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

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3
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117 views

Is this pde Elliptic?

I have seen that there is a classification of elliptic pde's. It says the pde $$au_{xx} + 2bu_{xy} + cu_{yy} + du_x + eu_y+f =0$$ is elliptic if $b^2 - ac < 0$. There is another definition that is ...
3
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152 views

Solving $ T' = 0 $ for distributions in $\mathbb{R}^n$

Denoting $ T \in \mathcal{D}'(\mathbb{R}^n) $ as distributions with $ T_f(\varphi) = \int_{\mathbb{R}^n} f\varphi\ dx $, I wish to prove the distribution solution of the equation $ T' = 0 $ ...
3
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118 views

Boundedness for Reaction Diffusion BVP with Arbitrary Exponent $\alpha$

Let $U\subset\mathbb{R}^{n}$ be open, $U_{T}$ and $\Gamma_{T}$ be the parabolic cylinder and boundary of $U$ for arbitrary $0\leq t\leq T$, respectively, and suppose $u$ solves $$ ...
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217 views

Solving systems PDE's

I have a bit of trouble solving a system of first order PDE's, that I get by solving a boundary issue problem in gravitation (here). I have six equations: ...
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122 views

invertible operator Sobolev space

Let $T:H_0^2(D)\rightarrow H_0^2(D)$ a linear bounded operator defined via the sesqulinear form $(Tu,v)=a(u,v):=\int_{D}\Delta u\Delta v dx$. I have shown that T is coercive and self-adjoint. How can ...
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166 views

Why is this function space dense in this Bochner-type space? (Rogers and Renardy)

Let $\phi(t) = \sum_{i=1}^N\beta_i(t)\phi_i(x)$, where $\phi_i$ is basis for space $V$, and $\beta_i \in C_c^\infty(0,T).$ Renardy and Rogers says that 1) Functions of the above form are dense in ...
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79 views

Is a solution to some partially differential equation homeomorphic (or diffeomorphic) to a solution of an equation with a different covariance group?

Consider some solution $\psi(x,t)$ to the linear Klein-Gordon equation: $-\partial^2_t \psi + \nabla^2 \psi = m^2 \psi$. Up to homeomorphism, can $\psi$ serve as a solution to some other equation ...
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109 views

an application of implicit function theorem?

I need to proof the following, could any one help me to proof step by step? $(t,\epsilon)\mapsto F(t,\epsilon):\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ be a twice continuously differentiable ...
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198 views

Show that the convolution of a spherically symmetric function with the heat kernel is also spherically symmetric

Let $\phi \colon \mathbb{R}^n \to \mathbb{R}$ be continuous with compact support. Furthermore, suppose that $\phi$ is spherically symmetric (i.e., assume $\phi(Tx) = \phi(x)$ for every orthogonal ...
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280 views

Wave Equations Concept questions

My Engineering Mathematics teacher have a very novel method of teaching. He thinks that while it's all good and well to learn the analytical side of math, he stresses more on the theory, than the ...
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263 views

When the weak derivative just is the strong (or classical) derivative?

When the weak derivative just is the strong (or classical) derivative? For instance, can we prove that weak derivate $Du\in C^\alpha$(or $C^0$) implies $u\in C^{1,\alpha}$(or $C^1$).
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234 views

how to uncouple and unreduce a system of first order PDEs

Suppose we are given a system of first order PDEs with constant coefficients. In particular, suppose we are given $k$ PDEs for $u_1,u_2, \dots u_n$ with respect to independent variables $x_1,x_2, ...
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110 views

An existence Theorem

Data: $\Omega \subset \mathbb{R}^{n}$ is an open connected (may be unbounded) set, and locally $\partial \Omega$ is aLipschitz graph. $S \subset \partial \Omega$ is measurabel and $H^{n-1}(S)>0.$ ...
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1k views

What is the difference between the domain of influence and the domain of dependence?

When analysing the wave equation $$u_{tt} = c^2 u_{xx}$$ in my PDE's module, I understand the 'domain of dependence' which is where the value $u(x_0,t_0)$ is only depends on the initial value of $x$ ...
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72 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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211 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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156 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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147 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 ...
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162 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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263 views

The canonical form of a nonlinear second order PDE

Can anyone help me find the canonical form of $$x^2u_{xy} - yu_{yy} + u_x - 4u = 0?$$ I don't know how to solve it because $a = 0$. I just got that it's hyperbolic since $a=0$ , $b =(x^2)/2$, $c= ...
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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,\ ...
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29 views

First order partial differential equation.

We need to solve the given first order partial differential equation : $(y-xu)u_x$ + $(x+yu)u_y$ = $x^{2} + y^{2}$ . I tried this : $\frac{dx}{y-xu}$ = $\frac{dy}{x+yu}$ = $\frac{du}{x^{2} + ...
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25 views

Partial derivative of $F{\equiv}0$

If you have a function $F{\equiv}0$ then is the partial derivative of $F$ with respect to any of its variables $0$? Specifically, when we have Charpit's equations for a PDE $F(x,y,u,p,q) = 0$, where ...
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12 views

Solving inverse differential equation

How to solve $$ \frac{1}{D_{x}^{2}+D_{x}D_{y}-12D_{y}^{2}}(ye^{3x+y}+x^{3})? $$ I understand that I need to use $$ ...
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23 views

How to use Duhamel's principle to solve wave equation

Let $x\in\mathbb{R}$,$t>0$, $$\frac{\partial u}{\partial t}+5\frac{\partial u}{\partial x}=e^{-t}sinx,$$ $$u(x,0)=(1-x^2)_{+}.$$ Using Duhamel's principle to solve it. By Duhamel's principle, the ...
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24 views

The kernel $k(x,y)=\frac{y}{y^2+x^2}$ is a solution of which equation?

The kernel $$k(x,y)=\frac{y}{y^2+x^2}$$is a solution of (A) Heat equation (B) Wave equation (C) Laplace equation (D) Lagrange equation Which are correct ? I tried through ...
2
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48 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). ...
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25 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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19 views

Domain of $A^{1/2}$ on $L^2(\mathbb{T}^2)$

Let $A$ be a densely defined unbounded self-adjoint operator defined on $L^2(\mathbb{T}^2)$, where $\mathbb{T}^2$ stands for the 2-torus. It is known that $A$ is positive, that is, $\langle Au, ...
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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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26 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 ...
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48 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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132 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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24 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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58 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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35 views

Separation of variables with harmonic, time-dependent boundary condition

I am studying the vibrating dynamics of beams. In its resting state, the beam is flat, on what I call the $x$ axis. The variable $w(x,t)$ measures the vertical deflection from that axis for any point ...
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19 views

Is one dimesional cubic NLS is globally wellposed in $H^{s}(\mathbb R), (0<s<1)$?

We consider the one dimensional cubic nonlinear Shr\"odinger equation (NLS): $$i\partial_{t}\phi (x,t) +\Delta \phi (x,t)= \pm |\phi (x,t)|^{2} \phi(x,t), \ (x, t\in \mathbb R),$$ $$\phi (x,0) = ...
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20 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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42 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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39 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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65 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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18 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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12 views

Global Bi-harmonic functions in a Riemannian manifold

Any help will be appreciated thanks! Consider $(\mathbb{R^n},g)$ to be a Riemannian manifold. For simplicity we can assume the manifold to be asymptotically euclidean outside a compact domain ...
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30 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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38 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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46 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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74 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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0answers
56 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) ...