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

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10
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1answer
344 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 ...
7
votes
2answers
360 views

Euler-Lagrange equations of the Lagrangian related to Maxwell's equations

Clarification on Lagrangian mechanics would be much appreciated: Suppose $$L(\phi,\,\,\phi_{,i},\,\,A_i, \dot A_i)=|\dot A+\nabla\phi|^2-|\nabla \times A|^2-c\phi+d\cdot A$$ Are the corresponding ...
5
votes
0answers
94 views

Solve PDE by getting two ODEs

My goal is to solve this PDE for $f:[-1,1] \times \mathbb{R}_{\ge 0}\rightarrow \mathbb{C}$ $$ \partial_t f(x,t) = -\partial_x^2 f(x,t) + g(t)V(x)f(x,t).$$ I would consider this PDE to be solved if ...
5
votes
1answer
193 views

Why is it important to study the eigenvalues of the Laplacian?

Why is it important to study the eigenvalues of the Laplacian acting on regions in $\mathbb R^n$? What information does this give us? What problems does this information help us solve? In particular, ...
4
votes
1answer
456 views

Semi-infinite heat/diffusion equation with time-dependent B.C. at x=0

I have a problem in which I need to solve the diffusion equation: $$ \frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2} $$ on a semi-infinite domain from $x=0$ to $x=\infty$. The initial ...
3
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0answers
145 views
+100

weak derivative questions

There are two definitions of generalized differentiation that seem relevant to the context of PDEs. (That is we generalize what objects can be differentiated but we stay in Euclidean space. There are ...
3
votes
1answer
730 views

What kinds of PDE can't be solved by separation of variables?

What kinds of PDE can't be solved by separation of variables? Except those without boundary and which are non-linear? Does it matters with the shape of solution domain? When should I use addition or ...
9
votes
1answer
318 views

Origin of the name 'test functions'

This is a very simple question really: where did the name 'test functions', used nowadays when speaking of infinitely differentiable and compactly supported functions, come from? More to the point: is ...
9
votes
5answers
2k views

Blow up of a solution

What exactly does blow up mean, when people say, for example, that a solution (to a pde (say)) blows up. Thanks.
7
votes
2answers
136 views

PDE: What is the most general solution of $F_xF_y=1$ for a real function, $F(x,y)$?

WolframAlpha gives the simple solution, $F(x,y)=cx+\dfrac{y}{c}+c'$ with two constants $c$ and $c'$ . Is this the most general solution?
6
votes
1answer
173 views

Characteristics in the theory of PDE's - What's Going On?

What's really going on as regards characterstics in the theory of PDE's? (Hopefully you wont have to check the links provided below, included for those who are interested really) For second order ...
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
366 views

A PDE exercise from Gilbarg Trudinger. problem 2.2

Prove that if $\Delta u=0$ in $\Omega\subset \mathbb{R}^n$ and $u=\partial u/\partial\nu=0$ on an open smooth portion of $\partial \Omega$, then $u$ is identically zero. I found a proof here, but I'm ...
5
votes
2answers
109 views

Looking for a first order perturbation of the Laplacian having 0 in its spectrum

I would like to find an explicit example of a linear elliptic operator having the following form: $$Lu=-\Delta u +b(x)\cdot \nabla u, $$ where $b\colon \mathbb{R}^n \to \mathbb{R}^n$, and such that ...
5
votes
1answer
563 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 ...
4
votes
2answers
167 views

how to solve this PDE

How to find general solution of the PDEs $$\frac{∂^2u}{∂x^2}-\frac{∂^2u}{∂y^2}=x^2y^y$$ the problem is the term $y^y$ in the equation. May i solve it by transforming into the canonical form? I have ...
4
votes
2answers
701 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 ...
1
vote
1answer
96 views

Questions about the hyperbolic system of equations

$$u_t+A(x,t,u)u_x=b(x,t,u) \tag 1$$ $$u=(u_1, \dots, u_n), b=(b_1, \dots, b_n)$$ $$A=[a_{ij}], i,j = 1, \dots, n$$ $$$$ We set the question if there are characteristic directions at the path of which ...
1
vote
0answers
98 views

