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

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4
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1answer
21 views

Fundamental solution of the Laplacian on the surface of a cylinder

Does the Laplace operator have a fundamental solution on the surface of a cylinder in $\mathbb{R}^3$? Intuitively, I can visualize a function that diverges to $\infty$ at a point, decreases to a ...
0
votes
1answer
20 views

Heat Equation Steady state question

Say you have a slab of material occupying the region $0\leq x\leq a$. Heat is supplied at a constant unit rate so the temperature T(x,t) satisfies $\partial T$/$\partial t$= $k$ $\partial^2 ...
1
vote
0answers
24 views

Variational methods : Why i can't apply this theorem?

Consider the following problem: Find a weak solution for $$u'' + u = \sin t ,\,\, 0 < t < \pi$$ $$u(0) = u(\pi)=0 $$ the corresponding functional for the problem is $\varphi(u) = ...
11
votes
2answers
175 views
+500

Positivity of principal eigenvalue for $L\phi=-\triangle \phi + \nabla \cdot( u \phi )$

EDIT: This question is still unresolved as of April 18. The two answers provide useful work in the right direction, but neither resolves the question. A counterexample should have $u \in ...
0
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0answers
12 views

FEM PDE's How do I find the best solution using abstract methods for FE.

I have a homework problem I am working on, and I need a little help or maybe a nudge in the right direction.... Question: (Purpose: Review abstract finite element methods.) Consider the ...
1
vote
0answers
20 views

Extension of function from $W^{1,p}(\Omega - \{x\})$ to $W^{1,p}(\Omega)$

Suppose $\Omega\subset \mathbb{R}^n$ is open, $p\geq 1$, $n\geq 2$ and $u \in W^{1,p}(\Omega-\{x\})$. Show that $u$ extends to a function in $W^{1,p}(\Omega)$. So far I have; clearly $u$ and each ...
0
votes
0answers
23 views

Solving $u_{yy} + (2-x)u_y - 2xu = 1$

I want to solve the pde $$ u_{yy} + (2-x)u_y - 2xu = 1 $$ so if I treat $x$ in the coefficients as arbitrary but fixed it is equivalent to solving the ode $$ y'' + (2-x) y' - 2x y = 1. $$ For the ...
0
votes
0answers
12 views

Newtonian potential is a solution of $\Delta u = f$

Firstly, I take a course in PDE. Each week, I have a hard time. The following are from my note. In particular, the lines are highlighted where I cannot understand. (I think I should write ...
1
vote
2answers
28 views

Laplace equation on semidisk

I am interested in the solution of the following boundary value problem on the semidisk $D=\{(r,\theta): 0<r<1, 0<\theta<\pi\}$: $$u_{xx}+u_{yy}=0 \mbox{ in } D, $$ $$u(1,\theta)=0 \mbox{ ...
1
vote
1answer
10 views

Change of variables in PDE

I need to use a change of variables in this PDE $f_{xx} - f_{yy} = 0$, using $s = (x + y)/2$ , $t = (x - y)/2$ I get $f_{ts} = 0$ But I'm asked to deduce that the general solution is of the form ...
0
votes
2answers
20 views

finite difference scheme for nonlinear partial differential equations

I have the following second order partial differential equation (PDE) on $[0,T] \times \mathbb{R},~ T >0 $: \begin{equation} \left(1 + \frac{1}{(1 + b f)^2}\right) \frac{\partial f}{\partial t} ...
2
votes
0answers
40 views

Sobolev spaces necessary and sufficient condition

I am studying PDE now using mostly Evan's book. I have been thinking about the following problem and I don't know whether there is a solution or not. Problem Formulation: Let $A$ be a subset of ...
0
votes
0answers
10 views

Solution to a system of partial differential equations for two functions in two variables

For two functions $f,g:\mathbb{R}^2 \rightarrow \mathbb{R}$ and coefficients $A,B,C,D:\mathbb{R}^2 \rightarrow \mathbb{R}$, consider the following system of coupled partial differential equations: ...
0
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0answers
8 views

this function is in some Holder space?

Consider $\Omega$ a open, bounded, and smooth domain of $R^N$ with $N \geq 3.$ And let $f: \Omega \times R \rightarrow$ a Caratheodory function . Supoose that $f$ is locally Lipschitz. Supose that ...
2
votes
0answers
18 views

Relationship between eigenvalues of differential operator and eigenvalues of its adjoint operator.

