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

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Finding the modified Green function for the Helmholtz equation

I've been wrestling with this question for quite some time now, and the result was like 20 leaves of paper packed with scribbling...anyway, here's the question: I need to find the solution to the ...
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24 views

$-p_{xx}+f(p)=0$ has a unique solution $p$

In a paper I am reading they have the following as a Lemma without proof. So I am trying to prove it myself. Suppose that $f:\mathbb{R}\to \mathbb{R}$ that satisfies the following: $f'(0) >0$ ...
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18 views

First order PDE with discontinuous coefficients

I want to consider the following equation $$u_t+\mathrm{sgn}(x)u_x=0,\,\,u(0,x)=u_0(x)$$ Now if $x>0$ or $x<0$ I can use the method of characteristics to obtain $u(t,x)=u_0(x-t)$ if $x>t$ and ...
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28 views

Textbook for Partial Differential Equations with a viewpoint towards Geometry.

Though similar questions have been asked at Good 1st PDE book for self study and Good reference texts for introduction to partial differential equation? but none of them really answer my query, so I ...
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26 views

Boundary Conditions for a Finite Difference Approximation of a Sixth Derivative

I am trying to use a finite difference scheme to numerically solve sixth order parabolic equations such as \begin{equation} u_t = u_{xxxxxx} \end{equation} with symmetry conditions \begin{equation} ...
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30 views

finding solution to a partial integro differential equation

I want to find a function (or a set of functions) such that $u(x,t)$ satisfies the following partial integro-differential equation with singular kernel \begin{eqnarray} &&u_x(0,t) = \int_0^t ...
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16 views

Existenence of the solution for a PDE-ODE system.

I have the PDE-ODE system below: $\frac{\partial c}{\partial t}= D \Delta c - \eta \nabla.(c\nabla v)+g(c,v)$ $\frac{dv}{dt}=-\alpha cv+\xi(c,v)$ with initial conditions and Neumann boundary ...
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28 views

mean curvature of a nonlinear parabolic PDE

I want to compute the mean curvature of a reaction diffusion equations of the form $$ u_t=u_{xx}-u^2 $$ Any suggestions is appreciated!
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15 views

First order partial differential equations in complex domain

Try to solve a first order linear partial differential equation $P(x,\partial)u(x)=f(x)$ in complex domain, while the operator is of the following form: $$ ...
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140 views

numerical method (implicit , backward difference or forward difference) for nonlinear pde

$\newcommand{\lbar}{\underline{\lambda}}$ In this linear PDE: \begin{cases} B_t+b^Q(r,t)B_r+\frac{1}{2}d^2(r,t)B_{rr}+(\mu(\lambda,t)+\alpha \sigma (t))(\lambda -\lbar)B_{\lambda} \\ ...
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16 views

Differential Equation for brownian bridge?

For the brownian motion, we know that probability density of the particle's position at time $ t $, $ \rho(x,t) $ satisfies the diffusion equation pde: $ \partial_t \rho = d \; \partial_x^2 \rho $. Is ...
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16 views

existence and uniqueness of reaction diffusion equation

I am trying to prove uniqueness of equation of the form $$ u_t=u_{xx}+f(u) $$ Can anyone suggest the standard literature for existence and uniqueness for equation like above?
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26 views

Comparison and maximum principle for parabolic pde

I was told the comparison principle can also be understood as the fact that the difference between a subsolution and a supersolution satisfies the maximum principle. I also know comparison principle ...
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27 views

Taking partial derivatives over multiple summations

I have the following equation obtained from one of the models. $\mathcal{H} = \sum\limits_{D} \sum\limits_{W}n(d,w)\sum\limits_{Z} p(z|d,w)[\log{p(d)}+\log{p(z|d)}+\log{p(w|z)]}$ I need to take ...
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57 views

Sine-Gordon Equation application

Is it true that Sine-Gordon is satisfied for geodesics on the central Pseudosphere ( rotated surface of Tractrix)? If so, please cite text-book or article references. $$ \alpha''(s)=sin ( \alpha) ...
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59 views

Loss of derivatives

In many books on pdes the expression "loss of derivatives" is used when some estimates on solution are proved. Can someone clarify to me (maybe with an example) the meaning of this expression? For ...
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23 views

The invariance of the Ricci tensor under diffeomorphisms and its non-ellipticity.

