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

learn more… | top users | synonyms (2)

8
votes
2answers
1k views

When can one expect a classical solution of a PDE?

When solving a PDE, there may be a classical solution or a weak solution (or distribution solution). But I am wondering that when people talk about "finding a solution" to some PDE, what do they refer ...
6
votes
1answer
252 views

Two different definitions of ellipticity

This is a question originating in another mathematics forum, matematicamente.it (in Italian). In literature one encounters the word elliptic in (at least) two different definitions. In what follows ...
5
votes
1answer
561 views

Green's functions of Stokes flow

In mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to specific initial conditions or boundary conditions. A fundamental solution for a ...
1
vote
2answers
389 views

Maximum principle of heat equation, without a bounded time interval

Is there a maximum principle for the heat equation $\partial_t u(x,t)=k \partial_{xx}^2 u(x,t)$ for $(x,t)\in[O,L] \times [0, \infty]$? If $u$ has a maximum it would occur at $t=0$, $x=0$ or $x=L$, ...
2
votes
0answers
115 views

Principal eigenvalue

How is the principal eigenvalue of elliptic differential operator defined? Is it just a spectral radius?
1
vote
2answers
346 views

What is a reference for the ( classical and well-known ) proof of Weyl's lemma?

What is a reference for the (classical and well-known) proof of Weyl's lemma that states: Let $U$ be an open subset of $R^n$. Then if $f\in L^1_{loc} (U)$ and if $\int_U ...
11
votes
2answers
3k views

Energy norm. Why is it called that way?

Let $\Omega$ be an open subset of $\mathbb{R}^n$. The following $$\lVert u \rVert_{1, 2}^2=\int_{\Omega} \lvert u(x)\rvert^2\, dx + \int_{\Omega} \lvert \nabla u(x)\rvert^2\, dx$$ defines a norm on ...
4
votes
2answers
697 views

How to prove Weyl’s asymptotic law for the eigenvalues of the Dirichlet Laplacian?

The following comes from Springer Online Reference Works: Consider a bounded domain $\Omega\subset\mathbb{R}^n$ with a piecewise smooth boundary $\partial\Omega$. $\lambda$ is a Dirichlet eigenvalue ...
5
votes
1answer
450 views

Eigenvalues For the Laplacian Operator

How do I show that the asymptotic speed of the eigenvalues $\lambda$ of the Laplacian Operator is $O(m^{2/n})$ where $m$ is the index of the eigenvalues and $n$ is the dimension of the space?
7
votes
3answers
935 views

Energy functional in Poisson's equation: what physical interpretation?

Let's consider this boundary-value problem: $$\begin{cases} -\Delta V = \rho & \rm{in}\ \Omega \\ V=0 & \rm{on}\ \partial \Omega \end{cases}.$$ We know that this problem has a ...
1
vote
1answer
3k views

Changing between Maxwell equations in differential and integral formats?

It takes me a long time to think about the equations even in one format and also to deduce things with Stokes. So how can you swap between the equations? I am looking more on the lines that suppose ...
2
votes
2answers
419 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 ...
2
votes
0answers
241 views

Two vague steps in the proof of Harnack inequality

I am reading the book Elliptic and Parabolic Equations and the proof is excerpted from page 133-136. In Theorem 5.1.3: it claims that ...
1
vote
0answers
238 views

Laplace Eigenfunction: Show Eigenvalue is Positive Using Fourier Transform

Problem: Let $ \lambda\in\mathbb{R}, u $ a smooth function, not identically zero, defined on a neighborhood of the unit disc satisfying $ \Delta u+\lambda u = 0 $ in the interior of the unit disc and ...
2
votes
1answer
184 views

Extension of the partial derivative

Let the open set $\Omega\subset{\mathbb R}^n$, and $k$ be a positive integer. $C^k(\Omega)$ will denote the space of functions possessing continuous derivatives up to order $k$ on $\Omega$, and ...
3
votes
3answers
203 views

What is a “domain” in the maximum-minimum principle?

