# Tagged Questions

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

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### How to tell if you have specified sufficient initial data for a differential equation?

I recently learnt that the following 'wave equation' is not well-posed $$\begin{cases} \partial_{tt}u=\triangle u, & (0,1)\times\mathbb R^d\\ u(0,x)=u(1,x)=0,&x\in\mathbb R^d \end{cases}$$ ...
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### Singularities in a PDE

This is more of a general question rather than anything specific but I was just wondering if someone could point me toward resources which discuss singularities in a PDE rather than in an ODE (by ...
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### Why is there no classification scheme for linear pde of third order or above

I am well aware of the classification of 2nd Order PDE into Elliptic, Parabolic or Hyperbolic type depending on the analogy with conic sections from analytic geometry. I have seen third order PDE ...
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### Please help: My MATLAB code for solving a 2D Schrödinger equation keep giving me weird output. [on hold]

I've been trying to solve the following Schrödinger equation numerically, -(\frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2})\Psi + \frac{\sinh^2(y) + \sinh^2(z)}{(\...
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### PDE: Classical Solution?

Part (a) is a standard problem which can be solved either using separation of variables or Fourier transform.I haven't seen a proper definition for 'classical solution' to answer (b). Any help is much ...
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### Heat Equation : Commutation of partial derivatives and summation

I'am having a problem when checking the validity of the solution i found for the heat equation: \begin{cases} U_{t}(x,t)=U_{xx}(x,t),\ {(x,t)\in (0,1)\times(0,+\infty)} \\ U(x,0) = x^2 - x\\U(0,t)=0\...
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### $-\Delta u = u^p$ in bounded domain

In my PDE lecture we had the following theorem and I am wondering how strong it is: Theorem Let $\Omega \subset \mathbb{R}^n$ be a bounded domain (with $\partial \Omega$ sufficiently smooth, lets ...
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### Why is uniqueness important for PDEs?

Every text on PDEs I come across will spend alot of time on showing the existence and uniqueness of solutions to a particular PDE. The importance of the existence of a solution to a PDE is obvious, ...
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### application of c*algebras to PDEs

I am preparing an introductory talk about c* algebras and I'd like to motivate C*algebras or show an application of them and I'd prefer an application in the field of partial differential equations. ...
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### How to derive a logarithmic potential from Newtonian?

Suppose we believe that the formula for Newtonian potential in $R^3$ is correct: $\varphi(\bar{x}) = \frac{1}{|x|} = \frac{1}{r}$, disregarding the constant. What is the justification of the fact ...
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### The set of all maximal ideals (Wiener Algebra)

I am trying to prove a proposition and in my proof I somehow need to find the set of all maximal ideals of a Banach Algebra. This is my working environment: Let $A(\mathbb{R}^2)$ be the (Wiener ...
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### Why this abuse of notation correctly solves the heat equation

Here's a stupid method I observed to solve the heat equation in $\mathbb R^d$, \begin{align*} \partial_tu=\Delta u,\quad u|_{t=0}=u_0. \end{align*} Pretend that $\Delta$ is a constant so this just ...
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### How to decide whether PDE is Homogeneous or non-homogeneous.

I am studying second order PDE. And I have seen homogeneous and non-homogeneous PDE. But I cannot decide which one is homogeneous or non-homogeneous. For examples; (1) $(D^3-3D^2D'+4D'^3)u=0$ ...
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### Separation of Variables and Linear PDEs

Separation of variables is a powerful method which comes to our help for finding a closed form solution for a linear partial differential equation (PDE). For example, we all know that how the method ...
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### PDE argument about passage to limit in Bochner space and weak derivative

I have a function $u \in L^2(0,T;H^1_0)$ with $u' \in L^2(0,T;H^{-1})$, all on some smooth bounded domain $\Omega$. In addition I have a sequence $u_n$ such that $u_n \to u$ in $L^2(0,T;H^1_0)$ and ...
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### Exercise involving sequence Palais-Smale and operator compact.

Let H be a Hilbert space and let K: H $\rightarrow{R}$ be $C^1$ and such that $\nabla K: H\rightarrow H$ sends bounded sets into precompact sets. Consider the functional $J: H\rightarrow R$ defined by ...
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### Solving non-homogenous PDE with forcing function (which diappears!) dependent only on time

Applying the method of eigenfunction expansion to the PDE $$u_t -c^2u_{xx}=F(t)$$ $$0<x<L, t>0$$ $$u(x,0)=f(x)$$ $$u_x(0,t)u_x(L,t)=0$$ for the homogenous part of this equation ($L[v(x,t)]=0$...
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### Name of a Sobolev/Poincare type inequality

It is a basic analysis exercise to verify that for $u \colon [0,1] \times [0,\infty) \to \mathbb R$, if $u(0,t) = 0$ for all $t$, then $$||u(\cdot,t)||_\infty \leq ||u_x(\cdot,t)||_2,$$ where the ...
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I don't know how to solve it $$\left\lbrace \begin {array}{lcc} u_{t} \left( x,t \right) =u_{{\it xx}} \left( x,t \right) +2\,{{\rm e}^{-t}} \left( x-1+\sin \left( \pi\,x \right) \right) &0&... 0answers 23 views ### Resonance in wave equation I have solved the non-homogenous equation by the method of eigenfunction expansion$$u_{tt} - c^2 u_{xx}=F(x)\sin(\omega t)0<x<L, t>0u(x,0)=u_t(x,0)=0u(0,t)=u(L,t)=0$$and got ... 0answers 25 views ### Problem 24 Chapter 2. Evans PDE 2nd edition This is a Problem from Evans PDE 2nd Edition,Chapter 2 Problem 24. (Equipartition of energy). Let u solve the initial-value problem for the wave equation in one dimension:$$\left \{ \begin{array}{...
Applying mass conservation to an element in a homogeneous, isotropic unconfined aquifer, show that the groundwater flow in the aquifer is given by Eq. 2. Assume a constant value of h (=ℎ0) for $𝑡≤0$. ...
### Problem from Evans PDE on $u$ and $v$ satisfying $u_t+u_x=d(v-u)$ and $v_t-v_x=d(u-v)$
This is a Problem from Evans PDE 2nd Edition,Chapter 2 Problem 22. Let $u$ denote the density of particles moving to the right with speed one along the real line and let $v$ denote the density of ...