Consider $(M,g)$ a compact Riemannian manifold. When viewed as a second order (non-linear) differential operator $$ \text{Ric} : C^{\infty}(\text{Sym}^2_+T^*M) \to C^{\infty}(\text{Sym}^2T^*M), $$ the ...
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49 views

Derivative nonlinear Schrodinger equation

I'm dealing with the following DNLS $$iu_t+u_{xx}=i(|u|^2u)_x$$ Let's consider the following transformation $w=\exp(-i\int_{-\infty}^x|u|^2dy)u$. I'm interested in the equation satisfied by $w$. I ...
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29 views

Elliptic PDE on the Whole Space

Can anyone suggest a reference for elliptic PDE on the all of $\mathbb{R}^d$, as opposed to some bounded domain $\Omega$, covering the standard topics of existence, uniqueness, and regularity. I ...
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38 views

Proof of Second Partials Test

How does one rigorously prove the second partials test without firstly assuming that $D(a,b)=AC-B^2$ that states the following: $ A=\frac {\partial^{2}f(a,b)}{\partial x^{2}},B=$$\frac ...
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21 views

harmonic functions: comparison of gradients

Consider $\Omega$ a open, bounded, convex domain in $R^n.$ I am trying to justify this: Let $u, v$ non negative harmonic functions in $\Omega$ (in the Sobolev sense). Suppose that $ u = v =0$ in ...
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36 views

Method of characteristics for systems of PDE (vs. Lewy's example)

Main question: How does the method of characteristics generalize for systems of first order PDE, as opposed to scalar PDE? Namely, is there such a generalization at all, and if so what information ...
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32 views

Finite difference scheme for hyperbolic system

I'm having a bit of trouble understanding the following, so it'd be great if anyone has any nice explanations! Thanks in advance! Consider the hyperbolic system $$u_t = Au_x + Bu$$ where $A$ and $B$ ...
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34 views

Research papers of monotone/pseudomonotone operators with applications to PDEs

I have recently been studying how coercive, pseudomonotone operators are used to prove the existence of solutions to elliptic boundary value problems. I have been studying the book "Nonlinear Partial ...
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62 views

Prove the maximum principle for a solution of the PDE $(-\Delta + \lambda)u=0$

Show that if $u$ satisfies $(-\Delta + \lambda)u=0$ in a smooth region $\Omega$ where $\lambda$ is a nonnegative real number, then u satisfies the maximum principle. We can just assume u is smooth, ...
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16 views

Growth condition in differential equation and vanishing solution at boundaries

In a discussion on solving a partial differential equation I lately read: "Under a standard growth condition on the solution at infinity, the resulting PDE is fully specified without boundary ...
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35 views

Finding gradient of an objective as a PDE

I am trying to find the gradient of the following optimization problem and then add to objective, but I got some trouble in computing. Could you please help me? Assume that we have an optimization ...
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56 views

Compute average and maximum value of a field over a streamline

I'm working on a code solving a set of PDEs. I have a vector field, $\vec{v}(x,\theta,z,t)$ (it's a velocity) and a scalar field, $c(x,\theta,z,t)$. I have a $2\pi$-periodicity in $\theta$. The ...
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62 views

Is this distributional laplacean a measure?

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. Suppose that $0<k<t<1$ and $\Phi$ is a mooth function satisfying: $\Phi(x)=0$ for $x\le k$, $\Phi(x)=1$ for $x\ge t$. Take $u\in ...
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32 views

Mollification of functions in $L^2(0,T;H^1)\cap H^1(0,T;H^{-1})$

Let $u \in L^2(0,T;H^1(\Omega))\cap H^1(0,T;H^{-1}(\Omega))$ where $\Omega \subset \mathbb{R}^n$ is a bounded domain. I know that (eg. from Wloka or Hunter's PDE notes) that there is a mollification ...
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129 views

Eigenvalues problem for generalized Kuramoto-Sivashinsky equation

I been working on Kuramoto-Sivashinsky Equation. In the process of analysis, I need to solve the following eigenvalues problem $$ -u_{xxxx}-\lambda u_{xx}=\beta(\lambda)u $$ where $\lambda$ is a ...
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46 views

Meaning of “Canonical System of the First Order”

I am learning about PDEs and came across the following. "Convert a partial differential equation of higher order into a canonical system of the first order" What does the above statement mean/imply? ...
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37 views

Solving PDE via Fourier Transform & Uniqueness

When a PDE is solved via Fourier transform, is there already a uniqueness assertion that comes for free? For example, if we Fourier the heat equation \begin{align} \partial_t u(x,t) &= \Delta ...
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31 views

Local Mollification of a $L^1$ function

Let $f$ be a $L^1(\Omega)$ function, where $\Omega\subseteq \mathbb{R}^2$ is a smooth and bounded domain. Then can we find a fuction $\phi\in W_0^{2,2}$ suct that $\phi>0$ in $\Omega$ and ...
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82 views

