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

learn more… | top users | synonyms (2)

1
vote
0answers
6 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 ...
0
votes
0answers
16 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 ...
1
vote
0answers
9 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 ...
-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
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\...
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{...
0
votes
0answers
29 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
14 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
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....
-5
votes
0answers
21 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{...
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
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
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 ...
1
vote
0answers
42 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 ...
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&...
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}{...
-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
24 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\...
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
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 ...
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
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 ...
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 ...
1
vote
1answer
34 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,...
2
votes
0answers
40 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{\...
0
votes
1answer
52 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 ...
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
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
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
1answer
24 views

Spectrum of the Laplacian for free space versus spectrum of Laplacian with boundary conditions?

In this question the spectrum of the Laplacian in free space is defined as $]-\infty, 0]$. So it seems the spectrum can be determined without regard to boundary conditions? If instead we consider the ...
1
vote
1answer
23 views

How to think of Sobolev spaces $W^{k, p}$ for a function that is no longer an element of $W^{k, p}$ for $p$ greater than some number?

Consider the function $u(x) = x^{\frac{1}{2}}$ on the domain $[0, 1]$. This function is an element of $W^{1, 1}$ and $W^{1, \infty}$ but not $W^{1, 2}$ as for $W^{1, 1}$, we have $\Vert\frac{\...
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
0answers
7 views

schwarz decomposition methods on shallow water equations [closed]

How can I apply schwarz decomposition methods on two dimensional shallow water equations mathematically. Thanks
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
vote
0answers
57 views

Norm of gradient of velocity field

If $\mathbf{u}(x,y,z,t)=(u,v,w):\mathbb{R}^3\times[0,+\infty)\to\mathbb{R}^3$ denotes a velocity field, what is the definition for $\|\nabla\mathbf{u}\|_{L^{\infty}}$? I know that $\nabla\mathbf{u}$ ...
0
votes
0answers
10 views

Questions about the regularity of the solution of the heat equation in a bounded domain

I have questions about the proof of the following theorem: Theorem 8 (Smoothness). Suppose $u\in C^2_1(U_T)$ solves the heat equation in $U_T$. Then $u\in C^\infty(U_T)$ Here is the statement and ...
1
vote
1answer
26 views

Infinite propagation speed for the Schrödinger equation

I've seen many articles making reference to the property of the infinite propagation speed for the solution of the linear Schrödinger equation; but i can't find a book giving a 'good' definition or a ...
0
votes
0answers
14 views

Definition of pseudo-differential operator

I'm trying to understand the defintion of pseudo-differential operators ($\psi do$) on manifolds. According to Hörmander, The analysis of Linear Partial Differential Operators, v. III, 18.1.20 (1994), ...
2
votes
0answers
44 views

Elementary proof of $C^\infty_c$ is dense in $L^p (L^q)$ mixed space

As it well known, $C^\infty_0 (\mathbb{R}^n)$(the space of infinitely differentiable functions with compact support) is dense in $L^p (\mathbb{R)^n}$ Here I want to consider the same result with ...
2
votes
1answer
22 views

Using scaling arguments to determine relationships between Sobolev spaces?

I was looking up how to find relationships between Sobolev spaces and I came across this post on MO in which the first comment talks about a scaling procedure for understanding the relationships: ...
0
votes
0answers
13 views

Curl pde with Dirichlet boundary condition in a simply connected domain

Let $\Omega$ be an open, bounded, connected, simply connected domain in $\mathbb{R}^3$, with boundary $\partial\Omega=\partial\Omega_1\cup \partial\Omega_2$. Suppose $\mathbf{H}\in H(\mbox{curl};\...
-2
votes
1answer
25 views

Partial differential equation solvable as ordinary differential equation [closed]

Solve the following partial differential equation for $u=u(x, y)$: $$U_{yy}+6U_y+13U= 4e^{3y}.$$
0
votes
0answers
6 views

Laplace Equation on Prolate Spheroidal Coordinates

I'm currently trying to solve Laplace's equation outside of a prolate spheroid. The scalar field has to vanish far from the spheroid. Because of the geometry I thought it might be convenient to use ...
2
votes
2answers
38 views

Weak problem formulation for PDE and boundary conditions

Consider the following example: $$ - \Delta u = f \mbox{ in } \Omega, $$ $$ u = 0 \mbox{ on } \Gamma, $$ Here $\Gamma$ is boundary of $\Omega$. To produce weak formulation we multiply by arbitrary $v$ ...
0
votes
0answers
30 views

Solution to wave equation for $u(t, 0) = \gamma(t)$

I am looking for a necessary and sufficient condition on a function $\gamma: \mathbb{R} \to \mathbb{R}$ such that there is a solution $u \in C(\mathbb{R}^2)$ for the wave equation $$\partial^2_t u(t, ...
7
votes
0answers
51 views

How to get the idea of the formula for the mean value property for the heat equation

From the mean-value property of the Laplace's equation, we have the following mean-value property: $$ u(x)=\frac{1}{a(n)r^n}\int_{B(x,r)}udy. $$ But for the mean-value property of the Heat equation, ...
2
votes
1answer
42 views

PDE boundary condition question regarding limits

Just as a bit of background, I'm working with the Black-Scholes PDE and I'm testing some things out by taking an initial condition for it as $\sin(S/50)$, where $S$ is the spot price (but that's ...