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

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2
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1answer
295 views

A-priori estimate help (something to do with Gronwall lemma)

Here $\Gamma(t)$ is a family of smooth compact connected and oriented surfaces in $\mathbb{R}^n$. $u$ solves the PDE $$\dot{u} + u\nabla_\Gamma \cdot v - \Delta_\Gamma u = 0$$ where $\dot{u} = u_t + v ...
2
votes
1answer
336 views

Why do we talk about Trace Operator?

What is the importance of a Trace operator in PDE . Although I have read the Wiki page on this but I am not able to connect it to the aspect of solving PDE's. Particularly why do we define Trace ...
2
votes
1answer
88 views

boundary conditions for Schr$\ddot{\textrm{o}}$dinger equation in 2D polars?

What are the boundary conditions at $r=0$ for Schr$\ddot{\textrm{o}}$dinger's equation for a quantum particle in 2D polars $(r,\theta)$, with potential $U=0$ for $r<a$ and $U=\infty$ for $r>a$? ...
2
votes
1answer
153 views

Why the extension is continuous?

I'm reading M.E.Taylor's PDE, Vol I, Chapter 5 Linear elliptic equations. I have some problem on Proposition 1.7. I will quote it here: Consider the following boundary problem for $u$: $$ \Delta u=0 \...
2
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1answer
145 views

Application of Fourier transformation

The problem requests to use Fourier transformation, which I totally have no clue how. It states as following: $u\in C^2_0$, prove $$\int_{\mathbb{R}^2}u_{xx}u_{yy}-u_{xy}^2 \,dx = 0$$ Any ...
2
votes
3answers
202 views

Non-uniqueness of the solution of the equation for a plucked string

I'm a bit confused about what is written in this PDF (in page 2). The author asserts that the differential equation $y'' +y = 0$ with boundary conditions $y(0)=0=y(\pi)$ has infinitely many solutions. ...
2
votes
2answers
420 views

solving a PDE in 2 variables without boundary conditions

how could i solve the PDE (without boundary or other initial conditions) $ 1= y\partial _{y}f(x,y) -x \partial _{x}f(x,y) $
2
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1answer
1k views

Linear PDE problem from Fritz John's book

Let $u(x,y)$ be a continuously differentiable function on the closed unit disc and is a solution to $$a(x,y)u_x+b(x,y)u_y=-u,$$ on the closed unit disc. Suppose $$a(x,y)x+b(x,y)y>0,$$ on the ...
2
votes
2answers
170 views

How to solve a PDE with quasi-periodic Poisson process?

For classic GBM stock price model, $$\frac{dS}{S} = \mu \cdot dt + \sigma \cdot dW$$ we have the solution: $$S(t)=S(0)\, \exp\left(\frac{\mu-\sigma^2}{2} t+\sigma W(t)\right).$$ During the ...
2
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1answer
149 views

eigenvalue of a variable coefficient operator

There are a couple of questions that I have not find in the book. if I have a linear operator acting $L^2$ to $L^2$ as identity minus Laplacian with a variable coefficient: $L:= I-a(x)Dxx$, assume a ...
2
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1answer
79 views

Sine-Gordon-like equation and solitons

The Sine-Gordon equation is: $$u_{xx}-u_{tt}+\sin(u)=0$$ where $u=u(x,t)$. My question is: if I have the following equation: $$u_{xx}-u_{tt}+\sin(u)^k=0$$ where $k=2,3,4,\ldots,N$ is it still possible ...
2
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1answer
196 views

How to show this weak formulation has unique solution?

Suppose $a$ is a bounded and coercive bilinear form on a Hilbert space $H$ and that $b$ is a bounded bilinear form on $H$ and $\ell$ is a bounded linear function also on $H$. How do I show that: For ...
2
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1answer
359 views

Deriving SDE(s) and Expectation from Given PDE

We want to solve the PDE $u_t + \left( \frac{x^2 + y^2}{2}\right)u_{xx} + (x-y^2)u_y + ryu = 0 $ where $r$ is some constant and $u(x,y,T) = V(x,y)$ is given. Write an SDE and express $u(x,y,0)$ as the ...
2
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1answer
370 views

The mean value property and local maximum

I have an exercise in P.D.E that I couldn't solve. Let $\Omega \subset \mathbb{R}^n$ be a connected open set and $u:\Omega \to \mathbb{R}$ a continuous function that satisfies the following ...
2
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1answer
328 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 ...
2
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1answer
1k views

Solving PDE using Method of Characteristics

I need to solve the following by using the method of characteristics $$u\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=1~,~u|_{x=y}=\frac{x}{2}$$ I have the following characteric ...
2
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1answer
543 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: $$u(x,t)=\sum^\infty_{n=-\infty}\cos k_nx(C_n\cos k_nt+D_n\sin k_nt)$$ ...
2
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1answer
576 views

Inhomogeneous PDE Help?

