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

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Numerical scheme to 1D advection equation

I am trying to numerically solve a system of equations which model the early universe in 1D. The equations I am stuck on are; $$ (1)\quad \partial\rho/\partial t + \partial(\rho v)/\partial x = 0 ...
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Switch integral and sum for Bessel function.

I haven't real knowledge in Bessel's function and I'd like to know how to switch integral and sum in these two equations. I've already tried a lot of ideas but nothing really works. The first one is : ...
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maximum principle for $p$-Laplace equation

Consider $\Omega \subset R^n$ a bounded domain. Let $\varphi \in W^{1,p}(\Omega) \cap L^{\infty}(\Omega)$. Let $u \in W^{1,p}(\Omega)$ with $\Delta_p u = 0$ in $\Omega$ with $u - \varphi\in ...
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1answer
22 views

singularity and degeneracy of an ODE

I have trouble distinguishing the difference of singularity and degeneracy in the context of ODE theory. Could anyone give me a couple of examples in illustrating the difference of singular point and ...
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19 views

The tangent hyperplane to the graph of harmonic function

This is an interesting question I found online about Laplace equation. We define a function $u$ :$R^N\to R$'s graph to be the set $\{x,u(x):\,x\in R^N\}\subset R^{N+1}$. Then I want to prove that ...
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How to find the wave equation for a given dispersion relation?

Let's assume I have general wave function written like this: $$ f(x,t) = \displaystyle\int_{-\infty}^{\infty}A(k)e^{i(kx-\omega(k) t)}dk $$ So it's composited from lots of plane waves and each of ...
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10 views

$L^2$ regularity of a convolution with newtonian potential.

I am reading Bertozzi, Majda Vorticity and incompressible flow and in page 71 72, we are concerned with recovering the velocity field of a flow from its vorticity. At some point we need to have the ...
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23 views

Show that $\max_{\bar{U}} |u|\le C(\max_{\partial U} |g|+\max_{\bar{U}} |f|)$

Let $U$ be a bounded, open subset of $\mathbb{R}^n$. Prove that there exists a constant $C$, depending on only $U$, such that $$\max_{\bar{U}} |u|\le C(\max_{\partial U} |g|+\max_{\bar{U}} |f|)$$ ...
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37 views

Solving cauchy hyperbolic second order pde

I'm currently taking a course in partial differential equations. I'm trying to solve the following problem (which is, as far as I can tell, a bit above the level of the course): $$\begin{align} ...
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1answer
41 views

Solving a one dimensional wave equation

Consider the partial differential equation $$\frac{\partial^{2}u}{\partial t^{2}}=c^{2}\frac{\partial^{2}u}{\partial x^{2}}$$ For the region $0<x<\pi$ where $t>0$. With the boundary ...
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Anyone knows a Good Textbook in Numerical PDES

I am planning on taking a course on numerical PDEs next semester. The course covers the following topics listed below. I am looking for a good book that covers these topics (or at least most of them). ...
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can someone help me with method of characteristics please [on hold]

Can someone help me with this question please? Please open this photo to see question. Thanks. Please need explain how you solve it. Thanks for looking. pg
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20 views

What physical information does the mean value property of heat equation convey?

I'm reading through the Evans' book on PDE, the chapter on heat equation. The definitions are the same as here. I see that mean value property of heat equation is useful for proving maximum principle ...
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24 views

Trace map $\gamma:H^1(\Omega) \to H^{\frac 12}(\partial\Omega)$ invertible on $\{ u \in H^1(\Omega) \mid \Delta u =0 \}.$

It is apparently true that the trace map $\gamma:H^1(\Omega) \to H^{\frac 12}(\partial\Omega)$, is invertible when restricted to the domain $$\{ u \in H^1(\Omega) \mid \Delta u =0 \}.$$ I've been ...
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23 views

Restriction estimates

What is the defining property of what someone in the harmonic analysis community would call a "restriction estimate?" I see sobolev norms, Fourier transforms, and inequalities relating these. The ...
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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 ...
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29 views

Basis of $H^1(\Omega)$ which is orthonormal wrt. $L^2(\partial\Omega)$ inner product?

