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

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1answer
14 views

PDE Method of characteristics with initial condition

I wanted to solve the following PDE with initial condition $$ u_t+tu_x=0, $$ $$ u(x,1)=f(x),$$ where $f(x)$ is a given function, using the method of characteristics. I explain what I have done. First ...
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1answer
21 views

How to show contradiction in the Hardy inequality when the singularity power is greater that 2.

Assume $ \Omega \subset \mathbb{R}^N$ is a smooth bounded domain. There is well known Hardy inequality that says For any $ u \in W_0^{1,2}(\Omega) $, $N\geq3$ we have $$ \Lambda \int_{\Omega} ...
2
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0answers
23 views

fourier transform for pde equation

I was solving the pde using fourier transform: $u_{tt}-u_{xx}+m^2u=0$ with initial values $u(0,x)=f(x)$ and $u_t(0,x)=g(x)$. I have received the answer $$U(t,k)=Ae^{-it \sqrt {k^2+m^2}}+Be^{it \sqrt ...
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34 views

Is this partial differential equation solvable?

Ok so I am asked to set up a partial differential equation and then motivate why it is solvable. I'm only 2 weeks into my course so we are not asked to solve anything yet. However, if someone would ...
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0answers
15 views

About PDE solution: H^1 norm bounded = bounded in L^2?

Let $\Omega \subset \mathbb{R}^d~(d=2,3)$ be an open bounded set with Lipschitz continuous boundary $\Gamma$. We assume that $\Gamma$ consists of two disjointed parts, i.e, $\Gamma = \Gamma_{c} \cup ...
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2answers
42 views

how come $C^{1}_{0}$ is not complete under norm $||.||_{1,2}$

why the space $C^{1}_{0}$ is not complete under the norm $||.||_{1,2}$? by some counter example $||u||1,2=(\int_{\Omega} (|\nabla u |^2+|u|^2))^{1/2}$
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28 views

Are the set of probability functions with compact support in a fixed closed ball complete under the Wasserstein norm?

Let $B_R$ be a closed ball of radius $R$ in the space $\mathbb{R}^d$. As the title suggests I have this feeling that the set of functions $$S:= \left\lbrace f:\mathbb{R}^d \to \mathbb{R} ...
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1answer
32 views

Fourier cosine series giving nonsense answer

I'm currently trying to find the cosine Fourier series of $f(x) = \left | \sin \frac{\pi n }{L} x\right |$ on the interval $0 < x < L$. I first started by calculating the first term of the ...
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15 views

Sketching partial differential equation

I have found a solution to my pde however I want to try and sketch it however I don't know where to start. pde
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23 views

Find the value of the partial derivative

Find the value of $\dfrac{1}{D_x^2-D_y^2}{\sin(x-y)};D_x=\dfrac{\partial }{\partial x};D_y=\dfrac{\partial }{\partial y}$ I am used to finding the value of $\dfrac{1}{F(D)}\sin ax$ where $D^2$ ...
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40 views

Solution of boundary value problem using Fourier series

I want to solve the following PDE using Fourier series. $u(x,y): \Omega \to \mathbb{R}$, $\Omega=(0,\pi)\times (0,2\pi)$ $u-3u_{xx}-u_{yy}= 3\sin(2x)-\sin(5x)$ $u_{xx}$ and $u_{yy}$ are second ...
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0answers
45 views

What is a very good book about Green's Functions?

I would really like to learn in a good way about Green's Functions. Their use, their derivations, how to use them to solve partial differential equations and so on. I'm a theoretical physicist so my ...
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0answers
24 views

Spherical Bessel expansion of Green function

Any reference/advice would be good. I can use eigenfunction to solve the Green function for $$\Delta u(x) + k^2 u(x) = \delta(x - y)$$ boundary condition given as $u = 0$ on $\partial B(1)$, unit ...
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0answers
178 views

Reformulation of Goldbach's Conjecture as optimization problem correct?

