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

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105 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 ...
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1answer
47 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 ...
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1answer
86 views

Chain or product rule for heat diffusion equation

A portion of the heat diffusion equation for a 1-D solid is given as: $$\frac{1}{r} \frac{\partial}{\partial r} \left(r \; k \frac{\partial T}{\partial r} \right)$$ Apparently this can be expanded ...
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1answer
39 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} = ...
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71 views

Semilinear Poisson Equation Using Direct Method of Calculus of Variations

The following problem comes from: http://people.physics.anu.edu.au/~gvn105/analyticMethPDE.pdf 12.9 Exercises 12.3: Let $\Omega$ be a bounded domain in the plane with smooth boundary. Let $f$ be ...
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1answer
28 views

PDE $ u_{x}+u_{t}+f(x)*u=0$

How would I solve this pde using characteristic line? $u_{x}+u_{t}+f(x)u=0$---arbitrary function f $u(x,0)=u_{0}(x)$---$u_{0}$ can be any value $u(0,t)=\varphi(t)$---non-homogeneous where $u(x,t)\ge ...
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1answer
36 views

Checking a solution of a PDE

I have the following PDE: \begin{equation} -yu_x + xu_y = 0 \quad\text{where } u(0, y) = f(y) \end{equation} I derived a solution as follows: \begin{align} -yu_x + xu_y =& 0 \\ \iff& ...
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1answer
28 views

Determining the expressions of the coefficients of the full Fourier series from the Complex Series

Let $l \gt 0$ be a positive real number, and $\phi:[-l,l]\rightarrow\mathbf{R}$ be the function defined by: $$\forall x\epsilon[-l,l], \phi(x)=e^x $$ (1) Calculate the coefficients of the Full ...
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1answer
156 views

Burger's Equation Shock Solutions

So what I'm confused about is how you go about finding shock waves. So suppose we are given the Cauchy problem for Burger's equation $u_t + uu_x = 0$ with $u(x, 0) = 1$ for $x \le 0$ and $u(x, 0) = ...
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1answer
63 views

Characteristics PDE with discontinuity

So there is an example problem in the textbook and I really just don't understand what's going on, both mathematically and conceptually. The problem is solve $z_x + 2zz_y = 1$ with boundary ...
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1answer
29 views

Symmetry results of solutions for elliptic equations

In the celebrated paper of Gidas, Ni, and Nirenberg, see http://web.math.unifi.it/users/magnanin/Dott/BibliografiaCorso/GidasNiNirenberg79.pdf ,certain symmetry results of positive solutions of ...
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0answers
18 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 ...
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0answers
42 views

Fourier transform for pde

I solved the following PDE: $u_{t}-(u_{t})_{x}+(b+1)uu_{x}=bu_{x}u_{xx}+uu_{xxx}$ numerically, using Fourier Transform method. For this i wrote it in the following way: ...
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0answers
28 views

Establishing a certain bound

If I know the bound of the certain derivatives: $$|\dfrac{\partial^k}{\partial x^{k_1}_1\cdots \partial x^{k_n}_n}V(x)|\leq M\epsilon^{-k} \exp(-m \epsilon^{-1}g(x)),$$ where $k=k_1+\cdots+k_n$ and ...
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1answer
92 views

Deriving the PDE for basket option

The payoff for basket option is max($w_1S_1+w_2S_2 -k,0)$. Using Ito's formula, I need to derive the PDE, where $dS_1 = rS_1dt + \sigma_1 S_1dW_1$ $dS_2 = rS_2dt + \sigma_2 S_2dW_2$ I need some ...
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2answers
60 views

Let $a,b,c \,$ be continuous functions defined on $\Bbb R^2$.

I am stuck on the following problem: Let $a,b,c \,$ be continuous functions defined on $\Bbb R^2$.Let $V_1,V_2,V_3$ be non-empty subsets of $\Bbb R^2$ such that $V_1 \cup V_2 \cup V_3 =\Bbb R^2 $ ...
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1answer
104 views

Combining two partial differential equations into one

I have the equation $$ \frac {\partial v}{\partial t}= \gamma \left(1-\frac{v}{v_0}\right)+\alpha \left(1-\frac{v}{v_0}\right)\rho-\beta (\rho-\rho_0) $$ and the mass conservation equation $$ ...
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0answers
53 views

Laplacian eigenvalue problem

I'm working through a PDE problem and we are given the eigenvalue problem $-\Delta u = \lambda u$ with $\frac{\partial u}{\partial n} = 0$ along the boundary given by the rectangle $\Omega = (0, \pi)$ ...
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1answer
75 views

Non-linear Partial Differential Equations

So I know how to solve linear partial differential equations but I am stuck on this new type of problem that is a nonlinear pde. The question is: Determine the solution of $\frac{\partial ...
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2answers
233 views

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 ...
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1answer
84 views

Separation of variables for a non homogeneous PDE $u_t-ku_{xx} = f(x,t),\quad u(0,t)=u(L,t)=0,\quad u(x,0)=\phi(x)$