Let $\Omega$ be a domain with $\partial\Omega$ bounded. Is it possible to find a smooth basis of $H^1(\Omega)$ and $L^2(\Omega)$ which is orthonormal wrt. the $L^2(\partial\Omega)$ inner product? ...
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How to solve $4x^2\cos y\sin y\partial{y}-3x\sin y\partial{x}+8\sin^2y\partial{y}=0$?

$$4x^2\cos y\sin ydy-3x\sin ydx+8\sin^2ydy=0$$ find the solution of this Bernoulli equation. How can I start?
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49 views

The rate of convergence for finite difference methods for Poisson's equation with piecewise constant data

I am solving the following PDE; $$ \nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = \rho, $$ where $\rho(0.5,0.5) = 2$ (zero elsewhere), $0\leq x,y\leq1$ and the ...
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23 views

Finding the fundamental mode of wave equation in a rectangle

Consider the two-dimensional wave equation: $$\frac{\partial^{2}u}{\partial t^{2}}=c^{2}\left[\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}\right]$$ for some $c>0$. I ...
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20 views

Solving 2D Laplace eigenfunction equation

I want to solve the equation $$\nabla^{2}\,{\rm P}(x, y) = \frac{k}{c^{2}}{\rm P}(x, y)$$ or $$\frac{\partial^{2}{\rm P}}{\partial x^{2}}+\frac{\partial^{2}{\rm P}}{\partial y^{2}}=\frac{k}{c^{2}}{\rm ...
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The solution of a PDE [on hold]

The PDE $$\begin{align} &u_{xx}+u_{yy}+\lambda u=0,&&0<x,y<1\\ &u(x,0)=u(x,1)= 0, && 0≤x≤1\\ &u(0,y)=u(1,y)=0, && 0≤y≤1\end{align}$$ has: A unique ...
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derivative by chain rule on wave equation pdes form [on hold]

If we have a wave equation $ U_x + U_y =0$, $y=Ce^x$, how can we prove that $U_x(x,Ce^x)=U_x + Ce^x$ and $U_y = U_x +yU_y$?
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33 views

Equilibrium Temperature in insulated rod from BVP

this problem is giving me some trouble. It is a review problem given to us to study for our Final Exam this week. I would love some help in understanding how to solve it so that I can study it and ...
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13 views

Classical solution of a non-homogenous Helmholtz equation

Let $G(x,k)$ be a fundamental solution for the Helmholtz equation in $\mathbb R^3$: $$ G(x,k) = \frac{e^{ik|x|}}{4\pi |x|}, \quad k>0. $$ For given function $\rho \in C_c(\mathbb R^3)$ define the ...
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39 views

What is the solution to poissons equation with a point source?

I am currently trying to solve the following PDE using various numerical methods - direct and iterative. $$ \nabla ^2 u = \rho $$ where $u = u(x,y)$ and $\rho(0.5,0.5) = 2$ (zero elsewhere). The ...
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2answers
43 views

Help with first order linear PDE with initial condition

I would like to solve the following pde: $2y\cdot \partial_x u(x,y)-3x\cdot\partial_yu(x,y)=0$ and $u(x,x)=e^{x^2}$ Without the initial condition I got the following result: ...
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nonlinear coupled partial differential equations

Is there any order for solving a set of nonlinear coupled partial differential equations analytically i.e. without a numerical algorithm. I cant solve the following set of equations $$ ...
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92 views

How would I compute this sum?

So I would to compute this integral which is coupled by a sum: $$ \int_{x = 0}^{x = \lambda} \sum_{k=-\infty}^\infty e^{-( \frac{x-k \lambda}{\sigma} )^2} dx$$ I was thinking about using parseval's ...
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first order equation problem

I have a first order PDE with the initial condition: (a) $$\displaystyle\frac{\partial f(x,t)}{\partial t}-\displaystyle(xt)\frac{\partial f(x,t)}{\partial x}=0$$ $$f(x,0)=\frac{1}{1+x^2}, ...
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Second order equation and regularity

Let $U$ be an bounded open set in $\mathbb R^3$ and ${V}\subset \subset{W} \subset \subset {U}$. Let $u \in {H^1}(U)$, and $f\in L^{2}(U)$ satisfies $$\int_{\mathbb R^3} \nabla u(x). \nabla \varphi ...
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43 views

Proof of an inequality in Sobolev space.

I want to show the next inequality: $$\| D^a u D^b v \|_{L^2(\mathbb{R}^n)} \leq C \| u \|_{H^s(R^n)} \| v \|_{H^s(R^n)}$$ (for $s>n/2$) Where $D$ is a differential operator. What I did so far is ...
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Domain of dependence of wave equation?

Is the solution is $t=R$? Because the domain of dependence of $x=0$ is $|x-0|=t$, so compared to $|x-0|=R$. I get $t=R$. Is that correct? I am not sure if my argument is sufficient. Can anyone help ...
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19 views

Fundamental solution to the bi-harmonic operator?