Problem: $$ \Psi_{xx} - \Psi_{tt} - \Psi = \exp(3t) \cdot \delta(x)$$ No boundary conditions specified. I solved the homogeneous portion, $\Psi_\mathrm{homogeneous}$, of this equation via ...
2
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1answer
1k views

Dirac's identity

Do somebody knows anything about the Dirac's identity? \begin{equation} \label{Dirac} \frac{\partial^2}{\partial x_{\mu}\partial x^{\mu}} \delta(xb_{\mu}xb^{\mu}) = -4\pi \delta(xb_0)\delta(...
2
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0answers
14 views

Why is there no classification scheme for linear pde of third order or above

I am well aware of the classification of 2nd Order PDE into Elliptic, Parabolic or Hyperbolic type depending on the analogy with conic sections from analytic geometry. I have seen third order PDE ...
2
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0answers
42 views

Methods for solving nth order semilinear elliptic PDEs

I am looking for names of methods, and examples of their use that can be used to find solutions for semilinear elliptic PDE equations of the below types: $$\frac{\partial^ny}{\partial x^n}+\frac{\...
2
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0answers
37 views

The eigenfunction of modified 1-laplace equation

Let $\Omega\subset \mathbb R^2$ be a bounded, smooth boundary domain. I am interested in the following operator $$ -\operatorname{div}\left(\frac{\nabla u}{\sqrt{|\nabla u|^2+\epsilon}}\right)=0. $$ ...
2
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0answers
45 views

Elementary proof of $C^\infty_c$ is dense in $L^p (L^q)$ mixed space

As it well known, $C^\infty_0 (\mathbb{R}^n)$(the space of infinitely differentiable functions with compact support) is dense in $L^p (\mathbb{R)^n}$ Here I want to consider the same result with ...
2
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1answer
22 views

Using scaling arguments to determine relationships between Sobolev spaces?

I was looking up how to find relationships between Sobolev spaces and I came across this post on MO in which the first comment talks about a scaling procedure for understanding the relationships: ...
2
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1answer
42 views

PDE boundary condition question regarding limits

Just as a bit of background, I'm working with the Black-Scholes PDE and I'm testing some things out by taking an initial condition for it as $\sin(S/50)$, where $S$ is the spot price (but that's ...
2
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0answers
49 views

Conservation Laws: Difference and Reasonability Weak solutions, Integral solutions, distributional solutions

Suppose we have for $T\in\mathbb{R}_{>0}$ a conservation law of the following general form \begin{align} \dot{u}(t,x)+(a(t,x,u(t,x)))_{x}&=0 && (t,x)\in(0,T)\times\mathbb{R}\\ u(0,x)&...
2
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1answer
48 views

How to show that a Schwartz distribution is in a Lebesgue or Sobolev space?

It is known that all $L^p$ spaces (and, consequently, all $W^{s,p}$ spaces) can be embedded in the space of Schwartz distributions $\mathcal D '$. There is a problem, though: how do I check whether ...
2
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1answer
35 views

Is Laplacian of product of functions (one is smooth) square integrable?

Let $\Omega\subset \mathbb{R}^d$: open, connected. Suppose $u\in L^2(\Omega)$ and $\Delta^n u\in L^2(\Omega)$ for $n=1,\dotsc,N$, where $\Delta$ is in the weak sense. Let $\zeta\in C_c^{\infty}(\...
2
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0answers
41 views

Optimizing potential

Let $X$ be a vector field on a Riemmanian manfiold $M$. I recently read that solving: \begin{equation} \operatorname{argmin}_{\phi}\int_M (\nabla \phi - X)^2 d\mu, \end{equation} where $\mu$ is ...
2
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0answers
36 views

Maple: Symmetries of strange modified heat-equation

I thought about the following PDE (the $u(x=0,t)$ is not a boundary condition! It really is a part of the PDE): $$\dfrac{\partial u(x,t)}{\partial t}=\alpha \dfrac{\partial^2 u(x,t)}{\partial x^2}+u(...
2
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0answers
25 views

Wave equation in a cube

Is it possible find a computable solution to the following homogeneous wave equation problem: Let $\mathcal{C}=\{(x,y,z)\in \mathbb{R}^3, 0< x,y,z < 1\} $ be the open unit cube. Find $u$ such ...
2
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0answers
21 views

Poincaré's inequality proof for $u \in W_0^{1,p}(\Omega)$.