Separation of variables for a non homogeneous PDE. I found this problem on this page http://www.math.psu.edu/wysocki/M412/Notes412_10.pdf Consider the problem on $(x,t) \in (0,L)\times (0,\infty)$ ...
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1answer
479 views

Proof of the Unsöld's Theorem (the sum of spherical harmonics)

There is an identity concerning spherical harmonics that plays a pretty important role in atomic physics. Thanks to wikipedia (http://en.wikipedia.org/wiki/Spherical_harmonic) I know that its name is ...
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1answer
76 views

Solving the LaPlace PDE by complete induction

I have a problem with an induction step that appears to be bothersome for me to comprehend it. Problem: Show that $f(x_1, \dots ,x_n)=(x_1^2+ \dots + x_n^2)^{1- n/2}$ for $(x_1^2 + \dots + ...
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1answer
34 views

Partial Differential Equations

This is actually a very easy problem but I am brand new to this subject and I just don't the mechanism on how to do it yet. The question is: Determine the solution of $\frac{\partial \rho}{\partial ...
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1answer
106 views

Solution to Anisotropic Heat Equation

I am trying to find the solution to a 1-D anisotropic heat equation. The domain is a line segment of length L (i.e., it's a line segment extending from $x = 0$ to $x = L$). The form of the equation ...
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1answer
50 views

Leibniz's rule in $W^{1,\,p}(\Omega)\cap L^\infty(\Omega)$

Is it true? If $\Omega\subset\mathbb{R}^n$ is bounded and $u,v\in W^{1,\,p}(\Omega)\cap L^\infty(\Omega)$, then $uv\in W^{1,\,p}(\Omega)\cap L^\infty(\Omega)$ and $$\nabla(uv)=u\nabla v+v\nabla u.$$ ...
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1answer
40 views

Show that this simple functional is not bounded below

Define $\varphi(u) = \displaystyle\int_{0}^{1} \displaystyle\frac{{|u'| }^2}{2} - \displaystyle\frac{{u }^2}{4} - hu \ dt$, $u \in H^{1}_{0}(0,1)$ where $h: [0,1] \rightarrow R$ is a continuous ...
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2answers
81 views

Graph (or manifold) Lipschitz satisfy the sphere (ball) condition?

Consider $\varphi: U\subset \mathbb{R}^{n-1}\to \mathbb{R}$ a Lipschitz function and $\Omega=Graph(\varphi)$, i.e., $$\Omega=\{x=(x_1,...,x_n)\in U\times\mathbb{R};x_n=\varphi(x_1,...,x_{n-1})\}.$$ ...
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1answer
20 views

The index of some perturbation about elliptic operator with Robin boundary condition

Let $I$ be an closed interval $[0, 1]$. $C^{2}(\bar{I})$ is the space of all $C^{2}$ functions on $(0, 1)$ with continuity at boundary and usual maximal norm. $C(\bar{I})$ is the space of all ...
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1answer
37 views

transport along a vector field

I have the spatial density u(x,t) of a substance, and I want to describe the simple transport of this substance along a given vector field phi(x,t). Am I correct that the corresponding equation is ...
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0answers
124 views

Newton boundary condition for second order pde

I have a few questions about Newton boundary conditions for a second-order partial differential equation: $$-\text{div}(a(x,u,\nabla u)) + c(x,u,\nabla u)$$ considered on a bounded connected ...
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0answers
71 views

$L^2(S;W^{1,p})$-regularity for solution of parabolic pde in Hilbert space setting

I have a question regarding the regularity theory for parabolic pdes. I am considering the following "minimal working example": Let $\Omega\subset\mathbb{R}^n$, $n\in\mathbb{N}$, be and bounded ...
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1answer
42 views

$f, \hat{f} \in L^{1}(\mathbb R) \implies \widehat {\text{Re}(f)}, \widehat{\text{Im}(f)} \in L^{1}(\mathbb R)$?

Let $f:\mathbb R \to \mathbb C$ such that $f(x)= (f_{1}(x), f_{2}(x))$; where $f_{1}(x)=\text{Re}(f(x))=\text{the real part of} \ f $ and $f_{2}(x)=\text{Im}(f(x))= \text{ the imaginary part of} \ ...
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1answer
182 views

Converting a PDE to a matrix form

I have to solve the problem $U_{xx}-U_{xy}+2U_y+Uyy-3U_{yx}+4U=0$ using diagonal matrix as described in this article page 44 section 3.2 But my problem is there the matrix A is symmetric matrix ...
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1answer
54 views

Differential of Lagrangian

My professor wrote this $\frac{\partial L}{\partial q}\dot{q}=\frac{d}{dt}(\frac{\partial L}{\partial \dot{q}})$. Due to the fact that I am very very very very bad at Math, could you explain me about ...
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1answer
66 views

Solve Partial differential equation(geometric optics)

Solve $x^2((u_x)^2+(u_y)^2)=1$ , $u(x,0)=0$ Use the characteristic equation The solution is $u(x,y)=-\ln\dfrac{\sqrt{x^2+y^2}+y}{x}$ I drove $\dfrac{dx}{dt}=2x^2p$ $\dfrac{dy}{dt}=2x^2q$ $Z=2t$ ...
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1answer
61 views

A maximum principle

Suppose that $\Omega\subset\mathbb{R}^n$ is bounded with smooth boundary $\partial\Omega$. Consider the elliptic boundary value problem for $\phi=\phi(x)$, $x\in\mathbb{R}^n$: ...
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0answers
38 views

Anyone help me with this PDE using Fourier Transform?

