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

learn more… | top users | synonyms (2)

0
votes
0answers
5 views

Fisher-KPP equation with a compactly supported initial condition

Let us consider the classical Fisher-KPP equation on the real line : $$ \partial_t u - \Delta u = u(1-u) $$ with an initial condition $u_0$ that satisfies : $u_0$ is smooth $u_0(x) = 1$ if $|x|\leq ...
0
votes
0answers
12 views

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 ...
2
votes
0answers
13 views

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 ...
1
vote
0answers
415 views

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, \begin{equation} -(\frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2})\Psi + \frac{\sinh^2(y) + \sinh^2(z)}{(\...
0
votes
0answers
23 views

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 ...
0
votes
0answers
30 views

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\...
1
vote
0answers
12 views

$-\Delta u = u^p$ 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 ...
21
votes
8answers
2k views

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, ...
-1
votes
0answers
11 views

Fixed Point method and existence of entropy solution of a Nonlinear Parabolic PDE [on hold]

Could someone point me to literature that addresses this kind of problem? Thanks
12
votes
1answer
474 views

Heat Kernel Property

Let $\phi$ be the Heat Kernel in $\mathbb{R}^n$, i. e. $$\phi (x,t)={(4\pi t)}^{-n/2}\exp\left( - \frac{\mid x \mid ^2}{4t}\right)$$ and let $u$ satisfy Heat equation. Show that: $$\frac{d}{dt}\...
1
vote
1answer
394 views

Simple heat equation, solution regularity

I have a small problem with a regularity result for a simple parabolic heat equation: Given a $C^2$ open subset of $\mathbb R^n$ called $\Omega$, and a time $T > 0$, i have the following heat ...
0
votes
1answer
27 views

Unable to reach the desired result by substituting a given solution into the Schrodinger equation

A Textbook question asks me to: From the time dependent Schrodinger equation: $$-\frac{\hbar^2}{2m}\nabla^2\Psi+V\Psi=i\hbar\frac{\partial\Psi}{\partial t}\tag{...
5
votes
1answer
384 views

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. ...
0
votes
1answer
17 views

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 ...
0
votes
0answers
30 views

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 ...
0
votes
1answer
32 views

How can I use this initial condition for the heat equation

How can I use the following initial condition for a partial differential equation describing heat diffusion? $$f(x) = \begin{cases} 0, & 0<x<0.45 \\ 1, & 0.45<x<0.55 \\ 0, & 0....
1
vote
0answers
43 views

Singularities in an Equation

these might sound like extremely trivial questions but since my background is more in probability and statistics I'm not too sure what to do, or even what to read up on to understand what to do. I ...
-5
votes
0answers
22 views

Problem 16 Chapter 2. Evans PDE 2nd edition [on hold]

This is a Problem from Evans PDE 2nd Edition,Chapter 2 Problem 16. Give a direct proof that if $U$ is bounded and $u\in C_1^2(U_T)\cap C(\overline U_T) $ solved the heat equation, then $$\begin{...
6
votes
2answers
90 views

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 ...
0
votes
1answer
483 views

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) $(D^3-3D^2D'+4D'^3)u=0$ ...
1
vote
3answers
56 views

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 ...
0
votes
1answer
16 views

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 ...
0
votes
0answers
14 views

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 ...
0
votes
0answers
41 views

mean evolution of 1D Fokker-Planck

Given the Fokker-Planck equation on 1D with drift term $D^{(1)}(x)$, and diffusion term $D^{(2)}(x)$, and the governing equation of probability density function $$\frac{\partial f(x,t)}{\partial t} = ...
1
vote
1answer
89 views

Is the naive solution of this PDE/BVP unique?

Problem statement Suppose I have a 2D or 3D equation of the form: $\vec{\nabla} \cdot \left[ \vec{\vec{a}}\left(\vec{x}\right) \cdot \vec{\nabla} f\left(\vec{x}\right) \right] = \vec{\nabla} \cdot \...
0
votes
0answers
25 views

Estimates on the integral of an inner product

Let $X$ be an inner product space. For vector-valued functions $F = (f_1,f_2), G = (g_1,g_2): [0,1] \to X^2$, we define the inner product $$(F, G) = \int_0^1 f_1g_1 + f_2g_2.$$ In particular, $$ ||F||...
0
votes
2answers
31 views

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 ($L[v(x,t)]=0$...
1
vote
0answers
74 views

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 ...
0
votes
1answer
36 views

How to solve this nonhomogeneous heat equation

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&...
1
vote
0answers
23 views

Resonance in wave equation

I have solved the non-homogenous equation by the method of eigenfunction expansion $$u_{tt} - c^2 u_{xx}=F(x)\sin(\omega t)$$ $$0<x<L, t>0$$ $$u(x,0)=u_t(x,0)=0$$ $$u(0,t)=u(L,t)=0$$ and got ...
0
votes
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}{...
0
votes
0answers
13 views

Groundwater related features and terminology. [on hold]

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 $𝑡≤0$. ...
0
votes
1answer
35 views

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 ...
-1
votes
1answer
27 views

Problem 23 Chapter 2. Evans PDE 2nd edition

This is a Problem from Evans PDE 2nd Edition,Chapter 2 Problem 23. Let $S$ denote the square lying in $\Bbb R\times (0,\infty)$ with corners at the points $(0,1),(1,2),(0,3),(-1,2)$. Define $$f(x,t):=\...
0
votes
1answer
25 views

PDE with a condition

Considering the heat equation, $$\frac{du}{dt}=\frac{d^2u}{dx^2}$$ if $$u(x,t)=t^{\alpha}\phi(\xi)$$ with $$\xi=x/\sqrt{t} \enspace then \enspace \phi \enspace satisfies \enspace \alpha\phi-(1/2)\xi\...
5
votes
1answer
51 views

Intuition for Fredholm operators?