I am trying to prove Poincare's inequality for $u \in W_0^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^n$ is an open bounded set and $1 \leq p < \infty$. This is Poincare's inequality: $||...
2
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0answers
35 views

The 3rd term of the energy estimates in chapter 7 Evans PDE

Hi I wonder someone could help me check my understanding of getting an inequality of the estimate for the 3rd term $\|u'_m\|_{L^2(0,T;H^{-1}(U))}$ correctly. This inequality need to be checked is ...
2
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0answers
42 views

Fractional powers of Markov generators

Let $H$ be the generator of a symmetric Markov semigroup on $L^2(\mathbb{R}^n).$ Why the fractional power $H^\alpha$ (defined on a proper domain) with $0 < \alpha < 1$ turn out to be the ...
2
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1answer
26 views

How to find other equations of lagrange for the Initial Value Problem

Find the solution of the Initial Value Problem $(x-y)\dfrac{\partial u}{\partial x}+(y-x-u)\dfrac{\partial u}{\partial y}=u$ where $u(x,0)=1$ . My try: $\dfrac{dx}{x-y}=\dfrac{dy}{y-x-u}=\dfrac{...
2
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0answers
29 views

PDE with Stochastic Coefficients

Does anyone have reference suggestions for pde's with stochastic coefficients? I've seen many papers on more advanced problems, but it would be great to have a reference discussing the basic theory ...
2
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0answers
29 views

A comparison principle for a nonlinear parabolic PDE

We know the following comparison principle holds for the diffusion equation: Suppose that $u(x,t)$ and $v(x,t)$ satisfy \begin{equation} \begin{cases} u_t\ge \Delta u+F(x,t,u), \ \ &x\in\Omega,\ \...
2
votes
1answer
40 views

Strange PDE solution

Given the linear equation $$u_t -xt u_x = x$$ $x\in\mathbb{R}$, $t>0$, with IVP $u(x,0)=u_0(x)$, my solution comes to $u(x,t) = u_0(xe^{t^2/2})+xt$, but Maple gives a much more complicated solution ...
2
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0answers
39 views

Eigenvectors of matrix equation $AX+XB^\text{T}=\lambda (CX+XD^\text{T})$

We have the matrix eigenvalue problem $$AX+XB^\text{T}=\lambda (CX+XD^\text{T})$$ Where $\lambda$ is the eigenvalue, $X$ is $m\times n$ and plays the role of the eigenvector, and $A$, $B$, $C$, and $...
2
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0answers
49 views

Hadamard counterexample Dirichlet-problem

Good afternoon, I try to understand the follwoing counterexample of Hadamard: The classical solution of $ \begin{cases} \Delta u = 0 ~~\text{ in }\Omega\\ u=g \text{ auf }~~ \partial \Omega \end{...
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0answers
22 views

PDE realated to heat equation with exponential additive term

I want to solve a PDE realated to heat equation with exponential additive term $${\partial u\over\partial t}={1\over x^2}{\partial\over\partial x}(x^2{\partial u\over\partial x})+e^u$$ I dervived ...
2
votes
1answer
38 views

Confusion with changing variables in second order DE

So in my physics assignment, we're given Schrodinger's equation: $$\frac{-\hbar^2}{2m}\frac{d^2\psi}{dx^2}+e\xi x\psi=E\psi$$ We're asked to substitute a function of the form $w(x)=Ax+B$ to arrive at ...
2
votes
1answer
144 views

Solving this non-linear PDE (which reminds of a linear parabolic PDE)

Problem: consider the following PDE: $$-u_t=\mbox{sign}(u) u_x+ \frac{1}{2}u_{xx},$$ with some boundary condition $u(T,x)=\delta_a(x)-\delta_{-a}(x)$, $a>0$ fixed, being $\mbox{sign}(u)\in \{-1,1\}$...
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0answers
38 views

Dirichlet Problem in Stochastic PDE Section of Probability Textbook.

I recently started learning about stochastic calculus and stochastic PDEs, and I don't know where to begin with some of the problems in my textbook. Any help with the following or a push in the right ...
2
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0answers
51 views

On spectrum of periodic boundary value problem

Consider the following boundary value problem on the infinite strip $(-\infty,\infty)\times[0,1]$ w/periodic conductivity $\gamma(x,y)=\gamma(x+2\pi,y)>0$: $$\begin{cases} \operatorname{div}(\gamma\...
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0answers
52 views
2
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1answer
50 views

Solving $4u_{tt}-3u_{xt}-u_{xx}=0$

Solving $\begin{cases} 4u_{tt}-3u_{xt}-u_{xx}=0\tag1\\u(x,0)=x^2\quad\text{and}\quad u_t(x,0)=e^x\end{cases}$ in $\mathbf R\times\mathbf R_{>0}$ First I factorized and get for the first line; $(...
2
votes
0answers
28 views

Poincare type inequality on unit square: can you improve on my constants?

I am trying to bound $\int_{[0,1]^2} u^2$ in terms of its gradient and boundary integrals, $\int_{[0,1]^2} |\nabla u|^2$, $\int_{\partial[0,1]^2} u^2$, with the best possible constants. So far I ...
2
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0answers
34 views

Sufficient Boundary Condition to a General PDE on a General Domain

We know that for an ODE of $n^{th}$ order we need $n$ different boundary conditions. In PDEs, for example, for Laplace equation $\nabla^2 U=0$ (which is a second order PDE) we need only one B.C. (e.g ...
2
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1answer
21 views

Power series solution to a PDE?

I have the following partial differential equation: $u_t = u_xu_y$ I know that the solution can be formed via power series. I want to find a solution of degree $2$ that satisfies an initial ...