I have this: $$\frac{\partial c}{\partial t} + p\frac{\partial c}{\partial z}+\lambda p\frac{\partial^{2} c}{\partial z\partial t}-\frac{\partial^{2} c}{\partial z^2}=0\quad(1)$$ $$c(z,0)=\delta(z)$$ ...
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1answer
24 views

$\int_{|x|<t} |\mathcal{F}^{-1}f(x) |dx \leq C t^{\frac{n}{2}}\|f\|_{L^{2}}$?

Let $f\in L^{2}(\mathbb R^{n}).$ Fix $t>0,$ My Question:How to show, $\int_{|x|<t} |\mathcal{F}^{-1}f(x)| dx \leq C t^{\frac{n}{2}}\|f\|_{L^{2}}$ ? [We note $\mathcal{F}$ denotes the ...
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1answer
77 views

Imbedding theorem for Sobolev space $D^{k,p}$

I can't find any reference on the imbedding theorem for the Sobolev space $D^{k,p}$, which is defined by $$D^{k,p}(\Omega)=\{u\in L_{loc}^{1}(\Omega)|D^{k} u\in L^{p}(\Omega)\}.$$ Is it the same as ...
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1answer
107 views

Cauchy-Riemann equations and harmonic functions

I have the following question which I am supposed to use the Cauchy-Riemann equations to prove: Let u and v have continuous second derivatives satisfying: $$\frac{\delta u}{\delta x} = \frac{\delta ...
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0answers
146 views

Hölder estimate of solution to linear parabolic PDE

Consider a standard Cauchy-Dirichlet problem for linear parabolic PDE: \begin{cases} u_t + \mathcal{L}u=-g, \ (t,x) \in [0,T)\times \Omega \\ u(T,x)=0 , \ x\in \Omega\\ u|_{\partial \Omega}=0 ...
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0answers
34 views

Help with semilinear PDE problem

I need some help: Let $T>1$, $\Omega_T=\{(x,y)\in\mathbb{R}\>|\>x\in(1,T)\}$, $\Gamma=\{(x,y)\in\mathbb{R}\>|\>x=1\}$ and $g\in C^1(\mathbb{R})$. Prove that there exists $T>1$ such ...
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94 views

Fractional Sobolev spaces and weighted L2 spaces

For $s\in[0,1]$ define function spaces $H^s(\mathbb{R})=\{u\in L_2(\mathbb{R}): (1+|\cdot|^2)^{s/2}\mathcal{F}u\in L_2(\mathbb{R}) \}$ (where $\mathcal{F}$ denotes the Fourier transform) i.e. the ...
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1answer
140 views

Solving the Laplace equation in terms of exponential of hyperbolic trigonometric functions

I'm solving the Laplace equation $U_{xx}+U_{yy}=0$ subject to BC's: \begin{align} U(0,y) &= 0 \\ U(a,y) &= 0 \\ U(x,0) &= 0 \\ U(x,b) & = \left\{x \text{ for } x \in ...
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1answer
85 views

Wave Equation Descending from 2D to 1D

I am stuck deriving the familiar d'Alembert formula using the Method of Descent, going from 2D to 1D. After using Kirchhoff's Formula, writing the solution independently of $y$ and some manipulations, ...
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27 views

Conjugating an operator with a gauge transformation; how is the kernel affected.

For the differential operator $$ D := i I \frac{d}{dx} + A(x) \colon C^\infty_T([0,\beta]),\mathbb{C}^m) \to C^\infty_T ([0,\beta],\mathbb{C}^m) $$ where $A(x)$ is Hermitian and $C^\infty_T ...
3
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2answers
445 views

dual of $H^1_0$: $H^{-1}$ or $H_0^1$?

I have a problem related to dual of Sobolev space $H^1_0$. By definition, the dual of $H^1_0$ is $H^{-1}$, which contains $L^2$ as a subspace. However, from Riesz representation theorem, dual of a ...
2
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2answers
145 views

Wave equation how to derive the form $u(x,t)=f(x+ct)+g(x+ct)$

I am referring Walter A. Strauss PDE book. There in solving the wave equation I have several parts which I don't understand. $U_{tt}=c^2U_{xx}$=(${\partial \over \partial t}$-$c \partial \over ...
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1answer
94 views

Schrödinger equation

How to prove that map $f\mapsto u$ from initial value to solution of Schrödinger equation is continuous map of $S(R^n)$ to $C^\infty(R^n,R)$? Thanks in advance.