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

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15 views

can someone help me with method of characteristics please

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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13 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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11 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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1answer
19 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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11 views

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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22 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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40 views

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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1answer
43 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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1answer
20 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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1answer
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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24 views

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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20 views

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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1answer
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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0answers
11 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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1answer
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
41 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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16 views

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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2answers
91 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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17 views

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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20 views

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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1answer
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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1answer
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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1answer
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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10 views

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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39 views

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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1answer
41 views

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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16 views

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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1answer
17 views

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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1answer
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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8 views

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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1answer
55 views
+100

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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24 views

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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1answer
44 views

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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8 views

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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1answer
19 views

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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1answer
16 views

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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1answer
16 views

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 ...
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26 views

Caucy Problem infinite domain- Question

Consider the following problem $$u_t-u_{xx}=p(x,t),-\infty<x<\infty,t\geq0$$ $$u(x,0^-)=0$$ The fundamental solution for this is $$u(x,t)=\int_{-\infty}^{\infty} \int_0^t ...
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35 views

How to treat this integration involving fractional Laplacian?

Let $N\geq 2,\,k\geq b>1$, Consider problem $$ \begin{cases} -\Delta u +u=g(u), \quad x\in\mathbb{R}^N\\ u\in H^1(\mathbb{R}^N). \end{cases} $$ where $g(t)\leq \frac{1}{k}t$, when ...
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2answers
74 views

Calculate weak derivative

I am supposed to calculate the weak partial derivatives of the function $f: B(0,\frac{1}{2}) \rightarrow \mathbb{R}, x \mapsto |\log(\|x\|_2)|^\alpha$ for all $\alpha \in \mathbb{R}$, where ...
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Partial differential version of $ G(y,m) = G( \frac{y}{m} + m^{1/4}, m^{1/2}) $

I figured there must be a relation between partial differential equations and parametric equations like the wave equation in physics. I was working on something and wondering if anyone could tell me ...
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1answer
14 views

Basic question about weak solution of p-Laplace equation

Let $p \geq 2$ and $B(x_0, d)$ an open ball in $R^n$ ($n \geq 2$). Let $u \in W^{1,p}(B(x_0,d))$ a weak solution of $\Delta_p u = f$ , $u = 0 , \ on \ \partial B(x_0,d)$, that is, $\int_{B(x_0,d)} ...
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23 views

Uniqueness of the damped wave equation

I want to prove the uniqueness of the following. \begin{cases} u_{tt} + u_t - u_{xx} = 0 & 0<x<b, t>0 \\ u(0,t) = u_x(b,t) = 0 & t\geq 0 \\ u(x,0) = f(x), u_t(x,0) = g(x) & 0\leq ...
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1answer
28 views

Bump function on set

Given an arbitrary compact set $K \subset \mathbb{R}^n$ and an open set $V \subset \mathbb{R}^n$, I want to construct a bump function $\zeta$ that is $1$ on $K$ and has its support in $V$, where I ...
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27 views

Solve for x and y in z=x*2^y with a known z [closed]

I'm trying to figure out register values for a program I'm writing. I have a spreadsheet where I'm attempting to reverse engineer mantissa and exponent values so I can get the necessary register ...
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1answer
32 views

Finite differences and conservation law

I am using a Finite Difference scheme to solve a simple PDE in conserved form: $$\partial_t u = \partial_x (\partial_x u +au\partial_x u) = (1+a)\partial_x^2u +a(\partial_x u)^2 $$ $$\frac{u_{n+1,j} ...
4
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1answer
103 views

Solving a PDE with Feynman-Kac Formula

I'm trying to solve this PDE using Feynman-Kac formula Now i follow the regular steps Here is where I don't know how to proceed. How do I calculate this expectation?
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19 views

Problem 5.10.21 in Evans' PDE.

Show that if $u,v \in H^s(\mathbb{R}^n)$ for $s > n/2$, then $uv \in H^s(\mathbb{R}^n)$, and: $$||uv||_{H^s(\mathbb{R}^n)} \leq C ||u||_{H^s(\mathbb{R}^n)}||V||_{H^s(\mathbb{R}^n)}$$ Does someone ...