Compact support

From PDE Evans, 2nd edition, page 204 Example 9 (Wave equation from the heat equation). Next we employ some Laplace transform ideas to provide a new derivation of the solution for the wave ...
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21 views

Hamiltonian form of PDE

I am quite new in this field and I was wondering what does exactly mean to write a PDE (in 2 or 3 dimensions) in an Hamiltonian form. More in detail, is there any standard procedure to write a given ...
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106 views

Finite Difference Spacing of Points for PDE's for Convergence of Explicit Forward-Stepping Scheme

I realize that this question could be pretty broad, but I'm wondering at least what the conditions are for my simulation. I'm developing an Explicit Forward-Stepping Finite Difference scheme to solve ...
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75 views

Solution to Schrödinger equation $ \partial_t f(x,t) = -\partial_x^2 f(x,t) + \delta(t)V(x)f(x,t).$

I want to solve $$ i\partial_t f(x,t) = -\partial_x^2 f(x,t) + \delta(t-t_0)V(x)f(x,t),$$ for any $V \in C^{\infty}[-1,1]$ and $f: [-1,1] \times \mathbb{R_{\ge 0}} \rightarrow \mathbb{C}$. I would ...
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52 views

Why is this bilinear form elliptic on $H^1_0$?

I have the following bilinear form: $$ a(v,w) = \int_\Omega \nabla v(x) \cdot \nabla w(x)dx + \int_\Omega \left( \sum_{i=1}^n \beta_i v_{x_i}(x) \right) w(x) dx $$ where $\Omega \subset ...
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24 views

eigen value problem with Robin Boundary Conditions at both ends

This is a problem from the book Partial Differential Equations by Walter.A.Strauss. Consider the eigen value problem with Robin Boundary Conditions at both ends: $-X''=\lambda X$ $X'(0)-a_0X(0)=0$ ...
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31 views

Imposing boundary conditions in a finite element sense

Say we have a PDE which is well posed with the boundary condition $|\nabla u|=r$ ($r$ constant) on $\partial\Omega$, where $\Omega\subset \Bbb R^n$ is uniformly convex. How would one impose this ...
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36 views

Explicit computation of spectrum of Hodge-Laplacian on forms

While I know of some explicit examples (sphere, flat torus) for the spectrum of the Hodge-Laplacian on 0-forms (i.e. the Laplace-Beltrami operator on functions), I haven't found anything for "actual ...
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27 views

When does weakly elliptic $\Rightarrow$ strongly elliptic?

While learning more about the analytic background for the Atiyah-Singer Index Theorem, I was curious about the following question (although not needed for the ASID): what are some general conditions ...
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30 views

Missing Step in a Paper of Struwe.

In this paper on Page 4, in the last line of the proof, the author asserts that if a radial function $u:\mathbb{R_t}\times \mathbb{R}^2\to \mathbb{R}$, smooth outside the origin $(0_t,0_x)$, admits ...
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54 views

Heat equation on a graph Laplacian

I would like to start with considering the time-dependent heat equation on a connected graph. To start, I will need to model it respect to time discretization. I mean I have to write something like: ...
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69 views

Relation between fractional integral operator and solution of poisson equation

For $0<\alpha<d$, fractional integral operator $I_{\alpha}$ is defined by $$I_{\alpha}f(x)=\int_{\mathbb{R}^d} \frac{|f(y)|}{|x-y|^{d-\alpha}} dy$$ for any suitable function on $\mathbb{R}^d$. ...
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38 views

Which came first, energy minimization or pde?

I'm interested in a historical perspective on pde. I would like to know more about the original derivation of pde. It seems like d'Alembert was working on the one dimensional wave equation $$ ...
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48 views

What is the name of such equation in fluid dynamics? How does it come out?

I read papers and see this equation. Honestly, I am not familiar with fluid dynamics. So I was wondering if this equation is common in fluid dynamics, or it is the special case in this paper. Could ...
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25 views

Greens function of a uniformly charged sphere

The potential $\phi(\boldsymbol{x})$ satisfies $\nabla^2\phi=f$ It may be shown that by defining an appropriate Green's function $g(\boldsymbol{x},\boldsymbol{\xi})$ that ...
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92 views

If $u:\cup_t \Omega_t \times \{t\} \to \mathbb{R}$ measurable, is $\tilde u:\Omega_0\times (0,T) \to \mathbb{R}$ measurable?

For each $t \in [0,T]$, let $\Omega_t$ be a bounded open domain. There is a diffeomorphism of class $C^2$ $$F_t:\Omega_0 \to \Omega_t$$ that maps the domains. Assume that that $F_t$ is differentiable ...