Uniqueness existence of solutions local analytical for a PDE

Consider the problem $$\begin{cases}u_{tt}=u_{xx}-2au_{x}+a^{2}u, (x,t)\in\mathbb{R}^{2}\\[8pt] u(x,0)=f(x), x\in\mathbb{R}\\[8pt] u_{t}(x,0)=g(x), x\in\mathbb{R} \end{cases}$$ where $a>0$. ...
1
vote
1answer
348 views

Wave propagation with variable wave speed

If we have $u_t + c(x,t) u_x = 0 \; \; $ describes uni-directional wave propagation in a medium with variable wave speed. a) Explain how to solve it by the method of charichtaristics for general ...
10
votes
2answers
223 views

Existence and uniqueness of Stokes flow

What are the solution existence and uniqueness conditions for Stokes' flow? $$\begin{gathered} \nabla p = \mu \Delta \vec{u} + \vec{f} \\ \nabla \cdot \vec{u} = 0 \end{gathered}$$ Maybe you could ...
8
votes
2answers
178 views

A Differential operator.

What are the fundamental solutions for the operator $$\mathcal D=a{\partial^2\over\partial x_1^2}+b{\partial^2\over\partial x_2^2}$$ on $\Bbb R^2 $ with standard cordinates $(x_1,x_2)$. Here ...
8
votes
2answers
215 views

Black-Scholes PDE with non-standard boundary conditions

I have the PDE $$ -\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0$$ with initial and boundary conditions: ...
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 ...
7
votes
1answer
227 views

Function in $H^1(\Omega)$ which cannot be extended to a greater Sobolev Space

The problem is like this: Consider the open set $\Omega \in \Bbb{R}^2$ by $\Omega=\{(x,y) : 0<x<1, 0<y<x^2 \}$ Is $\Omega$ with Lipschitz boundary? (i.e. the boundary is ...
6
votes
1answer
220 views

Difficulties in solving a PDE problem

This is an exercise in "Variation et optimisation des formes", chapter 3, Ex. 3.8. The preliminaries are: $$D=(0,1)^2,\ f \in L^2(D),\ x_{ij}=(i/n, j/n),\ 0<i,j<n,$$ $$\Omega_n = D\setminus ...
6
votes
1answer
230 views

Capacity theory beginner resources

I'm currently studying a book on shape optimization: Variation et optimisation de formes: Une analyse géométrique By Antoine Henrot, Michel Pierre. The book introduces at some point capacity, and uses ...
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 ) ...
5
votes
2answers
142 views

Definition of the principal symbol of a differential operator on a real vector bundle.

I'm trying to understand the construction of the dirac operator on a manifold, but actually I guess that doesn't really matter for the question at stake. I'm interested in understanding a definition ...
5
votes
1answer
112 views

How does the Hahn-Banach theorem implies the existence of weak solution?

I came across the following question when I read chapter 17 of Hormander's book "Tha Analysis of Linear Partial Differential Operators", and the theorem is Let $a_{jk}(x)$ be Lipschitz continuous in ...
5
votes
1answer
272 views

How should a numerical solver treat conserved quantities?

Since the good old Noether Theorem ultimately states that any physical system will exhibit conserved quantities, how should they be treated best in numerical solvers? On the one hand, one can observe ...
5
votes
1answer
499 views

How to solve this Initial boundary value PDE problem? [SOLVED]

Today I came across a question on PDE which makes me really frustrating. The question is to solve this initial boundary value problem using method of separation variables: $$u_{tt}=9u_{xx}\text{ for ...
5
votes
1answer
207 views

Claims in Pinchover's textbook's proof of existence and uniqueness theorem for first order PDEs

The reference here is Pinchover & Rubinstein's An introduction to partial differential equations, pages 36-37. It's about the existence and uniqueness of a solution to the equation $a(x,y,u)u_x + ...
5
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1answer
373 views

How to use FEM to solve a PDE

I come from a Computer Science background and I've been around searching for a concise tutorial that tells me how Finite Element Method is used to solve a certain partial differential equation. ...
5
votes
1answer
453 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?
4
votes
2answers
64 views

Bounded data means bounded solution to parabolic PDE

Let $u_0 \in L^\infty(\Omega)$ and $f \in L^\infty((0,T)\times\Omega).$ Consider $$u_t - \Delta u = f$$ $$u|_{\partial\Omega} = 0$$ $$u(0) = u_0$$ or more generally replace $\Delta$ with a suitable ...
4
votes
1answer
63 views