I am considering $L\phi = -\triangle \phi + u \cdot \nabla \phi$ and its "adjoint" operator $L^* \phi = -\triangle \phi - \nabla \cdot (\phi u)$ on a bounded domain $\Omega \subseteq \mathbb{R}^n$. ...
0
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0answers
18 views

On a article of Ambrosetti and Rabinowitz: About the regularity of weak solution for a boundary problem

Consider $\Omega \subset R^n$ $(n \geq 3) $ a bounded and smooth domain. Let $f: \Omega \times R \rightarrow R$ a Caratheodory function. Supose that exists $0 \leq \alpha < (N+2)/(N-2)$ and $c,d ...
1
vote
1answer
38 views

Solve the following wave equation

"Solve the wave equation: \begin{cases} u_{tt}(x,t)=c^2u_{xx}(x,t), 0<x<\pi, t>0 \\ u(0,t)=t, u(\pi,t)=(1+\pi)t,\\ u(x,0)=0,\\ u_{t}(x,0)=\sin(x)+x+1 \end{cases} Hint: Consider $u_s(x,t)$ a ...
2
votes
1answer
78 views

Second Derivative of Eigenvalue

Consider the potential $V_t=tx$ and the corresponding Schrödinger operator $H_t=- \frac{\partial ^2}{\partial x^2}+V_t$ on $L^2([0,R])$ with Neumann or Dirichlet boundary conditions. Let ...
2
votes
1answer
24 views

Intregation by parts

Let $u=(-R_2\theta,R_1\theta)$ where $R_1,R_2$ are the usual Riesz Transforms in $\mathbb{R}^2$, $\mathbb{T}^2$ denotes the torus, $\theta\in C^{\infty}(\mathbb{T}^2)$ and ...
0
votes
1answer
12 views

Solving a PDE: basic first-order hyperbolic equation $u_t+cu_x=0$

So I have to solve the first-order hyperbolic equation $u_t+cu_x=0$ and $c$ as a constant. It is a PDE, since there is the time and spatial variable, but I'm overwhelmed by the maths given in books of ...
0
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0answers
19 views

PDE subdifferential question

Let $ H: \mathbb R^{n} \rightarrow \mathbb R $ be convex. We say $\nu$ belongs to the subdifferential of $H$ at $p$, written $\nu \in \partial H(p)$, if $H(r)\ge H(p)+ \nu \cdot (r-p)$ for all $r \in ...
0
votes
1answer
17 views

What are good resources for learning Numerical methods for Partial Differential Equations?

I'm having an undergraduate course on Numerical Solutions to Ordinary and Partial Differential Equations. I need online resources to supplement my study preferably videos and books. I want to build a ...
1
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0answers
17 views

How to prove comparison principle for parabolic PDE (nonlinear)

Suppose $F:\mathbb{R} \to \mathbb{R}$ is smooth with $F(x) > 0$ for $x > 0$ and $F:(0,\infty) \to (0,\infty)$ continuous and increasing with $F(0) = 0$. Consider the PDE $$u_t = \Delta F(u) ...
3
votes
3answers
79 views

$\Delta u = f, f \in L^q \Rightarrow u \in W^{2,q}$ References

I'm looking for references for the following theorem. I will very grateful Theorem: [Calderón Zigmund] If $u$ is a solution of \begin{equation} \Delta u = f \quad \mbox{in} \quad B_2 ...
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0answers
33 views

Finding an upper bound on the gradient of the solution to the heat equation

I have a function $u:\mathbb{R}^n\times [0,T]\to \mathbb{R}$ that solves the heat equation $u_t=\Delta u$, is bounded, and $u(x,0)=g(x)$. I need to show that $$\max|\nabla u(x,t)|\leq ...
3
votes
2answers
183 views

What is the *standard duality argument?

What is the standard duality argument? I saw this foor exemplo in the following statement. The case $p < 2$ follows from the standard duality argument. To prove Theorem: [Calderón Zigmund] If ...
0
votes
0answers
19 views

How can I write this in Divergence form

Consider the PDE $u_{xx}-(yu_y)_x-y(u_x)_y+yu_y+(y^2+\frac{1}{H^2(x)})u_{yy}$ I need to write this in divergence form. That is, I need to write it in the form $\sum_{i,j}\frac{\partial}{\partial ...
0
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0answers
8 views

Finite Difference Scheme for a PDE on non-rectilinear coordinates

Consider poisson's equation on the domain $0 \leq x \leq 1$ $0 \leq y \leq H(x)$. Change the coordinates to $\xi=x$, $\eta=y/H(x)$. Construct a FDS that gives a positive definite symmetric matrix. ...
1
vote
1answer
42 views

How to determine where a non-linear PDE is elliptic, hyperbolic, or parabolic?

I'm trying to understand the classification of PDEs into the categories elliptic, hyperbolic, and parabolic. Frustratingly, most of the discussions I've found are "definition by examples.'' I think ...
0
votes
1answer
25 views

An application of the mountain pass lemma

I am trying to show the existence of classical solution for the following problem using the mountain pass theorem : $$ \left\{ \begin{array}{ccccccc} u^{''} + \lambda u + u³ = 0 (0<t<\pi)\\ ...
0
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0answers
14 views

Pseudo-monotone operators research paper question

Hi I just want to know if anyone can see how the result (2.34) is obtained in the following research paper http://caa.epfl.ch/publications/9-Boccardo-Dacorogna1984.pdf. Thanks, I know that it is a ...
0
votes
0answers
18 views

Elliptic regularity of Dirichlet problem

Suppose $\Omega\subset\mathbb{C}$ is a simply connected domain with $C^\infty$ boundary. Consider the following Dirichlet problem $$\Delta u |_{\Omega}=0 $$ $$u|_{\partial\Omega}=f$$ Under what ...
0
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0answers
7 views

