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

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1answer
21 views

Maximum Principle for the PDE $\Delta u - a^2u=a^2$

I have this Dirichlet probem \begin{align*} \Delta u - a^2u&=a^2\quad\text{on}\;\, \Omega \subset \mathbb{R}^n \\ u&\equiv 0\quad \, \text{ on }\partial \Omega, \end{align*} where $a^2$ is ...
3
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1answer
43 views

Help with a proof about heat equation

The question is Suppose $U=\Omega \times (0,T)$ where $\Omega \subseteq \Bbb{R}^n$ is a bounded domain. Let $u\in C_1^2(U)\bigcap C(\bar U)$ satisfy $u_t \le\Delta u + cu$ in $U$ where $c \le 0$ ...
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0answers
46 views

When is one or more PDEs equivalent to one or more ODEs?

I'm relatively new to PDEs and ODEs. It seems that PDEs are generally more difficult to solve than ODEs, and so I intuitively have the feeling that one needs more information/knowledge/theorems in ...
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0answers
42 views

How to apply Sobolev inequalities?

I'm struggling with an application of Sobolev inequalities in Evans' book. He presents his argument like this: For $4<p<5$ we have $2(p-1)=2(p-4)+6=2(p-4)+ 2^*$ and therefore $$\left( ...
0
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1answer
27 views

PDE model of metal rod at temperature=1 plunged into a bath of temperature=0

Consider a metal rod (0 < x < l), insulated along its sides but not at its ends, which is initially at temperature=1. Suddenly both ends are plunged into a bath of temperature=0. Write the PDE, ...
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0answers
33 views

A question about a system of PDE

It is well known that under suitable conditions, the symmetry of mixed second partial derivatives reads: $$\frac{\partial^2 f}{\partial x \partial y}=\frac{\partial^2 f}{\partial y \partial x}.$$ ...
0
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1answer
36 views

An application of strong maximum principle for harmonic functions

This should be an easy application, since it is given as a remark without proof in Evan's pde book. Let U be connected and u harmonic in U. Then u is positive everywhere in U if u is positive ...
0
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1answer
13 views

Construct an exactly smooth function as a cutoff of half ball with vanishing normal dirivative

$\newcommand{\pt}{\partial}$ Suppose $B_r^+:=\{x=(x_1,x_2)\in B_r(0)\subset R^2|x_2\geq0\}$, can we construct a $C^\infty$ smooth function $\phi$, $0\leq\phi\leq1$, such that $$ \phi\equiv1 \text{ in ...
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0answers
22 views

Issues with finite-difference implicit solution of Advection-Diffusion-Reaction eqn

Objective: I am trying to numerically solve $C(x,y,t)$ from the following advection-diffusion-reaction equation in 2D space (x,y) and time. I will be testing my numerical solution with an approximate ...
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0answers
11 views

Reference request for a mixed boundary value problem

Let $\Sigma$ be a compact Riemannian manifold with boundary and assume that $\partial\Sigma=Y_1\sqcup Y_2.$ Let $\Delta$ be the nonnegative Laplacian on $\Sigma.$ I am looking for the reference for ...
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1answer
15 views

standard mollifier (comparing the definition in Evans and wiki)

Hi I am looking at the definition of standard mollifier $\eta$ in Evans, and the $\eta$ from wiki enter link description here Have a very basic question, is the $\eta$ in Evans also compactly ...
2
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1answer
55 views

A specific 1st order PDE which looks almost like a linear PDE

I have a PDE on the following form: $$ \frac{\partial f}{\partial t}(t, x) + \mu \frac{\partial f}{\partial x}(t, x) + \lambda [f(t, x+1)-f(t, 1)]= 0 $$ where $\lambda$ and $\mu$ are positive ...
0
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1answer
47 views

Numerically solving a system of partial integro-differential equations in Matlab [closed]

Given the following system of partial integro-differential equations - $\frac{dS(t)}{dt}=\Lambda-\mu S(t)-\beta S(t)F(t),\\ \frac{\partial I(t,\omega)}{\partial t}+\frac{\partial ...
0
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1answer
42 views

different generalized functions?

I am trying to solve a PDE that's order 1 in time $t\ge0$ and order 2 in space $x\ge0$. The solution $u(x,t)$ exists, is unique and possesses the following properties: $u(x,t)\ge0$ for all ...
-1
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2answers
34 views

Difference quotients and weak derivatives (Evans 5.8.2 theorem 3) [closed]

Could anyone give a proof on the remark below the theorem? Basically it is problem 11 I think the proof relates to example 19.19 (p.375) in enter link description here But I really do not ...
2
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1answer
45 views

Why is the wave equation second order

It is very intuitive that any function of the form $y=f(x+vt)$ would describe a wave in two spatial dimensions and time. From that it is easy to use the chain rule, letting $w=x+vt$ and doing: $$ ...
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0answers
19 views