How find a solution to this PDE $\frac{xf'_{x}}{f'_{y}}+\frac{yf'_{y}}{f'_{x}}+x+y=C$

let $C$ is give the constant ,if the function $f(x,y)$ such $$\dfrac{xf'_{x}}{f'_{y}}+\dfrac{yf'_{y}}{f'_{x}}+x+y=C$$ Find the all $f(x,y)$ I found this problem one solution: ...
4
votes
1answer
156 views

Some Scaling Estimate for Heat Kernel

NOTE. I have rewritten the question to summarize my current progress on this question. The bounty is for completing what I have done so far, or by offering a more elegant solution probably based on ...
4
votes
1answer
153 views

Need help understanding this proof of regularity of traveling wave solutions to the Gross-Pitaevskii equation

These are actually 4 question about a proof given in this paper. Any hint to solutions for any of these questions would be much appreciated! Lemma 1. Assume $v$ is a solution to the equation ...
4
votes
0answers
129 views

Is there any theorem that tells us how many ICs or BCs are needed for getting the determine solution of a PDE or a set of PDEs?

It's a shame that, though I've taken the "Equations of Mathematical Physics" class for one semester and solved numbers of PDEs with Mathematica, I'm still unclear about how many initial ...
4
votes
1answer
157 views

Transient diffusion with compact support throughout (not just initially)

[Pardon my lack of rigor; I am an engineer by training. Also, for convenience, allow me to make this question as concrete as possible.] Assume the simplest linear diffusion equation: $\alpha ...
4
votes
1answer
641 views

The extension of smooth function

If $U$ is a bounded domain in $\mathbb R^n$ whose boundary is smooth, and $f$ is a smooth function on $U$ whose partial derivatives of all orders have a continuous extensions to $\bar U$. For an ...
4
votes
1answer
518 views

understanding of the classical definition of Green's function

I learn the classical definition of Green's function from Hunter's Applied Analysis. Consider the second-order ordinary differential operators $A$ of the form $$Au=au''+bu'+cu,$$ where $a,b$ and $c$ ...
3
votes
3answers
138 views

Initial-value problem for non-linear partial differential equation $y_x^2=k/y_t^2-1$

For this problem, $y$ is a function of two variables: one space variable $x$ and one time variable $t$. $k > 0$ is some constant. And $x$ takes is value in the interval $[0, 1]$ and $t \ge 0$. At ...
3
votes
2answers
113 views

If a Laplacian eigenfunction is zero in an open set, is it identically zero?

This seems like it should be easy but I can't seem to figure it out. Suppose $f$ satisfies $\Delta f + \lambda f = 0$ in $\mathbb{R}^n$ (where $\lambda > 0$ is a constant), and in some open set ...
3
votes
0answers
72 views

$u\in W^{1,p}(B)$ implies $u\in W^{1,p}(\partial B_t)$ for almost $t\in (0,1]$?

Suppose that $u\in W^{1,p}(B)$ where $B=B(0,1)\subset\mathbb{R}^N$ and $p\geq 1$. It was showed on this post (as an application of Fubini's theorem) that for almost $t\in (0,1]$ $$u_{|\partial ...
2
votes
1answer
34 views

a question about Fourier transforms

I know it s simple but how to show that $\mathcal F(f(-t))w=\mathcal F(f(t))(-w)$ ? $\mathcal F(f(-t))w=\int^\infty_{-\infty}f(-t)e^{-iwt}dt$ $if -t=x\to -dt=dx$ ...
2
votes
1answer
104 views

Example of linear parabolic PDE that blows up

Does anyone have an example of a linear parabolic PDE that blows up in finite time in a Sobolev space setting? How does one show blow up for that particular example? The one in Evans unfortunately is ...
2
votes
1answer
88 views

A question about a Sobolev space trace inequality (don't understand why it is true)

Let $\Omega$ be an open set with boundary $\partial\Omega$. Let $u \in H^1(\Omega)$. There exists a $\lambda \in \mathbb{R}$ such that $$\int_\Omega |\nabla u |^2 + ...
2
votes
1answer
640 views

PDE Question in Evans

The question has been posted here previously, however, I cannot quite put all the information together from the responses there. Hopefully you can help me now. The problem is as follows: Let $U$ ...