Question I think I managed to reformulate a stronger version of Goldbach's conjecture as an optimization problem: $$ \frac{\partial F_n}{\partial a_n} = \frac{\partial F_n}{\partial \overline a_n} ...
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24 views

Fundamental solution of the Poisson equation with variable exponent

Let the variable exponent $p(x)$, where $p(x) \in C(\overline{\Omega})$, I want to know the fundamental solution of $$-(\Delta u)^{p(x)}=\delta_0.$$
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1answer
28 views

Solution to piecewise heat equation

We have piecewise diffusion equation $u_{t}(x,t)=\left\{\begin{matrix} D_{1}u_{xx}(x,t) &x\in(0,1) \\ D_{2}u_{xx}(x,t) & x\in (1,2)\end{matrix}\right.$, with IC: $u(x,0)=2x+1$ and BC : ...
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0answers
21 views

Bounding a Subsolution of the Heat Equation

As the title suggests, I'd like to bound a subsolution of the heat equation. I have \begin{align*} u_t - \Delta u &\le 0 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text{ in } \mathbb R^n \times (0,\infty) \\ ...
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1answer
25 views

$L^1(0,T;(L^1_{loc}(\mathbb{R^N}))^N)$ - Confused about notations used for space and time dependent vector fields

I found this notation - $L^1(0,T;(L^1_{loc}(\mathbb{R^N}))^N)$ - in a paper of DiPerna and Lions concerning vector fields space and time dependent, "Ordinary differential equations, transport theory ...
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43 views

Barenblatt solution for diffusion on the whole line

Question It is my second course on PDEs and the teacher asked us to find solutions like: $$u(x,t) = t^\alpha · f\left(\frac{|x|^2}{t^\beta}\right)$$ for the diffusion (with $k=1$): $$u_t - u_{xx} = ...
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11 views

Greens function for Poisson equation with Neumann boundary condition

Consider Poisson's equation with Neumann boundary condition: \begin{eqnarray*} -\triangle u & = & f,\quad x\in\Omega\\ \frac{\partial u}{\partial n} & = & g,\quad x\in\partial\Omega ...
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1answer
13 views

How do you choose basis functions in finite element analysis?

I'm having some trouble understanding the underlying mathematics in finite elements. I know it's widely used to solve PDEs especially in structural applications. From what I understand you discretize ...
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37 views

Reference request: about inverse Laplacian operator

I am currently studying some problems about inverse Laplacian and the Yosida approximation and wishing to learn more about it. Here is a post about one of the problems that I am interested in. ...
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1answer
54 views

The Burgers equation $u_y + u u_x = 1$ with $u=0$ on the parabola $y^2=2x$

For the PDE $u_y + u u_x = 1$, sketch a plot of $\Gamma$ and a few representative curves, including the envelope curve. Conditions: $u=0$ on the curve $y^2=2x$, and $y,x>0$. Express $u$ as a ...
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2answers
22 views

A general solution for a 2d pdf (ode)

I have the following 2 dimensional PDE: $$ \partial_{x_1}^2 u(x_1,x_2)+\frac{1}{x_1^2}\partial_{x_2}^2 u(x_1,x_2)+\frac{1}{x_1}\partial_{x_1}u(x_1,x_2)=k $$ where $k>0$ is a constant, and ...
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0answers
35 views

The inverse of Laplacian for different orders.

This question is related to my previous question here Let $u,v\in C_c^\infty(\Omega)$ and $\Omega\subset \mathbb R^N$ is open bounded with smooth boundary. Let $\Delta$ denote the Laplacian operator, ...
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1answer
39 views

Solving a system of ODE that arose in solving Burgers' equation

Consider the Burgers' equation $$\partial_t u = \alpha u\partial_xu$$ Intend to solve this using Fourier Galerkin method. So When I convert this into $N$th Fourier partial sum, I get a system of ODE's ...
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5 views

Looking for references of sobolev spaces involving time

I am looking for references which introduce the the sobolev spaces involving time. What I have at the moment is only a short chapter from Evan's PDE-book. Is there any other similar literature but ...
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35 views

