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

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3
votes
1answer
98 views

How do I apply this maximum principle?

I have the maximum principle: $$\text{If } \psi\geq 0 \text{ on }\Gamma. \text{Then }L\psi\geq0\text{ implies } \psi\geq 0 \text{ in } \bar{D},$$ where $D=(0,1)\times (0,T], \Gamma$ is parabolic ...
0
votes
1answer
25 views

Why are we discarding solutions to this heat equation?

Dirichlet problem on unit disc in polar: $u_{rr} + (1/r) + (1/r^2)u_{\theta\theta} = 0$ $u(1,\theta) = f$ Period in $\theta$ gives $u(r,\theta) = \sum R_n(r) e^{in\theta}$ Inserted into our PDE ...
1
vote
0answers
40 views

Real Hardy space question

Please see this related question for the definition of the grand maximal function and the class of normalised test functions $\mathcal{T}$. I will refer to them in this question. Let $f\in ...
-2
votes
1answer
59 views

Why isn't there a general comparison principle for higher order equations?

I am studying elliptic equations. For second order equations (linear and nonlinear), comparison principle has many applications, e.g. to show uniqueness of the (weak) solutions, to construct super- ...
1
vote
1answer
64 views

Function Approximation

I need to solve the following equation $$-\frac{\partial S(x,y,t)}{\partial t}=ax^2+bx\frac{\partial S(x,y,t)}{\partial x}+c\Big[\frac{\partial S(x,y,t)}{\partial ...
0
votes
0answers
33 views

Solving partial differential equation with mathematica.

I am trying to solve the heat equation in cylindrical coordinate. The object of inspection is a thick ring with height $\{z_0,z_L\}$ and radii $\{r_1,r_2\}$ within a heat generating environment. I ...
0
votes
1answer
17 views

Green identity for measures with compact support

Let $\Omega\subset\mathbb{R}^N$ be a bounded, smooth domain. Assume that $\mu \in \mathcal{M}(\Omega)$ has compact support in $\Omega.$ Let $u\in W_0^{1,1}(\Omega)$ be a solution of $$ \left\{ ...
1
vote
1answer
35 views

Function spaces for the 1-dim heat equation.

Consider the standard 1-dim heat equation: $u_t(x,t)-\alpha u_{xx}(x,t)=0$, where $u:\mathbb{R}\times\mathbb{R_+}\rightarrow \mathbb{R}$, with initial conditions $u(x,0)=g(x), x\in\mathbb{R}$ and ...
0
votes
1answer
20 views

Laplace Equation with non-const Dirichlet Boundary Conditions

I'm struggling to get a Laplace problem with inhomogeneous boundary conditions solved. My memories are very rusty, and it almost works out, but I've got my brain twisted in some way. So I'm kindly ...
0
votes
0answers
36 views

Problems with understanding a proof.

I have some problems with understanding the Div-Curl lemma's proof. More precisely I don't understand the first part of the proof of the Theorem 1.1 in the article below: ...
0
votes
1answer
38 views

Solve a system of equations.

I have a system of equations: \begin{align} & x_{21} (\frac{\partial}{\partial x_{11}}f_{1111})( x_{11} , x_{21}, y_{11} , y_{21} ) + \frac{y_{21}}{x_{11}^2} (\frac{\partial}{\partial ...
0
votes
0answers
18 views

Covariance Integral Operator and its inverse

can someone help me with the following question? I want to consider parameter estimation problems for a single parameter q using the likelihoodmethod, where the state $y$ is constrained by a pde. But ...
0
votes
0answers
25 views

Space-dependent diffusivity and finite-differences

I want to implement a finite difference code of this simple diffusion equation with space-dependent diffusivity: $$\partial_{t}u =D\partial_{x}^{2}u+\partial_{x}D\cdot\partial_{x}u$$ I go for a ...
2
votes
0answers
45 views

Meaning of “Canonical System of the First Order”

