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

learn more… | top users | synonyms (2)

0
votes
1answer
37 views

PDE - how to transform this pde to an easier one using change of varriable

I have this PDE its actually an ADE and I can put it in one of these forms. All characters in all four equations are constants with exception of x, y, t and C. I have listed the equations in order ...
3
votes
1answer
75 views

The regularity of elliptic equation

Define $$ Lu:=-\partial_j(a_{ij}\partial_iu)+b_i\partial_iu+cu $$ where $a_{ij}$, $b_i$, and $c$ are all of $C^\infty(\bar \Omega)$ and we assume $\Omega$ is open bounded in $R^N$, smooth boundary, ...
0
votes
0answers
25 views

Partial derivatives vs. chain rule

Apologies if this is a very stupid question. I am trying to understand the use cases of partial derivatives vs. the chain rule. From what I can tell, partial derivatives are used to find the ...
1
vote
1answer
28 views

System of four first-order partial differential equations

Question: Consider the wave equation: $$\frac{\partial^2 u}{\partial x^2}=\frac{\partial^2 u}{\partial t^2}$$ for a smooth function $u(t,x)$ Letting $v=\frac{\partial u}{\partial x}$ and ...
1
vote
0answers
34 views

Regularity of solutions to non-elliptic differential equation

I am interested in the regularity of the solutions to the following partial differential equation on $\mathbb{R}^2$: $$ \left( a \frac{\partial}{\partial x} + \frac{\partial^2}{\partial y^2} \right) u ...
1
vote
0answers
82 views

Method of characteristics of a system of first order pdes

Consider the system of first order PDEs $ \left\{ \begin{eqnarray} \frac{\partial}{\partial t} v_1 + \frac{\partial}{\partial x_1} p_1 + \eta(x_1) v_1 = 0 \\ \frac{\partial}{\partial t} v_2 + ...
1
vote
0answers
25 views

Poincare inequality on balls to general open subset

Let $M$ be an n-dim compact Riemannian manifold with $Ric \geqslant -(n-1)$, it's well known that the following Poincare inequality holds for any function $f\in W^{1,2}(M)$ $$ (\int_B ...
3
votes
1answer
52 views

Why are elliptic/parabolic/hyperbolic PDEs called “elliptic”/“parabolic”/“hyperbolic”?

I see that the form of a (e.g.) parabolic equation is $$Au_{xx} + 2Bu_{xy} + Cu_{yy} + Du_x + Eu_y + F = 0$$ with $B^2-4AC=0$ whereas the equation of a parabola is $$Ax^2 + 2Bxy + Cuy^2 + Dx + Ey + ...
0
votes
0answers
25 views

Using taylor formula to compute laplace operator

Suppose $u\in C^2(\Omega)$ and $x\in \Omega\subset \mathbb R^N$. I am trying to prove that $$ \Delta u(x)=\lim_{r\to 0} \frac{2N}{r^2} \left[\frac{1}{\alpha(N)} \int_{\partial ...
0
votes
1answer
31 views

The Green re-constraction of Possion equation

Let's follows Evans's PDE book notation. We say $G(x,y)$ is the green function build on $\Omega\subset \mathbb R^N$ where $\Omega$ open bounded nice boundary. Then if $u\in C^2(\bar\Omega)$, we ...
1
vote
0answers
77 views

Frobenius theorem method

Question: Let $A(z)$ be smooth with nonvanishing derivative $A'(z)$, and consider the system with first-order PDE's, $$\frac{\partial u}{\partial x}=\frac{\partial K}{\partial ...
0
votes
0answers
21 views

Non homogeneous heat boundary value problem.Show that $B_n(t)=e^{-c^2\lambda_n t}(b_n+\int _o^t q_n(s)e^{c^2\lambda_n s}ds)$

$u_t=c^2u_{xx}+q(x,t)$ Given $t>0,0< x \le L$ $u(0,t)=0=u(L,t),t> 0$ $u(x,0)=f(x)$ Solving the homogeneous equation gives $u_t=c^2u_{xx}$ We can write the Fourier sine ...
1
vote
1answer
83 views

Solving Partial Differential Equation $ \Delta u = 4 $

How can I solve this equation: $ \Delta u = 4 \\ u(x,x)=2x^2 \\ u_x(x,x) = 2x$ where $u=u(x,y)$ using substitution: $ \Phi ^{-1}(s,t) = (x-y,y) $? My attempt to solve this: $v=u \circ \Phi ...
4
votes
2answers
76 views

Weak formulation for nonhomogeneous problem $-\Delta u = 0$

I am wondering about the definition of weak solution to the nonhomogeneous problem $$-\Delta u = 0 \text{ in }\Omega$$ $$u = g \text{ in }\partial\Omega$$ given $g \in H^{\frac 12}(\partial\Omega)$. ...
4
votes
1answer
52 views

The constraint subset of $H_0^1(\Omega)$ is a $C^1$-submanifold.