Alot of the material I'm reading lately seems to mention Fredholm operators and the 'Fredholm alternative' and operators being 'Fredholm of index $0$'. Can someone give me a high level overview of ...
0
votes
1answer
53 views

What does 'dissipative PDE' means?

Can you give me an idea what is meant with dissipative partial differential equations? I am no phycist (and do not know the difference between initial energy to final energy), but wikipedia told me ...
1
vote
1answer
36 views

solving PDE equation like Helmholtz equation in 2D

In my project I need to solve following equation analytically could anyone help me ? As I read the other questions, my equation seems like Helmholtz equation $$ \triangledown^2 u(x,y) - k_cu(x,...
0
votes
0answers
45 views

PDE $yu_{x}+xu_{y}=0, u(0,y)=e^{-y^2}$ - why the solution is determined only in the region ${x^2 ≤ y^2}$?

I am solving the PDE $$\left\{\begin{matrix} yu_{x}+xu_{y}=0\\ u(0,y)=e^{-y^2} \end{matrix}\right.$$ My results are below. In the answer to this problem there is a statement saying that "a sketch of ...
1
vote
1answer
40 views

Intuition behind steps in formulating Finite Element Method

Let's consider the classic elastostatics case where the strong form of the PDE is: $\sigma _{ij,j}+b_i =0$ on V By multiplying through by weighting functions and integrating we can create an ...
3
votes
1answer
31 views

Where does the name “tracking type problem” come from?

In PDE-constrained optimization problems, the distributed control problem $$ \begin{array}{ll} \displaystyle \min_{y,u} & J(y,u) = \frac{1}{2}\|y-y_d\|_{L^2(\Omega)}^2 + \frac{\alpha}{2}\|u\|_{L^...
15
votes
5answers
27k views

What is difference between Finite Different Method, Finite Element Method and Finite Volume Method for PDE?

Can you help me explain the basic difference between FDM, FEM and FVM? What is the best and why? Advantage and disadvantage of them?
0
votes
1answer
20 views

finding solution to $u_x + 2u_y + (2x − y)u = 2x^2 + 3xy − 2y^2$

I tried to solve this equation with the coordinate method but I got a bit different answer compared to the suggested one by the solution manual. Where I am making the mistake in my solution? My ...
2
votes
0answers
41 views

Methods for solving nth order semilinear elliptic PDEs

I am looking for names of methods, and examples of their use that can be used to find solutions for semilinear elliptic PDE equations of the below types: $$\frac{\partial^ny}{\partial x^n}+\frac{\...
1
vote
0answers
23 views

finding solution to $au_x+bu_y=f(x,y)$ where $a \neq 0$

I can't get the solution in the required form of $$u(x,y)=(a^2+b^2)^{-\frac{1}{2}}\int_{L}fds +g(bx-ay)$$ "where g is an arbitrary function of one variable, L is the characteristic line segment from ...
2
votes
1answer
48 views

Modified Wave Equation: Bound $\int u^2 \, dx$

I'm studying for a qualifying exam and I can't figure this problem out: Suppose $B \subset \mathbb R^n$ is the unit ball centered at the origin and that $u$ is a smooth solution of \begin{align*} ...
0
votes
2answers
46 views

equation $u_x+u=e^{2y+x}$ (part of the solution to $u_x+u_y+u=e^{x+2y}$)

I solved/analyzed the below PDE $$\left\{\begin{matrix} u_x+u_y+u=e^{x+2y}\\ u(x,0)=0 \end{matrix}\right.$$ and have a question to the one of the steps involving the integration, see below ...
2
votes
0answers
37 views

The eigenfunction of modified 1-laplace equation

Let $\Omega\subset \mathbb R^2$ be a bounded, smooth boundary domain. I am interested in the following operator $$ -\operatorname{div}\left(\frac{\nabla u}{\sqrt{|\nabla u|^2+\epsilon}}\right)=0. $$ ...
1
vote
0answers
21 views

The eigenvalue of Laplace problem on a domain with a line removed

Let $\Omega\subset \mathbb R^2$ be given such that $\Omega$ is open bounded with nice boundary. Let $\Gamma$ be a finite segment in $\Omega$. We also define $\Omega_1:=\Omega\setminus \Gamma$. ...
2
votes
2answers
553 views

Solve Heat Equation using Fourier Transform (non homogeneous)

I know how to solve heat equation where it's like $u_t=k\cdot u_{xx}$ (using Fourier Transform or using Separation of Variables) but this exercise is really difficult for me. I have this: $$u_t(x,t)=...