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

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2
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1answer
24 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{...
0
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1answer
28 views

Solving $\sin(\sqrt{\lambda}L) + \beta \cos(\sqrt{\lambda}L)\sqrt{\lambda} = 0$

I'm working with the ODE $$-\frac{d^2u}{dx^2}=\lambda u$$ and trying to find eigenvalues and eigenfunctions corresponding the boundary conditions $$u(0)=0, u(L)+\beta \frac{du}{dx}(L)=0$$ Assuming ...
0
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0answers
18 views

Differential Equation Invariant under Isometric Mapping

I started reading a book on Finite Elements ("Finite Elements", Braess) and one section describes elliptic PDE's of the form $Lu = f$. The author goes on to say "If a differential equation is ...
3
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2answers
41 views

“Inverse” Helmholtz Decomposition

So I am trying to write a report on the Helmholtz decomposition theorem on $\mathbb{R}^3$. The theorem states that under certain conditions, every vector field $\textbf{F}:U \subseteq \mathbb{R}^3 \to ...
0
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1answer
34 views

Tensor product space dense in $H_0^1$?

Given $C_c^{\infty}((a_i,b_i))$ for $i \in \{1,2\}.$ Is it then true that $$C_c^{\infty}(a_1,b_1) \otimes C_c^{\infty} (a_2,b_2)$$ is dense in $H_0^1((a_1,b_1) \times (a_2,b_2))$? Clearly, by ...
0
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0answers
13 views

Ineqality involving operators

I need your help to solve the following problem: We define the operatorss $a$ and $b$ with domain $\mathcal{C}_0^{\infty}(\mathbb{R}^n)$ by: $a=\partial_x+\frac{1}{2}\partial_xV(x)$ (where $V$ in ...
-1
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0answers
50 views

Is it possible to learn partial differential equations without an analysis course?

I need to learn partial differential equations and it seems that I need to take a course in analysis beforehand. Is it possible to learn PDE without analyais and only having learned ODE? Are there any ...
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0answers
25 views

What does the Schwarz lemma state?

What does the Schwarz lemma state ? Apologize, I googled it without success, I only found something related to complex analysis, what I ask is in the theory of PDE. It should state something like; ...
0
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2answers
22 views

Question about Langevin equation

The Langevin equation is given by: $dq=pdt,\ dp=-\nabla V(q) dt-pdt+\sqrt{2}dW$ I want to know what does the variables $p,\ q,\ t,\ V,\ W$ represent . Can someone help me ? Thanks.
1
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0answers
39 views

Why is the maximum principle not valid here?

Why is the maximum principle not valid here ? $u:\mathbf R^n_{+}\times (0,1]$ and $u(x,t)=1-\frac{1}{\left(4\pi (t-1/2)\right)^{n/2}}e^{\frac{-|x+1|^2}{4t-2}}$ then $u$ is bounded by $1$ and ...
3
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1answer
57 views

Exponential boundedness for PDE

Suppose we have $\kappa$ bounded and the following equation: $${\partial^2\kappa\over\partial t\partial\theta}=\kappa^2{\partial^3\kappa\over\partial\theta^3}+2\kappa{\partial\kappa\over\partial\...
-1
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0answers
12 views

Question about degénérate potential

I want to know why the potential given by: $P(q_1,q_2)=q_1^2q_2^2$ is a dégenerate potential at infinity? Can someone help me? Thanks
0
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1answer
26 views

Find the solution of the Cauchy problem for the pde

Find the solution of the Cauchy problem for the pde $a\dfrac{\partial u }{\partial x}+b\dfrac{\partial u}{\partial y}=1$ with the initial condition $u(x,y)=x$ on $ax+by=1$. By Lagrange's Equations ...
1
vote
1answer
17 views

Show that IVP has a solution which remains bounded as $t\to \infty$.

Find the solution of the Cauchy problem for the pde $\dfrac{\partial u }{\partial t}+x\dfrac{\partial u}{\partial x}=x$ $u(x,0)=2x$ and show that it has a solution which remains bounded as ...
4
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1answer
46 views

the proof of variational principal for the principal eigenvalue (checking orthonormal subset)

Hi I am looking at part 3 of the proof in Evans Chapter 6. I have difficulty understanding "Furthermore from (6) and (7) we see that $(\lambda_k^{-1/2} w_k)$ is an orthonormal subset of $H_0^1(U)$. ...
0
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1answer
23 views

Why do we call this 'homogeneous oscillation'?

