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

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3
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71 views

About the symmetric nature of Green's function.

What is the significance of Green's function being symmetric ? How do I understand intuitively ?
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209 views

Elliptic PDE regularity

I have $u \in H^1_0(\Omega)$ where $\Omega$ is bounded which solves some elliptic PDE of the form: $$-\Delta u + h(u) = f$$ in $\Omega$, where $f \in L^2(\Omega)$ $$u = 0$$ on $\partial\Omega$, say. ...
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152 views

Trying to solve Navier equations with Maple

I've been trying for ages now to get this to work, it's a system of partial differential equations (Engineering, Plate theory) $$pde1: = \frac{{{\partial ^2}}}{{\partial {x^2}}}u\left( {x,y} \right) ...
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142 views

Integral representation of directional derivative

We consider the Dirichlet problem $$ \begin{array}{c} \Delta\psi(x) + v(x)\psi(x) = 0, \;\;\; x \in D \\ \psi|_{\partial{D}} = f \end{array} $$ in some bounded region $D$ with smooth ...
3
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161 views

Using games to approximate solutions to PDE's

Hopefully the mathematics community can help me out this one, I'm currently studying my senior capstone at my college, and decided to do some research on a chapter in Stanley Farlow's book "Partial ...
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110 views

Solving a two-dimensional system of conservation laws

I have $$\begin{align} u_x + u_t &= 0\\(\rho u)_x + \rho_t &= 0 \\ \left(Eu\right)_{x}+E_{t}+pu_{x}&=0\end{align} $$ satisfying these boundary conditions: $u\left(x,0\right)=x,\ ...
2
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32 views

PDE question: heat equation (third order??)

I am familiar with the usual heat equation, however, my lecturer gave me this problem and it does not look like anything I have ever seen (in my whole entire life and I am not just being dramatic). ...
2
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32 views

Initial&Boundary Value problem-Fourier

$$u_{t}=u_{xx}, \hspace{5mm} x>0, t>0$$ $$u(0,t)=0 \hspace{3mm} u(x,0)=f(x)$$ We want that the solutions are bounded. We are looking for solutions of the form $$u(x,t)=X(x) \cdot T(t)$$ ...
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21 views

Partial Differential Equations Black Scholes Problem

Part 1) Consider the Black-Scholes problem $$\frac{\partial A}{\partial t}+\frac{\sigma^2B^2}{2}\frac{\partial^2A}{\partial B^2}+rB\frac{\partial A}{\partial B}-rA=0 ...
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17 views

For $f\in L^1_{loc} (\Omega)$, $f=0$ almost everywhere in $\Omega$ provided $\int_{\Omega}f(x)\Phi (x)dx=0 , \forall \Phi \in C_{c}^{\infty}(\Omega)$

I need to show that $f=0$ almost everywhere in $\Omega$ provided $$\int_{\Omega}f(x)\Phi (x)dx=0 , \forall \Phi \in C_{c}^{\infty}(\Omega)$$ Here is how I have decided to proceed. Suppose there ...
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22 views

Kolmogorov equations

I'm having a difficulty to understand the difference between the Kolmogorov Forward and Backward Equation in how they describe probability density rather then their mathematical formulation (I know ...
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19 views

Show a certain modeshape satisfies Helmholtz equation with boundary conditions

I need to show that a mode shape $\psi(x,y,z)$ given by: $p(x,y,z) = \psi(x,y,z)e^{-jk_0\lambda z}$ where $k_0 = \omega_0/c_0$, $c_0$ is the sound speed, and $\lambda$ is an unknown, non-dimensional ...
2
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45 views

Nonlinear Propagation of Discontinuities

I understand that ahead of the discontinuity (wave front) the initial condition ($u_o(x) \equiv u(x,0)$) is smooth. However the green box goes on to state that $u(x,t)$ is smooth ahead of the wave ...
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125 views

Prove difference quotient converges to weak derivative in $L^p$

I am trying to solve the following exercise: Let $U$ be an open set in $\mathbb{R}^n$ and let $V$ be a compact subset with Lipschitz boundary. Assume that $f$ is in $L^p(U)$ with $1<p<\infty$ ...
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19 views

Solving the parabolic equation

I have a parabolic equation $$u=(axu)_{xx}-((bx+c)u)_x$$ and I know the solution is $$\frac{b}{a(e^{bt})} (\frac{e^{-bt}x}{x_0})^{(\frac{h-a}{2a})} exp(\frac{-b(x+x_0 e^{bt})}{a(e^{bt}-1)}) ...
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54 views

A proposition of Urysohn's Lemma in real analysis

I have a problem proving the following theorem, which appears in Evans' PDE text book: Let $K$ be a compact set in space $R^n$, and $K\subseteq \cup_{i=1}^{k} U_{i},k\geq2$, where $U_i$ are open ...
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18 views