Verify function is a Solution to Wave Equation in 3D Spherical Coordinates

The wave-equation is given by $$\nabla^2E=\frac{1}{c^2}\frac{\partial ^2E}{\partial t^2} $$ And I'm trying to prove that this wave $ E(r,\theta,\phi)= \frac{A_o}{r}sin(\theta) cos(\omega t - ...
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1answer
43 views

Interior estimate for derivatives of harmonic function

I'm learning PDE from the book of Trudinger and Gilbarg and I'm attempting to prove the following theorem: Let $\hspace{0.1ex}u\hspace{0.1ex}$ be harmonic in $\hspace{0.1ex}\varOmega \subset ...
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1answer
49 views

Computing the inverse of Laplacian operator.

I am considering the following equation: $$ f(t):=\int_\Omega[(I-t\Delta)^{-1}\Delta(I-t\Delta)^{-1}u]\cdot u\,dx $$ where $u\in C_c^\infty(\Omega)$ and $t\geq 0$ a real number. $I$ is the identity ...
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1answer
14 views

Is L^2-norm of Laplace operator equivalent to 2-seminorm?

Let's say $\Omega$ is a bounded domain in $\mathbb{R}^2$. Is it true that $C_1\|\Delta u \|_{0,\Omega} \leq |u|_{2,\Omega} \leq C_2\|\Delta u \|_{0,\Omega}$? The left inclusion is obvious by ...
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16 views

Lax pair for Painleve V equation

Deformations of this linear system (which contains only fuchsian singularities) $$\frac{dY(z,t)}{dz}=(\frac{A_0}{z}+\frac{A_1}{z-1}+\frac{A_t}{z-t})Y(z,t)$$ are isomonodromic if and only if ...
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26 views

The first eigenvalue for Dirichlet boundary condition positive?

Let $M$ be a compact, n dimensional Riemannian manifold with boundary. Then we know that $W^{1,2}(M)=W^{1,2}_0(M)$, the latter is the completion of $C_0^{\infty}(M)$ function w.r.t $W^{1,2}(M)$-norm. ...
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1answer
19 views

Subsequence of $L^{2}(\Omega)$ - bounded sequence weakly * converging to a measure

I was reading a well available article in the internet: "THE COMPENSATED COMPACTNESS METHOD APPLIED TO SYSTEMS OF CONSERVATION LAWS by Tartar"; there at Page-266, it is written: "$L^{1}(\Omega)$ is ...
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1answer
32 views

Solving $u_{xx}+u_{x}^{2}=ku$ for various k

Here is a nonlinear ODE: $u_{xx}+u_{x}^{2}=ku$ for various k Attempts For $k=0$, we have $u=log(x+c_{1})+c_{2}$. For $k\neq 0$, 1)Divide by $u_{x}$ and integrate both sides to get: ...
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24 views

Exact solution of $u_{t}=a_{1}u_{xx}+a_{2}\Phi(t)u_{x}^{2}$, where $\Phi(t)$ is cdf normal

The PDE for $(x,t)\in \mathbb{R}\times (0,1]$ is $$u_{t}=a_{1}u_{xx}+a_{2}\Phi(t)u_{x}^{2},$$ where $a_{1},a_{2}$ are constants and $\Phi(t)=\int_{-\infty}^{t} \phi(\frac{x-0.5}{0.2})$. I picked ...
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1answer
45 views

how to solve a pde whose coefficient is the function itself

I am studying differential geometry, Walker metric in three dimension. I try to find the geodesic equations of a Walker manifold and I need to solve the following PDE. Unfortunately, I didn't take any ...
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20 views

Solving Hyperbolic PDE using MATLAB (finite difference Method)

I am given the PDE $u_t + u_x = 0, x \in (-2,3), t > 0$ with initial condition $ u(x,0) = 1 - |x|$ when $|x| \leq 1$ and $u(x,0) = 0$ when $|x| \geq 1$. With boundary condition $u(-2,t) = 0$. How ...
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1answer
36 views

Existence of a solution to the poisson equation with a Radon measure on the right hand side.