Second order PDE and its equivalent First order Representation

For a second order hyperbolic PDE of the form $u_{tt}-u_{xx}=f$, one could define the variables $v=u_t$ and $w=u_x$ to write the equivalent first order form as $\left[ \begin{array}{ccc} 1 & 0 ...
2
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0answers
33 views

A quick question in PDE and evaluation of norms, need verification

Given $I\subset \mathbb R^N$ open bounded. Then we define two quantities $$ Q_1=\sup\left\{ \int_\Omega u\,\text{div}\phi\,dx, \,\phi\in C_c^\infty(\Omega;\,\mathbb R^2), \,\|\phi\|_{L^\infty}\leq ...
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0answers
25 views

Example of function in $H^1(U)$ which is not continuous, where $U \subset R^2$ has a smooth boundary. [duplicate]

Does anyone have a nice geometric example of function in $H^1(U)$ which is not continuous, where $U \subset R^2$ and has a smooth boundary. I want something that is easy to remember.
2
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1answer
40 views

Computing the total energy of Nonlinear Schrödinger (NLS) equation

NLS: $$ i\, u_t + \frac 12 u_{xx} \pm \lVert u\rVert^2u=0 $$ Show that the following energy of the nonlinear Schrödinger (NLS) equation is constant $$ E=\int\limits_{-\infty}^\infty \left( ...
4
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1answer
65 views

Explicit solution for equation

The claim is that this equation has an explicit solution. $$\frac{\partial}{\partial t}c(x,t)=\frac{a}{\pi}\int_{\mathbb{R}}\frac{c(y,t)-c(x,t)}{(y-x)^2}dy.$$ What can one do to find this solution? ...
3
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0answers
48 views
+50

Are there existence results for the heat equation on unbounded Lipschitz domain?

I am looking for a reference/ ideas on the following problem. Let $\Omega\subset\Bbb R^2$ be a Lipschitz domain (if it helps, the domain can be piecewise smooth with only one "kink", for example the ...
0
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0answers
14 views

Where are the boundary conditions in the solutions of $\lambda F - AF = f$

This question is somewhat related to the previous post Solutions of a linear equation for an elliptic operator satisfying a boundary condition. Here $$AF(x): = \sum_{i,j} a_{ij}(x) \partial_i ...
0
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1answer
33 views

Differentials of Multivariable Functions

A soft drink can is h centimeters tall and has a radius of r cm. The cost of material in the can is 0.0015 cents per cm$^2$ and the soda itself costs 0.002 cents per cm$^3$. The cans are currently 10 ...
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0answers
28 views

Well-posedness of nonlinear PDE system

The surface is parametrized by two variables $\sigma_1$ and $\sigma_2$. Moreover, this surface evolves in time. As a result, coordinates of the surface are: $\vec{F} =[x(\sigma_1,\sigma_2 , t), ...
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0answers
25 views

Approximate solutions to differential equations

If one has a differential equation for $y(x)$. If this differential equation has two solutions one for $x\ll a$ and the other for $a\ll x$, where $a$ is constant real value. My question is at what ...
2
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1answer
46 views

Solving a PDE problem

I have an analytical problem: I need to prove the following $$\lim_{\varepsilon\rightarrow0}\frac{i}{2\pi}\int_{\partial B_{x_i}(\varepsilon)}\frac{\partial u}{\partial z}dz=0 ;$$ where $u$ is a ...
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0answers
29 views

prove this poisson inequality

I'm reading the section on Poisson's equation on the book Partial Differential Equations by Evans, on page 24 he writes the following inequality for $|I_{\epsilon}|$, which I'm having a hard time ...
0
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1answer
19 views

Vanishing viscosity solution is an entropy solution

Actually, I have a problem in understanding a step in the proof of Theorem $2$ of section $4$ of chapter $11$ of L.C. Evans' PDE. I have understood the step: $\mathbb \phi (u^{\epsilon})_{t} + ...
3
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1answer
54 views

Differentiation calculation

$L(E)$ espace fonction continuous and linear $$\begin{array}{llll} \psi:& L(E)\times E&\longrightarrow& E\\ &(u,x)&\longrightarrow &u(x) \end{array}$$ proved the application ...
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0answers
28 views

What are the general steps to turn a PDE into a dynamical system $\dot x(t)= Ax(t) + Bu(t)$

It is said that every boundary value PDE such as the heat equation can be turned into dynamic system of the type $\dot x(t)= Ax(t) + Bu(t)$ with appropriate I.C. Can someone elaborate as to how to ...
1
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1answer
44 views

proof of existence of a solution with $ f \in L^1$

Let $\Omega$ be an open domain in $\mathbb {R^n}$ and $f \in L^1(\Omega)$ then how can we prove there is a weak solution $u \in ‎‎ W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$ for the problem ...
0
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1answer
55 views

Second order pde in 2 variables [closed]

How I find the general solution of $u_{xx}-u_{yy}+\dfrac {2u_x}x=0$ equation by $w=x^nu$ transform?
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0answers
29 views