For a pde, a solution of the form $u(x,t)=u_{*}(kx-\omega t)$ are sometimes called wavetrains. Here, $k$ is the wavenumber and $\omega$ is the frequency. Let $k=0$, then, I've heard that we are ...
0
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1answer
27 views

Passing from classical formulation to weak formulation for a general PDE

I am reading a paper dealing with a general elliptic PDE that I need to transform from classical formulation to weak formulation: $$\left\{\begin{matrix} - \sum_{i=1}^n \sum_{j=1}^n (a_{ij} u_{x_i})_{...
1
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0answers
29 views

Mean Value Property for harmonic functions

I would appreciate any insights on this matter; Let us consider the mean value property for harmonic functions in three dimensional euclidean space. This suggests that the value of a harmonic ...
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0answers
12 views

Optimal Transport of a Translated Density

I've been writing an implementation to the Monge-Kantorovich optimal transport problem in Euclidean space with a quadratic cost function. The outline is that we have densities $f(x)$ and $g(y)$ s.t. $...
0
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1answer
21 views

Find the solution of the Cauchy problem for the pde

Find the solution of the Cauchy problem for the pde $x\dfrac{\partial z }{\partial x}+y\dfrac{\partial z}{\partial y}=z$ on $D=\{(x,y,z):x^2+y^2\neq 0,z>0\}$ with the initial condition $x^2+y^2=...
1
vote
1answer
36 views

variational principle for the principal eigenvalue

I am reading the proof of theorem 2 in chapter 6 Evan PDE. I have difficulty verifying the following part of the proof, i.e. 3 questions here. 1) The assumptions $u\in H_0^1(U)$ and $u\in L^2(U)=1$ ...
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0answers
18 views

Ask for an example of the following type of parabolic PDE with analytic solution

I aim to find an example of the following type semi-linear PDE with analytic solution to test my numerical method: $$\begin{cases} u_t=Lu+g(t,x,u,u_x), &(t,x)\in [0,T]\times \mathbb{R}\\ u(0,x)=f(...
1
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1answer
24 views

Proof that Laplace equation permits no local maxima

I glanced at a proof of the uniqueness theorem for Laplace's equation which implicitly relied on the non-existence of local maxima in the solution, and realized I didn't know how to prove that very ...
1
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1answer
31 views

Diffusion equation involving dirac delta term

I've ran across the following diffusion equation: $$\frac{\partial c_i(r,t)}{\partial t}- a \nabla^2c_i(r,t)=b \delta[x-x_1(t)]$$ where $a$ and $b$ are constants related to the context, $\delta$ is ...
1
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0answers
11 views

Estimate the drift and diffusion function numerically

I have a 1D problem as following $$\frac{\partial f}{\partial t} = \frac{\partial}{\partial x} \Big[ \frac{1}{2} \frac{\partial (g(x) f)}{\partial x} -\mu(x)f \Big]$$ I have a time-series of ...
1
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0answers
20 views

How to extend a function to be periodic and smooth?

Assume we have a function f(x) that is twice differentable on [0, L]. Let us define F(x) = f(x) on [0, L], F(x) = -f(-x) on [-L, 0], and F(x + 2L) = F(x) outside of [-L, L]. Thus, F(x) is ...
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0answers
20 views

definition of a traveling wave (solution)

today I've got a problem with understanding what is meant by the term "traveling wave solution" of a PDE in a more general sense. To be more precise, I've got a following Poisson equation \begin{...
2
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1answer
36 views

Is this a general identity for the Resolvent? [solved: integral representation of the resolvent]

And if so, what is it called? $$i(H-\lambda - i\epsilon)^{-1}\phi = \int_0^\infty e^{-\epsilon t} e^{i\lambda t}e^{-iHt}\phi\,\text dt$$ as in Reed-Simon XIII.7 example 1. It is stated there for $H=-i\...
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0answers
28 views

Intuition behind steps in formulating Finite Element Method

Let's consider the classic elastostatics case where the strong form of the PDE is: $\sigma _{ij,j}+b_i =0$ on V By multiplying through by weighting functions and integrating we can create an ...
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0answers
15 views

Alexandrov Maximum Principle and $W^{2}_p$ estimates

I'm reading an article of N. V. Krylov: About an example of N. N. Ural'tseva and weak uniqueness for elliptic operators, Nonlinear partial differential equations and related topics, 131–144. This ...
1
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1answer
38 views

A Young's inequality used to bound curvature terms

I've been having a look at the Gage and Hamilton's The Heat Equation Shrinking Convex Plane Curves (here). In particular I've been working in the Lemma 4.4.2 and some further results where they find ...
0
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0answers
20 views

Why do degenerate PDEs require weighted Sobolev spaces

Is there a reason that weighted Sobolev spaces are required for degenerate PDEs other than the fact that when one sets up the weak formulation of the PDE the weights are naturally present, so it is ...
0
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1answer
21 views

Solving Neumann problem with only one boundary condition.