I am learning about PDEs and came across the following. "Convert a partial differential equation of higher order into a canonical system of the first order" What does the above statement mean/imply? ...
0
votes
1answer
50 views

Green's representation on a compact domain

This is from page 17-18 of Trudinger and Gilbarg Let $\Omega$ be a domain for which the divergence theorem holds. Let $\Gamma(x-y)$ be the normalised fundamental solution of the Laplace's equation, ...
0
votes
1answer
46 views

Applications of PDE and laplace equation

The edge r = a of a circular plate is kept at temperature f(theta). The plate is insulated so that there is no loss of heat from either surface. Find the temperature distribution in steady state. I'm ...
2
votes
0answers
33 views

Solving PDE via Fourier Transform & Uniqueness

When a PDE is solved via Fourier transform, is there already a uniqueness assertion that comes for free? For example, if we Fourier the heat equation \begin{align} \partial_t u(x,t) &= \Delta ...
2
votes
1answer
37 views

References on estimating capacities (Newton, Martin etc) for sets & alternative formulations.

By G-capacity for capacitable set K I mean: $Cap(K)=[inf\{\int\int G(x,y)d\mu(y)d\mu(x):\mu$ probability measure on K$\}]^{-1}$. where G(x,y) is any kernel eg. the Green kernel. Q1:We've calculated ...
0
votes
0answers
27 views

Solutions of a PDE in two dimensions

I should solve the following PDE: $$\dfrac{\partial u(x,y,t)}{\partial t}=a\nabla^2u(x,y,t)+b\nabla u(x,y,t)+g(x,y,t)-c u(x,y,t)$$ where $(a,b,c)$ are constants, $$u(x,y,0)=u_0$$ $$u(x,y,t)=0$$ for ...
2
votes
0answers
28 views

Local Mollification of a $L^1$ function

Let $f$ be a $L^1(\Omega)$ function, where $\Omega\subseteq \mathbb{R}^2$ is a smooth and bounded domain. Then can we find a fuction $\phi\in W_0^{2,2}$ suct that $\phi>0$ in $\Omega$ and ...
4
votes
2answers
76 views

Intuitive interpretation of $\frac{\partial S(a,t)}{\partial t} = -\frac{\partial S(a,t)}{\partial a}$

I'm trying to visualize what the following equation is saying: $$\frac{\partial S(a,t)}{\partial t} = -\frac{\partial S(a,t)}{\partial a}$$ where $S$ is a probability-density, but I think you can ...
0
votes
0answers
36 views

On an interesting boundary condition

So I am tackling an interesting boundary condition, where $B(Du)=0$, for $x\in\Omega$, where $B$ is the signed distance function to $\Omega^*$ (where $\Omega,\Omega^*$ are convex domains in $\Bbb ...
4
votes
3answers
82 views

Pair of PDEs to be solved together

I have the following pair of equations to be solved together to find the functions $H_{x}$ and $H_{y}$, which are the components of a vector $\bar{H}(x,y)=H_{x}(x,y)\hat{x}+H_{y}(x,y)\hat{y}$ in ...
6
votes
1answer
102 views

Non-examples for the Kato-Rellich Theorem

The Kato-Rellich Theorem is a classical result stating that if $A,B$ are unbounded operators on a Hilbert space with $A$ self-adjoint, $B$ symmetric, $\mathcal D (A)\subset \mathcal D(B)$ and $$ ...
4
votes
0answers
71 views

Second degree partial differential equation with variable-change

Edit: @Etienne mentioned that I did a typo, writing $u_y' = -xye^{-y}$ instead of $u_y' = -xe^{-y}$. I've corrected that in the calculations and now it's closer to being correct! Though I still miss ...
2
votes
1answer
46 views

Elliptic partial differential equations and elliptic operators

I'm starting to study elliptic partial differential equations and I just want to know if there are any connections between the following concepts: An elliptic partial differential equation is given ...
0
votes
0answers
16 views