Linear PDE of second order

I have the following problem: $ \rho_{tt} +a\rho_{xt}-c^2\rho_{xx}=bv_{xx}$, where $\rho=\rho(x,t)$, $v=v(x,t)$ and $a,b,c$ are constant. My attempt to solve such an equation is to treat $v$ as any ...
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39 views

Kahler-Einstein Metrics in Physics - Topic Suggestions

I am hoping to get some topic suggestions for a presentation I have to give in a couple of weeks. The course the presentation is for is called Kahler-Einstein metrics. I would really like the ...
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38 views

Reformulate this PDE in different notation

I would like to rewrite this general PDE \begin{equation} \alpha\partial_tu+\beta\partial_xu+\gamma\partial_{xx}u+\delta u=\varepsilon \end{equation} in this form $$c\left(x,t,u,\frac{\partial ...
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33 views

Global Holder regularity from DiGiorgi-nash-moser

For a strictly elliptic differential operator $L=\sum_{i,j} D_i (a_{ij}D_ju) $ with strictly elliptic condition on open bounded smooth domain $\Omega$,and bounded coefficients, DiGiorgi-Nash-Moser ...
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63 views

Continuous solutions to a first order PDE

Im giving the pde as follows $(x+y) \partial _x u+(y-x)\partial_y u=0$. First I need to show that a continuous solution must be constant and then deduce that the difference of any two continuous ...
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16 views

Analytic solution to wave equation on a hollow cylinder

Is it possible to find an analytic solution for the modes of vibration of a hollow cylinder, assuming azimuthal symmetry? That is can the following PDE be evaluated: ...
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53 views

Is $\Delta^{-1}$ a bounded operator?

Is the inverse Laplacian $\Delta^{-1}: H^{m+2}(M)\mapsto H^m(M)|1$ a bounded operator? Where $M$ is a compact manifold and $H^m(M)|1$ means its elements $f \in H^m(M)$ and ...
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28 views

Infinitesimal Generator of A One Parameter Group

This is a small problem which drives me crazy. Let $\varphi(x,y,t)=(F_1(x,y,t),F_2(x,y,t))$ be a one paramter transformation group on $\mathbb{R}^2$. Let ...
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18 views

Eigenvalue Function of Laplace Equation discretizes by nine-point stencil

I'm trying to plot the eigenvalue function of the Laplace equation $$-u_{xx}-u_{yy}=0,\;(x,y)\in (0,1)^2$$ with $$u(x,y)=0$$ on the boundary of the unit square. I have the nine-point stencil ...
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43 views

Constructing a function using the Fourier transform

Pick an integer $n\ge 5$ and let $f\in C_{C}^{\infty}(\mathbb{R}^{N})$. We want to use the Fourier transform to formally construct a function $u\in L^{\infty}(R^{n})$ that solves ...
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71 views

Limit under the integral sign and partition of unity

Let $U \subset \mathbb{R}^N$ be a bounded open set and let $\{ U_j \}_{j=1}^\infty$ be an open covering of $U$ such that $U = \bigcup\limits_{j=1}^\infty U_j$. Suppose $\{ \psi_j \}_{j=1}^\infty$ is ...
2
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41 views

Wave Equation with outgoing wave boundary conditions

I need some help with this problem: I have a to solve the wave equation with two initial conditions and with outgoing wave boundary conditions; i.e., $$\begin{cases} u_{tt}-u_{xx} & =0\\ u(x,0) ...
2
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37 views

Help with this eingenvalues problem

I'm trying to find the Laplacian eigenvalues for the Dirichlet problem in a right isosceles triangle $T\subset\mathbb R^2$ where its smaller side has length $c$. I saw in an article that the ...
2
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33 views

Why does an infinite Neumann boundary condition become a Dirichlet condition?

Often when I read a paper I see a statement of the type: Our boundary condition at the surface is $\frac{\partial f}{\partial x} = \alpha$. In the limit of $\alpha \to \infty$ this is equivalent ...
2
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41 views

Sturm-Liouville problem with vibrations - probably easy for most.

Trying to do this one... A model for the transverse vibrations of a stretched string with variable density ρ and tension τ (both continuous and strictly positive on the closed interval [0,l]): PDE: ...
2
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28 views

Extension of Sobolev function

Let $D$ be a convex bounded domain in $\mathbb{R}^{n-1}$. Let $A:D\to\mathbb{R}^{+}$ be a Lipschitz continuous function. Let $\,\Omega\,$ be a bounded domain in $\mathbb{R}^{n}$ of the form ...
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53 views

Show that the space $C^{0, \gamma}(U)$ is complete

How can we show that the space $C^{0, \gamma}(U)$ is complete?? I have tried the following: So that the space is complete, the following has to stand: $$\forall \epsilon >0, \exists n_0 ...
2
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34 views

How to solve this initial-boundary value problem for a PDE

Consider $$u_{tt}-a^2u_{xx}+u_t+a u_x=0,\quad 0<x<\infty,\quad t>0,(*)$$ where $u_t=\frac{\partial u}{\partial t}$ and etc. It is not so hard to use the method of characteristics to solve it ...
2
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23 views