I am not sure about what the hint means. If $\Delta u =\frac{1}{2 \pi}(1+\log|x|)$. Since $\log|x|$ is a fundamental solution of $\Delta u =0$. Does that mean $\frac{1}{2 \pi}(1+\log|x|)$ is a ...
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41 views

Prove the energy is constant in a PDE?

I calculated the $$ \begin{align} \frac{dE(t)}{2\,dt} & = \int_\Omega u_tu_{tt}+DuDu_t+u^3u_t\,dx \\ & =\int_\Omega [u_t(u_{tt}-\Delta u)+u^3u_t] \, dx+\int_{\partial \Omega} u_t ...
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Meta Euler Lagrange

Consider the following generalized standard Euler Lagrange Problem which is to maximize the quantity $$ \int_{x_1}^{x_2} L(x,y,y', y'' ... y^{(n)}) dx $$ It can be solved by first determining a ...
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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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How do I accurately find a point within a grid of 4 based on varying values?

I am specifically trying to find a source of heat, with heat sensors giving readings from 4 different points eg below: The diagram above represents a 4 x 4 grid where the numbers are heat sensors ...
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Interior boundary value problem for the Helmholtz equation

Let $D \subset \mathbb R^d$ be a $C^2$ bounded domain. I consider the following boundary value problem for the Helmholtz equation $$ (\Delta+k^2)u = 0 \quad \text{in $D$}, \\ u|_{\partial D} = ...
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Finding (exactly) the electric potential, in presence of non-constant dielectric

In a medium with homogenous dielectric, the electric field can be solved as an instance of Poisson's equation, but this is not the case in general. I can find the variational form and solve with ...
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30 views

Partial Derivatives versus Proper Derivatives

I'm having some difficulty understanding exactly what a partial derivative is. I had been content with the definition $$\frac{\partial F}{\partial x_i } = \lim_{\Delta x \rightarrow 0} \frac{F(x_0, ...
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Inverse Fourier Transform of $1/k^2$ in $\mathbb{R}^N $

This comes up in the context of finding the Green's function of Poisson's equation for $\mathbf{x} \in \mathbb{R}^n $ $$ \nabla^2 G(\mathbf{x}) = \delta(\mathbf{x})$$ Attempt by using Fourier ...
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A problem on infinite domain diffusion equation

Consider the following problem $$u_t-u_{xx}=p(x,t), -\infty<x<\infty,t>0$$ $$u(x,0)=0$$ $$u\rightarrow0 \text{ as } x\rightarrow \pm \infty$$ This can be solved using many sub problems as ...
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Nonlinear Schrodinger Equation

Consider the equation $$i u_{t} = u_{xx} + au - bu|u|^{2},$$ where $a, b > 0$ are real constants and $|u|^{2} = uu^{*}$. (a) Find the dispersion relation for the equation and discuss the behavior ...
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How do I prove this differential equation has 3 dimensional solution?

$$\dfrac {\partial\boldsymbol u}{\partial t}+(\boldsymbol u\cdot\nabla)\mathbf u=\boldsymbol f-\dfrac1\rho\nabla p+\nu\Delta\boldsymbol u.$$ How do I prove it? I don't know where to start. Thanks in ...
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What is numerical flux function?

I am learning "Numerical Approximations of Hyperbolic Systems of Conservation Law". I can not find answers for the following questions: 1.What is the numerical flux function? 2.How can one find ...
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use of Strichartz estimates

I have been learning(understanding) the Strichartz estimates. My naive Questions: (1) Can you give some ideas, how Strichartz estimates plays role in solving nonlinear dispersive equations(e.g. ...
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Calculate $\int_{B_R(0)}u$ for a very weak solution $u$ of a Dirichlet boundary problem

Let $B_R:=B_R(0)\subseteq\mathbb{R}^n$ be the open ball with radius $R>0$ around $0$, $f\in\mathcal{L}^\infty(B_R)$ be radial and $u$ be a very weak solution² of $$\left\{\begin{matrix}-\Delta ...
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Uniqueness of the fundamental solution of a 2nd order linear parabolic PDE

I'm reading Avner Friedman, Partial Differential Equations of Parabolic Type. Let \begin{eqnarray} L:=a_{ij}\dfrac{\partial}{\partial x_i\partial x_j}+b_i\frac{\partial}{\partial ...
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Meanvalue formula of harmonic functions - translation invariance of surface measure

In almost every proof of the mean value formula for harmonic functions (e.g. Evans 2.Theorem 2) one finds the following calculation $$ \frac{1}{|{\partial B(0,1)}|}\int_{\partial B(0,1)} D ...