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

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1answer
20 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 ...
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 ...
0
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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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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 ...
-4
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0answers
19 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 $U_y = U_x +yU_y$?
1
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1answer
32 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 ...
0
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0answers
9 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 ...
0
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1answer
37 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
32 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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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 ...
-1
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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}, ...
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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 ...
2
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1answer
40 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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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 ...
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 ...
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 ...
2
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0answers
38 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
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
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 ...
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 ...
1
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1answer
52 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
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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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. ...
0
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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
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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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0answers
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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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 ...
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0answers
4 views

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 ...
1
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1answer
13 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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0answers
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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0answers
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 ...
3
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1answer
30 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
102 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 ...
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0answers
14 views

How to construct a integral solution to poisson equation with green functions

How to construct a integral solution to poisson equation with green functions
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0answers
20 views

Comparison principle for heat equation with smooth nonlinearity

Let $f : \mathbb{R} \to \mathbb{R}$ be $C^\infty$ and satisfy $f(0)=0$. Suppose $u_1,u_2 : \mathbb{R}^d \to \mathbb{R}$ are $C^2$ and satisfy $$\frac{\partial u}{\partial t} - \Delta u = f(u)$$ for ...
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0answers
9 views

method of characteristics in a nutshell

Good morning everybody, I need a quick reference for the following inhomogeneous first-order pde... namely $$f(x,y,z)=A\partial_x\varphi+B\partial_y\varphi+C\partial_z\varphi,$$ where $\varphi\in ...
2
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0answers
34 views

solution of multidimensional PDE

I'm looking for a way to find a solution 'f' to the following PDE. $$ y \frac{\partial f}{\partial r} + g_1(r)\left(z\frac{\partial f}{\partial y} - y\frac{\partial f}{\partial z}\right) + ...
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0answers
18 views

Weak time derivative for functions $u \in L^2(0,T;L^2(\Omega))$

The weak time derivative of a function $u \in L^2(0,T;H^1)$ is defined to be $u' \in L^2(0,T;H^{-1})$ satisfying $$\int_0^T \int_\Omega u(t) \varphi'(t) = -\int_0^T \langle u'(t), \varphi(t) \rangle$$ ...
1
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1answer
18 views

Solving characteristic base curves initial value PDE

I'm trying to solve the characteristic base curves of an initial value problem. $$ \left\{ \begin{matrix}\ xy\frac{\partial u}{\partial x} + (2y^2 - x^6)\frac{\partial u}{\partial y} = 0 ; ...
0
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1answer
26 views

Diffusion-Reaction PDE - radial coordinate

I am trying to obtain an expression for the concentration $C$ based on this stationary equation : $\frac{\partial C}{\partial t} = \frac{1}{r} \frac{d}{dr} \left(r \frac{\partial C}{\partial ...