Deriving lower bound for eigenvalues of laplace operator

Let $u$ be a function such that $$ \Delta u + \lambda u = 0 \quad \mbox{ on } \quad D, \quad u = 0 \quad \mbox{ on } \partial D $$ and let $w$ be a function such that $\Delta w + \beta w < 0$ ...
1
vote
0answers
23 views

normalized mean curvature flow with convex initial hypersurface has finite velocity

I can't understand the prove in [Xi-Ping Zhu] Lectures on mean curvature flows. The statement as follow. Lemma 3.5 (page 32) There exists a positive constant $C$ such that ...
2
votes
1answer
102 views

Show that $L[v^2] := \Delta(v^2) + \frac{2}{w} \sum_{i=1}^n w_{x_i} (v^2)_{x_i} \ge 0$

Let $u$ be a function such that $$ \Delta u + \lambda u = 0 $$ for some $\lambda \in \mathbb R$, also let $w$ be a function such that $$ \Delta w + \beta w < 0. $$ for some $\beta \in \mathbb R$. ...
1
vote
1answer
39 views

Compatibility of initial and boundary conditions

Suppose we consider the heat equation $$\partial_t u = \Delta u, x \in \operatorname{int}D^2, t > 0$$ where $D^2$ is the closed unit disc in $\mathbb{R}^2$, subject to Neumann type boundary ...
1
vote
0answers
49 views

Fourier transform help for solving $u_t+u_{xxxx}+u_{xx}=0$

I just started to learn a little bit of fourier analysis in solving PDEs. I want to find a solution $u(x,t)$ to $u_t+u_{xxxx}+u_{xx}=0$. My attempt: Applying the fourier transform to both sides gives ...
1
vote
1answer
40 views

Constructing a specific normalised test function.

Given the following class of normalised test functions: \begin{equation} \mathcal{T}=\{\phi\in C_c^{\infty}(\mathbb{R}^n)\ |\ \text{supp }\phi \subseteq B(0, 1),\ \|D\phi\|_{\infty}\leq 1\} ...
2
votes
0answers
74 views

Compact support

From PDE Evans, 2nd edition, page 204 Example 9 (Wave equation from the heat equation). Next we employ some Laplace transform ideas to provide a new derivation of the solution for the wave ...
3
votes
1answer
45 views

complex calculation in Schrödinger equation

I'm studying a paper with the following Schrödinger equation. $$i\,y_t+\Delta\,y-F(y)=0$$ subject to Dirichlet boundary conditions where $F$ is supposed to be of the form $F=\displaystyle ...
0
votes
1answer
21 views

Dirichlet boundary value problem in convex domains with discontinuous boundary values

Consider $\Omega$ an open, bounded, and convex domain in $\mathbb{R}^n$. Let $g \in L^{2}(\partial \Omega)$ such that the problem $$ \left\{ \begin{array}{ccccccc} \Delta u = 0, \ \text{in} \ ...
5
votes
2answers
245 views

Solutions to the PDE :$2V\frac{\partial I}{\partial V}+2W\frac{\partial I}{\partial W}=I$

While working on engineering problem, I came across this PDE: Let $c_1,c_2$ be two real numbers. Find a continuous function $I:\mathbb{R}_{\geq 0}\times\mathbb{R}_{\geq 0}\rightarrow \mathbb{R}$ such ...
1
vote
1answer
83 views

linearize a nonlinear ode

Could anyone suggest me how to linearize the following system of nonlinear odes (special attention to (2) \begin{align} -cU'&=-U''+UV\tag{1}\\ -cV'&=-k(k+1)V^{k-1}(V')^2+(k+1)V^k ...
7
votes
1answer
93 views

Ellipticity of Ricci tensor, does it depend on coordinates?

