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

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23 views

Calculating the Legendre transform of a function explicitly

This is a portion of one of the questions in Evan's PDE book. Let $H: \mathbb{R}^n \to \mathbb{R}$ be defined by $H(p) = \frac{1}{r}|p|^r,$ for $1 < r < \infty$. I want to show that $$L(q) = ...
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15 views

Numerical Analysis - Richardson Method

About the Richardson Method, how should we initialize the function? I am trying to solve: $\frac{\partial u}{\partial t}-\frac{\partial^{2}u}{\partial x^{2}}=0$ where $x\in [0,2]$ and $u(x,0)=sin(2\pi ...
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1answer
34 views

Fourier series coefficients in PDEs

I have a problem that involves solving a PDE using separation of variables. For context, here is the question: $u(x,t)$ is the displacement of a string at position $x$ and time $t$, which is ...
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2answers
36 views

Newton potential for Neumann problem on unit disk

Problem: Show that $$G(x, y) = \frac{1}{2\pi} \left( \log \lvert x - y \rvert + \log \left\lvert \frac{x}{\lvert x \rvert} - \lvert x \rvert y \right\rvert \right)$$ is a Green's function for the ...
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0answers
28 views

Question about the solution to the heat equation?

The question I am attempting to solve is to show that the solution to the heat equation of a rod of length $10$ with initial temperature distribution given by $u(x,0)=f(x)$ is $$\frac{a_0}{2} ...
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26 views

Fundamental solution of Laplacian in 2d doesn't agree with 3d version physically?

The fundamental solution of the Laplacian, with $(x, y) \in \mathbb{R}^3$, is $$ \Phi(x, y) = -\frac{1}{4\pi|x - y|} $$ So, say, the further away $x$ is from some point source $y$ the lesser the ...
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1answer
22 views

Constructing a PDE given solution with arbitrary differential function?

I have the following multivariable function: $u = x + y + F(xy)$ where F is an arbitrary differentiable function. I would like to construct a first order PDE with u(x,y) as the following solution. ...
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15 views

Complicated exponential function inverse laplace transform

The problem is from wave equations for describing flexural vibration of Euler-bernoulli beam. The equations are listed in the following pics. I. equations and 16 elements to be analyzed II.cF:wave ...
2
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1answer
29 views

Poincare's inequality with difference quotient

For the classical Poincare's inequality, if $u \in H^1_0(\Omega)$, then $$\int_\Omega u^2 \,dx \le C \int_\Omega |\nabla u|^2 \,dx.$$ Do we have something similar with the difference quotient? That ...
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0answers
44 views

Product of two $W^{1,p}_0$ functions

I have a fairly simple question. Suppose that $p>n$ and $u,v\in W^{1,p}_0$, how can I prove that the product is also in $W^{1,p}_0$ ? Of course, we have to employ Morrey's inequality. My idea was: ...
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1answer
21 views

Bounded solution to the heat equation on a strange domain.

Find a solution to the heat equation $u_{xx} = u_t$ that is bounded for $0\leq x < \infty$, $-\infty < t < \infty$. Would $u(x,t) = f(x)$ since if there was any time dependence the solution ...
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1answer
14 views

What does it mean for a boundary to be analytic in the context of a PDE?

I am reading a paper where they assume the boundary of a domain is "Analytic". They never define it. Is this a standard definition, and, if so, what is it?
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17 views

Can we show that each element of the Sobolev space $H^k(D)$, with $D\subseteq\mathbb R^d$ being a bounded domain, has a continuous representative?

Let $d\in\left\{2,3\right\}$ $D\subseteq\mathbb R^d$ be a bounded domain $\lambda$ be the Lebesgue measure on $D$ $H^k(D)$ be the Sobolev space Can we show that each element of $H^k(D)$ has a ...
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0answers
15 views

Show that the Dirichlet problem has infinitely many solutions.

I need some help regarding this assignment. Let $\Omega = \{x \in \mathbb{R}^3: \lvert x \rvert \geq 1 \}$ and consider the Dirichlet problem $$ \Delta u = 0, \ \ \ x \in \Omega, $$ $$u = ...
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0answers
11 views

Transforming a PDE to an ODE by discretizing the size-axis of the integro-partial differential equations

I have this issue on modelling a process. It is a system of ODEs except for one equation related to the population balance of crystals in a mixture that is a PDE. The author of the publication ...
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1answer
26 views

Method of Characteristics for PDEs Yielding Different Answer Than Expected? (Can I get a check on work?)

I was interested in solving the following system: $x^2u_x - xyu_y = -y^2$ I used the method of characteristics to reduce my problem into the following: $\left(\frac{dx}{x^2}\right)$ = ...
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0answers
14 views

Linear PDE with variable coefficients method of solution?

