=\int_UL(Dw,w,x)dx$. (Hint,:Look for a Lagrangian ...
2
votes
1answer
208 views

Laplace's equation in rectangle geometry

Consider Laplace's equation in a rectangle with length and width of a and b respectively, with following boundary conditions: All the boundaries with $x < a/2$ have Drichlet boundary condition ...
2
votes
1answer
286 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 ...
1
vote
1answer
85 views

The functional $I[w] = \int_U \frac{1}{2} |Dw|^2 - fw \, dx$ is weakly lower semicontinuous

I am studying calculus of variation, and I need to prove that $I[w] = \int_U \frac{1}{2} |Dw|^2 - fw \, dx$ with $f \in L^2(U)$ is weakly lower semicontinuous on $H_0^1(U)$. In classes, I only ...