I have to find the solution of the problem $$ u_t(t,x)=u_{xx}(t,x)+2e^{t-x} \text{ in set } R_+\times R_+ $$ With boundary conditions: $$ \begin{cases} u(0,x)=7\cos x &\text{ for } x>0\\ u_x(...
0
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1answer
29 views

2d laplace equation seperation by variables with confusing boundaries

Solve Laplace's equation (using separation of variables) $U_{xx}+U_{yy}=0$ for $U(x,y)$ on square $0 \leq x \leq L$, $0 \leq y \leq L$ subject to following boundary conditions: $$U(x,L)=0$$ $$U_x(0,...
0
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0answers
19 views

Constructing the general solution of a PDE using eigenfunctions

As I am learning how to solve PDE, I came into a rather bothersome problem. Let's say that I have to solve the Laplacian of a field $u(x,y)$, where I have $$\nabla^{2}u=0,$$ where $$\nabla^{2}=\frac{\...
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0answers
32 views

Existence of solutions of $\Delta u = f$ such that $|u|_{L^\infty} < \infty$ if $|f|_{L^\infty} < \infty$.

I am struggling with figuring out the details of proposition 7.1. in the paper Curvature and Uniformization - R. Mazzeo and M. Taylor. Setting is as follows. Let $\Omega$ be a noncompact Riemann ...
0
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0answers
64 views

Maximum Principle for Poisson Equation for functions in sobolev spaces

I saw this topic Maximum Principle for Poisson Equation which says about Maximum Principle for Poisson Equation but the solution u is smooth. Can I apply maximum principle if u belongs to a sobolev ...
0
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0answers
18 views

Definition of complete integrals - existence of envelope?

Let suppose a partial differential equation: $$\Phi(x,y,z,\partial_x z,\partial_y z)=0\qquad (1)$$ In some books I have found the following definition: Let $\Lambda$ and $\Omega$ be two open subsets ...
0
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1answer
39 views

boundedness of inverse (Evans PDE)

First, I have difficulty understanding why $$\|u_k\|_{L^2(U)}>k\|f_k\|_{L^2(U)}$$ is being assumed in theorem 6 chapter 6.2 Evans. Second, the last sentence of the proof says (30) implies $\|u\|_{...
3
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2answers
70 views

Proving the range of operator is closed

I have a hard time understand (2) of the Fredholm alternative in Evan's Appendix. To prove the image of $I-K$ is closed, what result from functional analysis is used? I am lost in understand the ...
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0answers
24 views

General solution for this PDE?

let $f$ be a function maps $\mathbb{R}^2$ to $\mathbb{R}$. let: $u=f^{(1,0)}(x,y)$ $v=f^{(0,1)}(x,y)$ which are partial derivatives w.r.t the first & second argument of $f$. solve $f(h, \...
0
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1answer
40 views

Solving PDE using characteristics

I am having trouble solving $u_t+u^2u_x=0$ with initial value condition $u(x,0)=$\begin{cases} 1, & \text{if $x<0$} \\ 1-x, & \text{if $0<x<1$}\\ 0, & \text{if $x>1$} \end{...
0
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0answers
29 views

Is it true that $|∇u(x)|^2\chi_\Omega=|\nabla (u \chi_\Omega)|^2$

Let $u\in L^\infty(\Omega)\cap H^1(\Omega)$ with $\Omega$ open, bounded and regular (as you wish) domain of $\mathbb{R}^N$. Is it true that $$ \int_\Omega |\nabla u(x)|^2 \mathrm{d} x= \int_{\mathbb{...
3
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1answer
55 views

What is the regularity of the eigenfunctions for self-adjoint operators with non-smooth coefficients?

There are well know result about elliptic operators, $L$, that guarantee that an operator of order $2k$ generates a basis for $H^{k}$ on smooth domains.( See Do eigenfunctions of elliptic operator ...
0
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2answers
36 views

Solving the 1-D diffusion equation

For the equation $$u_t = Du_x$$ where $D$ is a diffusion constant, we can define the system $$u_x=v$$$$u_t=Dv_x$$ However, how does one solve for $v$? $\frac{\partial u}{\partial x}=v \iff {\partial ...
0
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1answer
33 views

Sturm–Liouville equation and the Eigenvalue general Problem (PDE)

As I am studying for my Partial Differential Equations exam, I came across Sturm–Liouville equation where it says that it's solutions $y(x)$ are the eigenfunctions of the general problem $Ly=λy$. I do ...
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0answers
17 views

Nonlinear PDE: regularity issues

I have this execrise I've been trying to solve Let $B=\{x \in \mathbb R^n: |x|<1\}, \quad a: \mathbb R^n \rightarrow \mathbb R , \quad a\in C^\infty(\mathbb R^n) \cap L^\infty(\mathbb R^n) $ $$\...
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0answers
14 views

Maxwell-type system: how to solve?

$\DeclareMathOperator{\curl}{curl}\renewcommand{\div}{\mathop{\mathrm{div}}}$ Consider the following system of equations for the unknown vector field $A$ in the unit 3d ball $B$ with boundary $S$: $$ ...
0
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3answers
94 views

Solution to $\frac{d^2f}{dr^2}+\frac{1}{r}\frac{df}{d r}=0$

I know that $f(r)=aln(r)+b$ where $a$ and $b$ are constants is a solution of $$\frac{d^2f}{dr^2}+\frac{1}{r}\frac{d f}{dr}=0$$ are there any other solutions to this, would appreciate it if someone ...