The maximum-minimum principle says that A harmonic function on a domain cannot attain its maximum or its minimum unless it is constant. Here is my question: If we restrict our attention in ...
5
votes
1answer
310 views

KDV PDE: energy constant in time

Show that if u solves the KDV equation $u_t + u_{xxx} + 6uu_x = 0$ for $x \in \mathbb{R}$, $t > 0$ then the energy $\int_{-\infty}^{\infty} \frac{1}{2} u_x ^2 - u^3 \,dx$ is constant in time. ...
1
vote
2answers
547 views

How do the solutions to the wave and heat equations converge in general?

I would like to check my understanding with someone if possible. When we cover the heat and wave equations, for instance, in "methods" courses at university, they normally restrict the initial ...
6
votes
2answers
2k 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
2answers
612 views

How to solve this PDE?

I was wondering how to solve $$a(x-1)\frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} =0 ?$$ $a$ is a constant. (1) I am still having trouble to understand the method of ...
1
vote
1answer
292 views

Solution to the Dirichlet problem is smooth up to the boundary if the boundary data is smooth?

where can I get a quick exposition to the boundary regularity problem for the Laplacian operator ? In other words, suppose $h:S^1 \to \mathbb C $ and let $H: \bar{D}\to \mathbb C $ be its complex ...
4
votes
3answers
360 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 ...
1
vote
3answers
178 views

Why is it useful to express PDE solutions as $L^2$-convergent series?

The existence of an $L^2$ orthonormal basis consisting of eigenfunctions of a Sturm-Liouville equation helps us to express the solutions of various ODEs and PDEs as infinite series. However, in the ...
5
votes
1answer
487 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.
6
votes
2answers
565 views

Gradient of a Harmonic Function

I was asked the following vector calculus problem: Let $D$ be the unit ball and let $S$ be the unit sphere in $\mathbb{R}^3$. Suppose that $F:\mathbb{R}^3\rightarrow \mathbb{R}^3$ is a $C^1$ ...
1
vote
0answers
207 views

Complete vs General Integral of first order PDE

The following is an excerpt from Landau's Course on Theoretical Physics Vol.1 Mechanics: ... we should recall the fact that every first-order partial differential equation has a solution depending ...
22
votes
2answers
944 views

Non-linear partial differential equation

I would like to find out if there is any specific method -apart from numerics- for finding solutions of a non-linear PDE of the form $$\nabla \times \mathbf{A} = \pm\lambda\mathbf{A} \tag{1}$$ under ...
1
vote
1answer
113 views

Solution for a PDE on $\Omega=[0,\pi]^3$

In Strauss's Partial Differential Equations, the eigenvalue problem $$-\triangle v=\lambda v,\qquad v|_{\partial \Omega}=0$$ is solved by separating the $x,y,z$ variables: $v=X(x)Y(y)Z(z)$, $$ ...
2
votes
1answer
193 views

Schrödinger operator: where is the generator to be defined?

The theory as I know it Let $\mathcal{H}$ be a Hilbert space and $(A, D(A))$ a self-adjoint operator acting on it. The Spectral Theorem (cfr. Reed & Simon Methods of modern mathematical physics, ...
1
vote
1answer
326 views

On the method of characteristics

By the method of characteristics, it is possible to prove the existence of local solution for the first-order linear non homogeneous equation: $\mathcal{L}_X f+g=0$ where $X$ is a non-singular ...
0
votes
1answer
357 views

Proving that the bi-laplacian of a radial basis function is the dirac delta

According to equation (2.14) of the paper "The Uniform Convergence of Thin Plate Spline Interpolation in Two Dimensions" a radial basis function $\phi(\parallel x \parallel)$ has the property $$ ...
4
votes
2answers
678 views

How can one relate inverse of a differential operator to an integral operator?

Informally speaking, the integral operator can be regarded as the inverse of some differential operator. In some very special case, finding the inverse of the differential operator is equivalent to ...
0
votes
1answer
114 views

Further thoughts on the energy estimate

I put my question about the energy estimate two days ago. And finally I can get $$\frac{d}{dt}\|x\|^2=2\|x\|\frac{d}{dt}\|x\|=\frac{d}{dt}\langle x,x \rangle=2Re\langle x, Ax\rangle$$ If I have the ...
6
votes
1answer
173 views