Well, I am afraid this is a silly question because I know the answer must be 'yes, it does'. But I don't see why. I put the problem in context. The ricci tensor can be regarded as a differential ...
1
vote
0answers
22 views

Existence of the Solution to a particular Parabolic PDE

Suppose we have the following parabolic PDE in $X(s, t)$: $\frac{\partial X}{\partial t} + sM_1 \frac{\partial X}{\partial s} + \frac{1}{2} s^2 M_2 \frac{\partial^2 X}{\partial s^2} + (M_3 - M_1)X = ...
1
vote
1answer
132 views

Existence and uniqueness for a system of first-order PDE

Let $y$ be a scalar and ${\bf t}=(t_1,\ldots,t_K)$ and \begin{align*} \frac{\partial y({\bf t})}{\partial t_k} &=f_k(y({\bf t}),{\bf t}) \qquad k=1,\ldots,K\\ y(t_{10},\ldots,t_{K0}) &=y_0 ...
0
votes
0answers
22 views

How do the fundamental solutions for pressure and stress in Stokes flow define flows themselves?

This questions is related to section 3.2 Pozrikidis' "Boundary integral and singularity methods for linearized viscous flow" book. If $\mathbf{p}$ and $\mathbf{T}$ are the pressure vector and stress ...
0
votes
0answers
33 views

Estimation of solution to $u_t=u_{xx}+x^3u_x$ using integrals

Let $$u_t=u_{xx}+x^3u_x$$ With: $$u(0,x)=u_0(x)$$ $$u(t,0)=u(t,l)=0$$ Find an energy approximation of $u$ on $(0,T) \times(0,l)$. By multiplting by $u$ we get: ...
3
votes
2answers
43 views

Variable substitution in second order PDE

Consider the the PDE $$A(x, y)\partial_{xx} u + B(x, y)\partial_{xy}u + C(x, y)\partial_{yy}u=h(x, y) $$ Now I want to make a variable substitution $\xi=f(x, y), \eta=g(x,y)$, so I can get $u$ as a ...
2
votes
2answers
79 views

What do mathematicians mean by “analytical solution of an equation”?

Given a PDE equations of the form: $\dfrac{\partial}{\partial t} u(t,x) = \left(\hat{L}+\hat{N_u}\right)u(t,x) \;\;\;\;\;\;\hspace{10mm}(**)$ where $\hat{L}$ is a linear operator and ...
0
votes
0answers
7 views

How might the method of lines be used to solve a pure boundary value problem for a time-independent PDE in two space dimensions?

So, this is from Heath, Scientific Computing: 11.5a. How might the method of lines be used to solve a pure boundary value problem for a time-independent PDE in two space dimensions? The basic ...
3
votes
0answers
46 views

Why is entropy = the Legendre transform?

Can someone give me a mathematician's explanation (and not a physicist's) as to why $$\int_{\Omega}\Psi^*(b(u(t))$$ is called the entropy where $\Psi^*$ is the Legengre transform of ...
0
votes
2answers
50 views

Weak solution Boundary Value problem

I have to prove that the following problem $$(P) \begin{cases} -u''-u=1\,\,\,\,\,\,\,\,\text{if}\,\,\, x\in(0,\pi)\\ u(0)=u(\pi)=0 \end{cases} $$ doesn't admit weak solutions. I'm proceeding by ...
1
vote
1answer
26 views

Sturm Liouville form

How do you put $u'' +c u' +d =0$ into regular SL-form? Can not see how it's an eigenvalue problem without a first order term. But the theorem states EVERY second order operator can be put into SL ...
0
votes
0answers
21 views

Curves not contained in any linear subspace

Let $\gamma: R \rightarrow R^n$ be a curve which is implcitly given through as the solution of a differential equation. I would like to show that $\gamma$ is not contained in any linear subspace ...
8
votes
1answer
64 views

Green's identity contradicts Helmholtz theorem

Let F be a vector field on a bounded domain $V \subset \mathbb{R}^3$, which is twice differentiable, let $S := \partial V$. According to the Helmholtz Theorem, F can be decomposed, such that: $$ ...