Solution of a parabolic PDE

I'm reading a paper where the following parabolic PDE is considered: $u_t(x,t)=u_{xx}(x,t)+b(x)u_x(x,t)+\lambda(x)u(x,t)$, with boundary conditions $u_x(0,t)=qu(0,t) \text{ and } u(1,t)=\int_0^1 ...
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21 views

Solving the PDE's naturally arising from requiring vector fields to commute

If one has two vector fields $X$ and $Y$ in $\mathbb{R}^4$ whose coefficients are indeterminate functions (some coefficients are being forced to be linearly related to others), then the condition that ...
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20 views
2
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17 views

How to find first-order quasi-linear PDEs form second-order quasi-linear PDE?

Transform $u_{tt} u_{xx}-u^{2}_{tx} + uu_{tt} + 1=0 $ into first-order quasi-linear PDEs. Attempt: $u_{tt}(u_{xx}+u)=(u_{tx}-1)(u_{tx}+1)$ To get $u_{tt} = u_{tx}-1\Rightarrow u_t = u_x ...
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39 views

if $\Delta u = |\nabla u|^{3/2}$ then $u \in C^{\infty}$

Suppose $u \in H^1_{\text{loc}}(\mathbb{R^2})$. I want to show that if $\Delta u = |\nabla u|^{3/2}$ in the distributional sense then $u \in C^{\infty}$. I know that since $\nabla u \in L^2$ (I'll ...
2
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22 views

Solution operator for Laplacian is a continuous operator $C^{1,\alpha}(\partial D) \to C^1(\overline D)$

I consider the Dirichlet problem $$ \begin{cases} \Delta u(x) = 0, \quad x \in D,\\ u|_{\partial D} = \varphi, \end{cases} $$ where $\varphi \in C^{1,\alpha}(\partial D)$ and $D$ is a ...
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43 views

Deriving a given result of a Proof

I am going over the proof of the Cauchy–Kovalevskaya theorem using analytic majorization (for the second order PDE case) and I am unable to derive the result given by equation $(4)$. The details are ...
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35 views

Inequality involving $H^s$ and $L^2$.

I have this inequality which I don't see how to prove it. We have $F \in C^s$, and $u\in H^s$. I want to show that: $$\| F\circ u \|_{H^s} \leq C(\| F \circ u \|_{L^2}+\sum_{r=1}^s \sum_{j=1}^r ...
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50 views

Proof of Heisenberg Uncertainty Principle Exercise

I'm not very knowledgeable in QM, and I know many physics books derive the uncertainty principle using commutators, but as an exercise in my PDE book (by Asmar), I should be able to derive it from one ...
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55 views

If $-\Delta u=f$ and $f$ is analytic, then $u$ is analytic as well.

I know the theorem that if $-\Delta u=f$ and $f$ is analytic, then $u$ is analytic as well. The book I know that contains the proof of this theorem all use the approach with respect to complex ...
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69 views

Regularity energy minimizing harmonic maps

I am using the book "Geometric Measure Theory- An introduction" by Fanghua and Xiaoping. I'm studying the proof of the following Lemma (Lemma 2.1.8 page 38). This chapter is dealing with the theory ...
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55 views

Equipartition of energy

Let $u$ solve the initial-value problem or the wave equation in one dimension: $$\begin{cases}u_{tt}-u_{xx}=0 & \text{in } \mathbb{R} \times (0,\infty) \\ u = g, u_t = h & \text{on } ...
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55 views

Derive $u(x,t)$ as a solution to the initial/boundary-value problem.

Given $g : [0,\infty) \to \mathbb{R}$, with $g(0)=0$, derive the formula $$u(x,t)=\frac{x}{\sqrt{4\pi}}\int_0^t \frac 1{(t-s)^{3/2}}e^{-\frac{x^2}{4(t-s)}}g(s)\,ds$$ for a solution of the ...
2
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0answers
49 views

Duhamel's principle in constructing heat kernel

I want to construct the heat kernel $k_t(x,y)$, which is the fundamental solution to the heat equation $(\partial_t + \Delta_x)u(t,x) = 0$. There is something that I don't understand about using ...
2
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0answers
43 views

Confusion about domains in PDE and “every $C^k$ manifold can be thought of as a smooth manifold”

I have read that every $C^k$ domain can be described instead by $C^\infty$ chart maps. So in this sense every $C^k$ domain is a $C^\infty$ domain. But consider standard PDE where people say thing like ...
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40 views

Comparison principle for heat equation with smooth nonlinearity

Let $f : \mathbb{R} \to \mathbb{R}$ be $C^\infty$ and satisfy $f(0)=0$. Suppose $u_1,u_2 : \mathbb{R}^d \to \mathbb{R}$ are $C^2$ and satisfy $$\frac{\partial u}{\partial t} - \Delta u = f(u)$$ for ...