Questions about a PDE: $-u_{txx}+u_{t}=u_{x}$, $u(x,0)=u_0(x)$, $x,t\in{\bf R}$

Consider the BBM equation: $-u_{txx}+u_{t}=u_{x}$, $u(x,0)=u_0(x)$, $x,t\in{\bf R}$. One may rewrite this equation as following $u_t=((I-A)^{-1}\partial_x)u$ where $Au=u_{xx}$ if $(I-A)^{-1}$ ...
2
votes
2answers
446 views

wave equation and superposition

If I have this equation: $$\frac{\partial^2u}{\partial x^2}=\frac{\partial^2u}{\partial t^2}$$ And this general solution: $$u(x,t)=\sum^\infty_{n=-\infty}\cos k_nx(C_n\cos k_nt+D_n\sin k_nt)$$ ...
5
votes
1answer
599 views

What connections are there between number theory and partial differential equations?

What connections are there between number theory and partial differential equations?
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 + ...
10
votes
1answer
1k views

How to compute the first eigenvalue of Laplace operator in an ellipse?

Let $\mathcal{E}$ be an ellipse in the $\mathbb{R}^2$ plane with center in $o=(0,0)$, given focal distance $c\geq 0$ and given area $A>0$. It is a fact that the eigenvalue problem for the Laplace ...
0
votes
0answers
160 views

The quantum harmonic oscillator [closed]

I want to ask how to solve the equation $$-\frac{{\hbar}^{2}}{2m}\frac{\partial}{\partial_{r}^{2}}u(r)=(E-\frac{1}{2}Kr^{2})u(r)$$ with $K$ being a constant.
3
votes
0answers
504 views

Finding the modified Green function for the Helmholtz equation

I've been wrestling with this question for quite some time now, and the result was like 20 leaves of paper packed with scribbling...anyway, here's the question: I need to find the solution to the ...
4
votes
1answer
468 views

(Question) on Time-dependent Sobolev spaces for evolution equations

I have got a question on so-called time-dependet Sobolev spaces - in particular as introduced in Evans book on PDE for the treatment of parabolic and hyperbolic PDE. Let us take a look at a linear ...
3
votes
2answers
2k views

Crank-Nicolson Local Truncation Error

I'm trying to derive the LTE for CN applied to the linear heat equation; $u_t = u_{xx}$. The problem is that I end up with terms of the form $\frac{{\Delta t}^k}{{\Delta x}^2}$ when using a two ...
2
votes
3answers
422 views

Resolution of a non-homogeneous heat equation

I'm looking for solution to this non-homogeneus problem. $\frac{\partial u}{\partial t}-\frac{\partial^2 u}{\partial x^2}=F(x,t)$ for $0<x<\pi$, $t>0$ $u(x,0)=0$ ...
1
vote
1answer
704 views

Eigenfunction expansion solution to a PDE with a constant non homogeneous term

I'm wondering if the method of finding a solution to a nonhomogeneous PDE by the method of eigenfunction expansion works if the nonhomogeneous term is a constant, rather than a function of the ...
1
vote
1answer
172 views

Question on Evans' treatment of elliptic 2n order equations

I am working through L.C.Evans' Partial Differential Equations -- the chapter on second-order elliptic equations. I have got a general question on symmetric vs. non-symmetric elliptic operators. ...
1
vote
1answer
164 views

PDE transport equation please help

transport equation: $$ \frac{d}{dt}u +cu = 0\qquad \mbox{ in } \mathbb{R}^n\times (0,\infty)$$ $$ u(x,0) = 0 \qquad \mbox{ on }\mathbb{R}^n\times \{t=0\} $$ b-const
5
votes
1answer
393 views

is $\nabla \cdot ( c^2 \nabla)$ a Laplace-Beltrami operator?

I asked this on mathoverflow, and was suggested to ask it here. Someone mentioned, in passing, to me that $u \mapsto \nabla \cdot ( c^2 \nabla u)$ is a Laplace-Beltrami operator. Does anyone have ...
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 ...
7
votes
2answers
471 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 ...
1
vote
2answers
544 views

Quasilinear PDE of Second Order - Some Analytical Direction Needed

I'm looking at this second order quasilinear PDE: $\alpha_{xx} - \alpha_{yy} - m\alpha^2 = 0$ Attempted Strategies: Fourier Transform resulted in convolution due to the $\alpha^2$ term, I don't ...