I am trying to solve the following PDE and have experienced some difficulty. $x^2u_x - xyu_y = -y^2$ I have divided both sides by $x^2$ to get the following: $u_x - (\frac{y}{x})u_y = ...
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1answer
27 views

$\eta_{\epsilon}*u$ satisfies heat equation

If $u(x,t)$ satisfies the heat equation then $\eta_{\epsilon}*u$ also satisfies it, with $\eta(x)=e^{\frac{1}{|x|^2-1}}$ for $|x|<1$ and $0$ else and ...
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0answers
8 views

Norms of Slobodekii and classical Sobolev space

For r to be a non-negative integer, we have $$\|u\|_{W^{r,p}(\Omega)} = \left( \sum_{|\alpha|\le r} \int_\Omega |\partial^\alpha u(x)|^p dx \right)^{1/p}.$$ For $0 < \mu < 1$, let $s = r + \mu$. ...
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0answers
10 views

Interpolationspace for H^{-1}\cap H^{1}

For which $\theta\in[1,\infty)$ does hold $(H^{-1}(\Omega),H^1(\Omega))_{1-\frac{1}{\theta},\theta}=L^2(\Omega)$ if $\Omega$ is a bounded domain with smooth boundary and is three dimensional. I don't ...
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20 views

Classifying this Partial Differential Equation?

I have the following partial differential equation:   $u_{xx} + 2u_{yy} + u_{zz} - 2u_{xy} - 2u_{yz} = 0$ To my knowledge, when classifying a partial differential equation, you analyze the ...
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21 views

How to find $\partial ^2 u/ \partial x^2$?

If we have $u=u(\xi , \eta )$ and $\xi=\xi (x,y)$, $\eta = \eta (x,y)$ and $$\frac{\partial u}{\partial x}=\frac{\partial u}{\partial \xi}\frac{\partial \xi}{\partial x} +\frac{\partial u}{\partial ...
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14 views

inhomogeneous transport equation

For given $u_0\in C^1(\mathbb R), f\in C([0,\infty)\times \mathbb R), a\in\mathbb R$ I am asked to solve $$u_t(t,x)+a u_x(t,x)=f(t,x), t>0, x\in\mathbb R$$ with initial condition $$u(0,x)=u_0(x), ...
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1answer
27 views

Poincaré-inequality

I learned this week about sobolev-spaces and thereby my prof. claimed the truth of an inequality (without proof) that seemed to him very clear and easy to see but to me it was not clear at all. He ...
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0answers
55 views

Why such coordinates are still called “isothermal” in the Lorentz case?

We know that in a two dimensional Riemannian manifold there always is, at least locally, a coordinate system in which $g_{ij} = \lambda^2 \delta_{ij}$ - such coordinates are called isothermal. There ...
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1answer
21 views

general conditions for reverse poincare inequality

I'd like to know when the reverse Poincare inequality is true: Given a bounded domain $\Omega$, and $f \in L^2(\Omega)$, under what conditions on $f$ (neglecting the trivial constant case) and/or ...
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11 views

reverse poincare inequality [duplicate]

I'd like to know when the reverse Poincare inequality is true: Given a bounded domain $\Omega$, and $f \in L^2(\Omega)$, under what conditions on $f$ and/or $\Omega$ is it true that there is a ...
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1answer
59 views

Why is the Laplacian of $1/r$ a Dirac delta? [duplicate]

How does one show that $\nabla^2 1/r$ (in spherical coords) is the Dirac delta function ? Intuitively, it would seem that the function undefined at the origin and I'm not able to construct a limiting ...
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1answer
17 views

Assuming a solution to a PDE

I am working on exercise involving the PDE $\frac{1}{2}{\sigma}^2 S^2 \frac{\partial^2 P}{\partial S^2}-rP+r\frac{\partial P}{\partial S}=0$ The solution I am looking at says to assume ...
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1answer
54 views

Solution for partial differential equation

How to solve this partial differential equation $$a(1-q)\frac{\partial}{\partial q}A[p,q]+(bp(q-1)+c(1-p))\frac{\partial}{\partial p}A[p,q]-(s+d(1-p))A[p,q]=0$$ where $a,b,c,d$ and $s$ are constants ...
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0answers
78 views

Derivation of a stochastic Navier-Stokes equation with multiplicative noise

Most of the literature is targeting a special stochastic version of the deterministic Navier-Stokes equation without giving a derivation of the considered equation. I'm searching for such a ...
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0answers
25 views

Problem understanding compact and Fredholm operators

I'm trying to understand the general interaction/duality between Fredholm and compact operators and I ran into the following: Let $L$ be the Laplacian or some elliptic operator on the Sobolev space ...
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16 views

Dirichlet problem with separation of variables

$\triangle u=0$, with $0<r<2$ and $0\leq \theta <2 \pi$ $u(r=2,\theta)=\sin \theta $ There are three conditions that we need to check with separation of variables, which are: a) The ...
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1answer
45 views

Is this a Schwartz function?