Solution procedure for poisson equation

Consider the Poisson equation in the rectangle $Q=\{(x,y):0<x<1,0<y<1\}$, $$u_{xx}+u_{yy}=F(x,y)$$ $$u(0,y)=0,\,u(1,y)=0$$ $$u(x,0)=\phi(x),\,\,u(x,1)=\psi(x)$$ My Question: Is ...
3
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1answer
37 views

Regularity of solutions to a transport equation

Currently I am working on a transport equation and have been able to prove the existence and uniqueness of a weak measurable solution to said equation. I am now working in trying to jot down (with ...
0
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0answers
48 views

solving a 2nd order PDE with constant coefficients

This question is followed up from this question system of non-homogeneous advection equations \begin{equation} \left\{ \begin{array}{lll} u_t+b_1 u_x=(r+l_1)u-l_1v,\\ v_t+b_2 v_x=(r+l_2)v-l_2u,\\ ...
1
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1answer
55 views

Boundedness of $A$ in the operator equation $Au = f$ of $-\Delta u(x)=f(x)$.

We consider the boundary value problem on a bounded, open domain $\Omega \subset \mathbb R^n$ determining $u : \Omega \rightarrow \mathbb R$ such that $$-\Delta u(x)=f(x), \qquad ...
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0answers
28 views

Numerical method for wave equation with nonlinear forcing in 1+1 D

I am looking for numerical method to solve the equation $$ \square \phi = \frac{\partial^2 \phi}{\partial x^2} - \frac{\partial^2 \phi}{\partial t^2} = \lambda \phi^3 \, $$ for real $\phi = ...
0
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1answer
21 views

Analytical solution to PDE exists?

I have the following PDE: $\delta_\epsilon(S(\epsilon)\phi(x,\epsilon))+\delta_x\phi(x,\epsilon) = -T(\epsilon)\phi(x,\epsilon)$ Deltas represent partial derivatives, for ease of notation. Does it ...
0
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0answers
65 views

How to do this estimate

If $a,b$ are two vectors in $\mathbb R^n$ satisfy the following relation \begin{equation} \frac{|a|^2}{1+(1-|a|^2)^{\frac{1}{2}}}\geq \frac{|a|^2-2\langle ...
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2answers
40 views

An intuitive definition of the frequency spectrum of a function.

In a PDE book I'm reading, the author introduces the Fourier transform by first introducing the Fourier series, and then the Fourier integral representation of a function. The Fourier integral ...
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1answer
28 views

general solution function using Method of Characteristics

Suppose I am given a function $f(x, y, z)$ that is such that $3 f_x + xf_y + 2yf_z = 0$ I want to know how to write down a general representation for functions with such a property. Proceeding ...
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2answers
210 views

Numerical methods (for ODE/PDE) that could take approximate solutions/good initial guesses, and further refine it to an certain accuracy

I am currently playing with an old analog computer, which could solve time-dependent ODE/PDEs pretty fast, without time-stepping; thus there is no convergence issues caused by time-stepping because of ...
0
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0answers
20 views

Closure for Legendre Polynomials

Suppose I have a PDE which can be solved by an expansion on the Legendre Polynomial basis, for an axisymmetric problem in spherical coordinates. For the derivation for such a problem, see these ...
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0answers
24 views

Monotonic convergence of Newton's method for boundary value problems

I’m interested in solving nonlinear elliptic boundary value problems of the type $$ -a\Delta u + f(u) = 0, $$ $$ u|_\Gamma = u_0 $$ by Newton’s method when its convergence is global and monotonic. ...
1
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2answers
39 views

Pdes definition of spaces

I am reading Temam's book Navier Stokes Equations and he defines $E(\Omega) = \left\{u \in L^2\left(\Omega\right), \ \operatorname{div}(u) \in L^2\left(\Omega\right)\right\}$. Later he says that if $p ...
0
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1answer
38 views

Heat equation with sin initial condition

How do i find the analytical solution of the heat equation: $$U_t = U_{xx} + \sin{\pi x}$$ subject to $u(0,t) = u(1,t) = 0$ and $u(x,0) = \sin(\pi x).$ I appreciate its a pretty common/general ...
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0answers
54 views

system of non-homogeneous advection equations

I would like to solve this system \begin{equation} \left\{ \begin{array}{lll} u_t+b_1 u_x=(r+l_1)u-l_1v,\\ v_t+b_2 v_x=(r+l_2)v-l_2u,\\ \end{array} \right. \end{equation} First , I would like to ...
0
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0answers
21 views

System of multi-valued Hyperbolic semi-linear inhomogeneous PDEs

Recently I am working on solving a hydrodynamic problem which is a system of multi-valued Hyperbolic semi-linear inhomogeneous PDEs, \begin{eqnarray} \partial_t U+ \sum_{i=1}^3 ...
1
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1answer
28 views

Computing weak derivatives on an open square

I am looking at the computation of weak derivative in the blog https://sunlimingbit.wordpress.com/2012/11/25/one-example-related-to-weak-derivative-2/ For equation (3), I have some confusions. Here ...