# Tagged Questions

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

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### 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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### 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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### 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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### 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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### 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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I'm trying to solve the characteristic base curves of an initial value problem. $$\left\{ \begin{matrix}\ xy\frac{\partial u}{\partial x} + (2y^2 - x^6)\frac{\partial u}{\partial y} = 0 ; x>0,y\... 1answer 111 views ### Diffusion-Reaction PDE - radial coordinate I am trying to obtain an expression for the concentration C based on this stationary equation : \frac{\partial C}{\partial t} = \frac{1}{r} \frac{d}{dr} \left(r \frac{\partial C}{\partial r}\right)... 1answer 89 views ### Solve laplace equation inside a rectangular My answer is U = Acos(nπx/L)e^-nπy/L I really have no idea how to solve the particular solution. Please advise me. 0answers 52 views ### Transforming 1D Burger's Equation into infinitely many coupled ODE's I've been working on the following problem but I can't justify my steps, would a savvy mathematician kindly tell me what, if any, violations I've made. Problem: Show Burger's equation can be written ... 0answers 57 views ### when does a partial differential equation have unique solution? [duplicate] The differential equation  xu_x + yu_y = 2u satisfying the initial conditions y = xg(x), u=f(x) with f(x) = 2x, g(x) = 1, has no solution f(x) = 2x^2, g(x) =1, has infinite number of ... 1answer 145 views ### Is fractional order Sobolev spaces reflexive? Let 0<s<1, we define$$ W^{s,p}(\Omega):=\left\{u\in L^p(\Omega),\,\frac{|u(x)-u(y)|}{|x-y|^{\frac{N}{p}+s}}\in L^p(\Omega\times\Omega)\right\} $$with norm$$ \|u\|:=\left(\int_{\Omega} |...
I'm looking for any insight into solving the following PDE: $$u_{tt}=c^2 (u_{xx}+u_{yy})-\sin(y)$$ $$u=0, y\in {0,\pi}$$ $$u_x=0, x\in {0,1}$$ $$u(x,y,0)=\cos(\pi x)\sin(3y)$$ $$u_t(x,y,0)=0$$ ...