Obtaining characteristic v on Cauchy Problem

$(x-y)p+(y-x-z)q=z$ Find the integral surface which the curves it passes are $z=1$ and $x^2+y^2=1$ Here is my try. $$\frac{dx}{x-y}=\frac{dy}{y-x-z}=\frac{dz}{z}$$ So we have ...
0
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0answers
14 views

$C^\infty$ approximations of $f(r) = |r|^{m-1}r$ [duplicate]

Consider $f(r) = |r|^{m-1}r$ where $m \geq 1$. Is it possible to find $C^\infty$ functions $f_n$, such that $f_n \to f$ uniformly on compact subsets of $\mathbb{R}$, $f_n' \to f'$ uniformly on ...
0
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0answers
11 views

What is the motivation for characterizing second order linear PDEs as hyperbolic, elliptic, or parabolic?

I see the connection between the PDEs and the equations of conic sections, but why is that important? I am under the impression that one of the big differences between the wave equation and the heat ...
0
votes
1answer
23 views

Find a bound for the summation $\sum_{j=k}^J jc^{k - j - 1} $

The problem: I've hit what might be a dead-end. If it is true, I would like to show that for $c \in (0,1)$ and $1 \leq k \leq J$, the sum $$ \sum_{j=k}^J jc^{1-j-k} = \sum_{n=1}^{J+1-k} (k-n-1) ...
1
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0answers
40 views

Applications of PDEs

I teach an undergrad ODE course. As I have completed basically all the material, I thought it would be nice to give the students a brief introduction to PDEs. At the end of the lecture, I said that ...
0
votes
1answer
25 views

Integrating the Gaussian kernel with absolute value

How do I integrate: $$\dfrac{1}{\sqrt{4\pi t}}\int_{-\infty}^{\infty}|y|e^{\frac{-(\xi-y)^2}{4t}}dy,$$ in terms of of the error function, erf$(x)=\dfrac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}dt$. I have ...
0
votes
1answer
12 views

Solving the reaction-diffusion equation for a single species

$$ \frac{\partial u}{\partial t} =k\Delta u+ru. $$ Where all of the bounds are $0$. Please help! Very new to PDE's and don't understand how to solve this. I know that I need to use separation of ...
3
votes
1answer
43 views

How to solve Cauchy problem?

I'm new to this problem. Here is the question. $$(y+xz)z_x+(x+yz)z_y=z^2-1$$ Find the integral surface which the curves it passes are $y=1$ and $z=x^2$ By Lagrange system i found $u$ and $v$. We ...
0
votes
0answers
22 views

Final value problem PDE

I need help unterstanding and solving the following problem $\rho c_p \frac{\partial\lambda}{\partial t} + k \frac{\partial^2\lambda}{\partial x^2} + 2 [T(x,t,q(t)) - Y(t)] \delta(x-x_{s}) = 0 $ ...
3
votes
3answers
128 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 ...
0
votes
1answer
28 views

Implicit Euler Scheme and stability

Find the fixed points of the implicit Euler scheme \begin{equation} y_{n+1}-y_{n}= hf(t_{n+1},y_{n+1}) \end{equation} when applied to the differential equation $y'=y(1-y)$ and investigate their ...
0
votes
0answers
30 views

Partial Derivative With Respect to $t$

What is $\frac{\partial v}{\partial t}$ if $v$ can be defined as $v(x,t,\zeta)=w(x(3t)^{-1/3},\zeta (3t)^{1/3})$?
2
votes
0answers
11 views

Finite Element Method Weak Formulation

I have a question about the weak formulation of a PDE in finite element analysis. Suppose we have the following two-dimensional PDE: $$ \Delta \cdot u(x,y) = q(x,y) $$ where $q$ is given, $u$ is ...
2
votes
1answer
46 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 ...
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0answers
28 views

Regularity of Dirichlet Eigenvalues on Lipschitz Domain

What kind of regularity do we generally have for weak solutions to the Dirichlet problem? $$(\Delta+\lambda)u=0 \textrm{ in }U$$ $$u=0 \textrm{ on }\partial U $$ where $U$ is a planar domain with ...
0
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0answers
6 views

Modified Symplectic Euler

Simple harmonic motion: $y'= -z $, $z'= f(y)$ and the modified Symplectic Euler equation are $$y'=-z+\frac {1}{2} hf(y)$$ $$y'=f(y)+\frac {1}{2} hf_y z$$ deduce that the coresponding approximate ...
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0answers
44 views

Show Green's function solves Poisson's equation

I am reading Walter Strauss's book Introduction to PDE. The definition of Green's function is as follows: The Green's function $G(x)$ for the operator $-\Delta$ and the domain $D \subset ...
1
vote
1answer
23 views

$1^{st}$ order PDE in population system

Here is the age-structured continuous population partial differential equation: \begin{equation} \left\{ \begin{array}{lcl} \frac{\partial p(a,t)}{\partial a}+\frac{\partial p(a,t)}{\partial t} = ...