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

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1answer
76 views

Cauchy-Riemann equations and harmonic functions

I have the following question which I am supposed to use the Cauchy-Riemann equations to prove: Let u and v have continuous second derivatives satisfying: $$\frac{\delta u}{\delta x} = \frac{\delta ...
0
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1answer
18 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 ...
4
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2answers
93 views

Purely “discrete” PDEs?

Usually, one formulates a system of continuous PDEs and then discretizes it in order to approximately solve it. Is there a view point that instead formulates a system of "discrete" PDEs, which ...
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0answers
39 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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0answers
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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0answers
19 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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2answers
40 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: ...
0
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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
41 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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2answers
262 views

General solution for the system of PDEs from the curl of a vector field equaling another

In my vector calculus class, when we were introduced to the curl operator the professor gave us this example: Is it possible to find a vector field $\mathbf{G}$ such that $$\mathbf{F} = \nabla ...
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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 ...
1
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1answer
161 views

Parabolic PDE $\to$ ODE on Banach space

Would someone please explain to me the concept of converting a parabolic PDE to an ODE on Banach space? If I have a PDE, say $$u_t = f(u_{xx}, u_x, u, p)$$ where $p$ is a parameter and the solution ...
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0answers
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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0answers
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 $U_y = U_x +yU_y$?
2
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1answer
32 views

Solve Basic partial differntial equation question [on hold]

Help me solve the partial differential equation. $$\frac{\partial^2z}{\partial x^2} + 2 \frac{\partial^2z}{\partial y^2} - 3\frac{\partial^2 z }{\partial x \partial y} = e^{2x-y} + e^{x+y} + \cos(x ...
1
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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 ...
5
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1answer
86 views

Transforming a PDE given basis vectors

I have a non-orthogonal coordinate system defined by $\mathbf x=\mathbf x(r,\beta,z)$, and so I can find the basis vectors as $$ \mathbf g_r=\frac{\partial \mathbf x}{\partial r},\quad\mathbf ...
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0answers
10 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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0answers
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 ...
2
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1answer
42 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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1answer
53 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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0answers
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}, ...
1
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1answer
40 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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0answers
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 ...
1
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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)} ...
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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1answer
334 views

A question about Fisher's Equation and the Traveling Wave Equation

I am dealing with the Fisher's Equation: Consider the PDE: $u_t=u_{xx}+u(1-u)$, where $-\infty<x<\infty$ and $t>0$.Prove that there exists $c^*>0$ such that for each $c>c^*$ there ...
0
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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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0answers
9 views

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 ...
3
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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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0answers
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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0answers
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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0answers
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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0answers
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 ...
3
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2answers
53 views

Separation of Variables (PDEs): What about $0$?

Question: [See the context given below.] $\rm\color{#c00}{(a)}$ When we divide by the functions $T$ and $X$ to obtain $(1)$, aren't we assuming that the functions will be non-vanishing on ...
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1answer
118 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 ...
2
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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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0answers
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
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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0answers
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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0answers
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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0answers
17 views

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. ...
2
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1answer
44 views

Solving a first order nonlinear PDE

I have to solve this equation : $$ \frac{\partial u}{\partial x} \cdot \frac{\partial u}{\partial y} = xy $$ with initial condition $u(x,0) = x$. I know it is easy with separation of variables, but ...
2
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0answers
33 views

Solving a system of two linear PDE: $u_x+v_x +u_y=0$ and $v_x+u_y-{1\over 2} v_y=0$

trying to solve the following cauchy problem: $$u_x+v_x +u_y=0\\v_x+u_y-{1\over 2} v_y=0\\u(x,0)=1-x,v(x,0)=x$$ my solution is: 1. multiply each equation by $t_1,t_2$ and sum the two equations like ...
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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 ...
0
votes
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 ...