I would like to know whether this function $f : \mathbb{R} \rightarrow \mathbb{R}$ $$f(x):=\frac{1}{\sum_{k=0}^{\infty} \frac{x^{2k}}{(k!)^2}}$$ is a Schwartz function? By applying the chain-rule ...
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0answers
19 views

Conditional expectation of a hitting time of a Brownian motion and Laplace transform

I am trying to solve the following problem: Suppose B is a 1-dim Brownian motion, let $\mathcal{T}_a = inf\{t: B_t = a\}, \mathcal{T}_{a,b}=min\{\mathcal{T}_a,\mathcal{T}_b\}$. For $a < 0 < b$ ...
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33 views

BC/IC for this type of PDE

For a 1st order PDE in 1D that takes the following form: $$\frac{∂u}{∂t}=\frac{∂f(t,x,u)}{∂x}+s(t,x,u)$$ How many boundary conditions(BC)/initial conditions(IC) are needed? I would normally think ...
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How to show that the C^2 solution to this PDE is in $L^\infty ([0,T] \times R^d)$

We ASSUME that a $C^2$ solution exists. Given that $u_0$ is a bounded $C^2$ function and $h:[0,T]\times R^d\times R^d \to R^d$ such that $|h(t,x,y| \le M+C|y|$ and $|h(t,x,y)-h(t,x,z)| \le C|y-z|$
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1answer
22 views

Dual spaces and weak solutions.

I have two questions: (1) $H^{-1}$ space is defined as the dual space of $H_{0}^{1}$, so is the dual space of $H^{1}$ also $H^{-1}$? Or is it correct to act an $T\in H^{-1}$ on a function $u\in ...
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11 views

Problem with the numerical PDE solving (possibly lattice-pinning)

I am solving quite complicated PDE's. The behavior was unexpected, and I started to simplify it. Finally I found out, that the problem is in the modified heat equation. The equation is: $$ ...
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14 views

Test space for weak formulation

From an applied functional analysis course where the weak formulation was covered, I think I have a high-level understanding of the concepts that are involved. To derive a weak formulation for a PDE, ...
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0answers
31 views

well posedness of 1D linear Schrodinger

I have the following problem: $iu_t - u_{xx} = f$ on the interval $[0,L]$ with $u(0,t)=u(L,t)=0$ and $u(x,0)=0$. I can show that $\|u\|_{L^2(x,t)}$ (space-time) is controlled by the norm ...
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1answer
24 views

How to find the solution of system of equations by using the method of manufacturing solution

I want to use the method of manufacturing solution MMS to find the exact solution of the following linear system of two equations $$\begin{cases} u_t -(u_{xx} + u_{yy}) = u + v\\ v_t -(v_{xx} + ...
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2answers
41 views

Construction of Sobolev space

I am reading about the construction of Sobolev spaces from $L^2$. In the book I read (Introduction to Partial Differential Equations, by Gerald B. Folland) that the operator used to construct those ...
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1answer
20 views

Regularity of solution to an elliptic PDE

Let $R$ be the shifted open unit cube \begin{equation} \Big(-\frac{1}{2}, \frac{1}{2} \Big)^3 \subset \mathbb{R}^3, \end{equation} and let $k \in \mathbb{C}$ be a constant with $\textrm{Re}(k) ...
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10 views

Proof of $A\ne \mathrm{conv}~ \mathrm{relbd} A\Rightarrow A$ is a flat or a half-space

Picture below is from the 16 page of Schneider R. "Convex Bodies--The Brunn-Minkowski Theory". I am fuzzy with the red line, what is the translate of $H^-$ ? And what is the effects of convexity ...
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1answer
63 views

Why is a differential equation a submanifold of a jet bundle?

I'm reading The geometry of jet bundles by D.J Saunders and struggle with the definition of a differential equation 6.2.23. on page 203. First of all, Saunders introduces a differential operator ...
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1answer
24 views

Solve the PDE using characteristics method

Consider the PDE $$3 \frac{\partial{u}}{\partial{t}} + t^2\frac{\partial{u}}{\partial{x}} = 0: t \lt 0 $$ with the initial condition $$u(x,0) = f(x): 0 \lt x \lt 1$$ Determine the characteristics ...
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13 views

Sobolev space membership of logarithmic function

Determine the largest $s\in(0,1)$ for which the following integral converges $$\int_0^1\int_0^1\frac{\Big|\log|x-\frac{1}{2}|-\log|y-\frac{1}{2}|\Big|}{|x-y|^{1+2s}}^{2}dxdy$$
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1answer
20 views

Numerically solving 1D 2nd order PDE Goursat problem

I am looking for help with solving the following Goursat problem (this is what the paper I am reading from calls it). I have been attempting a numerical solution in matlab but I do not fully ...
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10 views

continuous dependence of Liouville equation

I am considering the following equation $\partial_t f+v\cdot\partial_x f+\partial_x\psi\cdot\partial_xf =0$ with initial data: $f(0,x,v)=f_0(x,v)$. There are some analysis like the uniqueness and ...