I was trying to guarantee the existence of a solution to the problem $$-\Delta u=\mu,\quad u\lvert_{\partial\Omega}=0$$ where $\mu$ is a signed Radon measure, i.e., $\mu(A)=\int_{A}f\,dx$, $f\in ...
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25 views

HOW to work out mixed partial derivatives

I dont understand how to work out the partial derivative using the chain rule, for exaple, $ u = \psi_{y} $ $v = -\psi_{x}$ $ \psi = (2x)^{1/2} f(\eta) , \eta = (2x)^{-1/2)}y$ so $ \psi_{y} = ...
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A problem about Frechet derivative:

Let A be a $n \times n $ matrix with real entries and eigenvalues with strictly negative real parts. Let $g \in C^1(R^n;R^n)$ with $ g(0) = 0 $ and with Frechet derivative $ Dg(0) $ satisfying ...
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1answer
24 views

mixed partials for PDE

Can someone please help me. if $ u = \frac{\partial\phi}{\partial y} , v = -\frac{\partial\phi}{\partial x}$ and $ \phi = (2x)^{\alpha} f(\eta)$ where $ \eta = (2x)^{\beta}y$ I need, to work out ...
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20 views

Is there such a thing as a “partial differential”, brother to “total differential”?

I am familiar with total differentials in the form $$ f = f(x,y,z) $$ $$ df = \frac {\partial f} {\partial x} dx + \frac {\partial f} {\partial y} dy + \frac {\partial f} {\partial z} dz $$ however, ...
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25 views

Solution of a PDE using Bessel Functions of the first an second kind $J_0(z), Y_0(x)$

Solution of a PDE using Bessel Functions of the first an second kind $J_0(z), Y_0(x)$. I want to p}{\partial rurobe that the solution of the PDE: $$g\big(\dfrac{\partial^2u}{\partial ...
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1answer
59 views

Momentum is quantised in compact spaces?

Background One of the first examples given when studying quantum mechanics is the particle on a cylinder, or particle on a ring. One finds that because of the periodic boundary conditions, ...
3
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1answer
33 views

Which assumptions on $Ω\subseteq\mathbb R^d$ do we need in order to show density of $C_c^∞(Ω)$ in $(L^p(Ω),\left\|\;\cdot\;\right\|_{L^p(Ω)})$?

Let $\Omega\subseteq\mathbb R^d$, $u\in\mathcal L^1(\Omega)$ and $$u_\varepsilon(x):=\frac 1{\varepsilon^d}\int_\Omega\rho\left(\frac{x-y}\varepsilon\right)u(y)\;{\rm d}\lambda(y)\;\;\;\text{for ...
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37 views

Differential Equation Simplification on American Put Option paper

I am currently reading "An Exact and Explicit Solution for the Valuation of American Put Options" by Song-Ping Zhu. The articel is available at ...
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11 views

Does a pde solution completely depend on the parameters

I was wondering whether pde solutions are completely determined by the pde parameters. For example let $f_{a,b}(x,y)$ denote a solution of a pde with UNKNOWN parameters $a,b$. Then, can we always ...
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12 views

Classification of second order linear PDE

I have been looking at the classification of the second order linear PDEs and came across two different definition If a PDE is defined as following: $$Au_{xx}+Bu_{xy}+Cu_{yy}+Du_x+Eu_y+Fu = G$$ ...
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50 views

Why can the solution of a SPDE $\partial_tu(t,x)=\cdots$ be viewed as a stochastic process indexed by $t$ with values in a space of functions of $x$?

Please consider a stochastic partial differential equation of the form $$\partial_tu(t,x)=F(t,x,u(t,x),{\rm D}u(t,x),{\rm D}^2u(t,x))+G(t,x,u(t,x),{\rm D}u(t,x))\partial_tB(t,x)\tag 1$$ where ...