This problem comes from the constraint problem in CoV. (the lagrange-multiplier case) Let $\Omega\subset \mathbb R^N$ be open bounded, smooth boundary. We define the sub-manifold $$ M:=\{u\in ...
0
votes
2answers
38 views

About normal derivative

Let $w:\Omega \subset \mathbb{R}^n \longrightarrow \mathbb{R}$, where $\Omega$ is an open, bounded, connected set, $w \in C^2(\Omega)\cap C(\overline\Omega)$ and $x_0 \in \partial \Omega$ such that ...
0
votes
0answers
21 views

Poincaré inequality by capacity estimate.

Let $(X,d,\mu)$ be a complete metric space with a doubling measure $\mu$. For any ball $\mu(B)< \infty$ and $\mu(A)=\sup\{\mu(K);K\subset A, K \text{ compact}\}$. For any two closed disjoint ...
0
votes
1answer
28 views

Using Direct method to prove Rayleigh Quotient Theorem

Define the elliptic PDE operator $Lu:=-\partial_j(a_{ij}\partial_iu)+cu$ where $A=(a_{ij})$ is and uniformly elliptic matrix and $c\geq 0$, i.e., $A\xi\cdot\xi\geq \theta \lvert\xi\lvert^2$ for ...
4
votes
1answer
91 views

Solving the PDE $\frac{\partial T}{\partial t} = K\frac{\partial^{2} T}{\partial x^{2}}$

I am trying to find solutions to the following PDE: $$\frac{\partial T}{\partial t} = K\frac{\partial^{2} T}{\partial x^{2}},$$ With boundary condition: $T(0,t)=T_{0}\cos(\omega t)$, where $\omega ...
0
votes
0answers
26 views

How to classify this problem?

$\begin{align} & considering\,x(R,a)\,and\,y(R,a): \\ & f(x)\frac{\partial x}{\partial R}+f(y)\frac{\partial y}{\partial R}=2R \\ & 2x\frac{\partial x}{\partial a}+2y\frac{\partial ...
0
votes
0answers
13 views

The heat kernel as the fundamental solution of the heat equation

If $P$ is a linear differential operator on $\mathbb{R}^N$, we say that $u\in\mathcal{D}'(\mathbb{R}^N)$ is a fundamental solution of $P$ if $$Pu=\delta $$ in the sense of distributions. The point ...
0
votes
1answer
30 views

Heat kernel property

We define the heat propagator on a Riemannian manifold $M$: $$e^{-t \Delta_g}: L^2(M) \rightarrow L^2(M)$$ $$e^{-t \Delta_g} f(x) = \int_M p(x,y,t) f(y) \,dV(y)$$ where $p(x,y,t)$ is the ...
1
vote
0answers
31 views

The $p$-Laplacian is strongly monotone

I am studying the solution of $p$-Laplacian by finding the minimizer of the following energy, among the space $W_0^{1,p}(\Omega)$, $p\geq 2$, $$ E[u]=\frac{1}{p}\int_\Omega \lvert\nabla ...
1
vote
1answer
64 views

wave equation on a square domain

I'm stuck on the following problem. Let $u(x, y, t)$ denote a solution to the linear wave equation $k^2(u_{xx}+u_{yy}) = u_{tt}$ with $k = 2$ on a square domain with corners at (0, 0), (0, 1), ...
0
votes
0answers
17 views

Maximum & minimum principle for harmonic functions

Let $\Omega \subseteq \mathbb{R}^n$ be an open, bounded, connected set and $u \in C^2(\Omega) \cap C(\overline\Omega)$ a harmonic function on $\Omega$. Then $\min \lbrace u(x) | x \in \partial \Omega ...
3
votes
0answers
38 views

Solving system of first-order PDEs with Frobenius theorem

I've been stuck trying to solve this system: $$\ \frac{\partial u}{\partial x} = \frac{-2xy^2}{u} + 3y $$ $$\ \frac{\partial u}{\partial y} = \frac{-2x^2y}{u} + 3x $$ Which must satisfy $ \ u(0,0) = ...
-1
votes
0answers
33 views

boundedness of the functions in Sobolev spaces. [duplicate]

The question was asked by me in the Functional analysis category by me in Mathstackexchange but no one has answered the question. Are the functions in $W_0^{1}(\Omega)$, $\Omega$ is a bounded subset ...
4
votes
2answers
76 views

Fourier Transform of Partial Derivative w.r.t x of [ x*f(x) ]

Can someone please help with the Fourier Transform of : Thank you in advance! ::Edit:: This is what I am trying to solve: $\frac{\partial(p(x, t))}{\partial t} = -A\frac{\partial(xp(x, ...
1
vote
0answers
64 views

Regularity energy minimizing harmonic maps

I am using the book "Geometric Measure Theory- An introduction" by Fanghua and Xiaoping. I'm studying the proof of the following Lemma (Lemma 2.1.8 page 38). This chapter is dealing with the theory ...
0
votes
2answers
48 views

Lack of homogeneous boundary conditions of a Sturm-Liouville problem.

In my exercise bundle about Sturm-Liouville problems and solving partial differential equations with the separation method there is an exercise that goes as follows: Calculate the temperature ...
0
votes
1answer
28 views

Newtonian capacity: do we have $\operatorname{cap}(A\cup B)+\operatorname{cap}(A\cup C)-\operatorname{cap}(A)-\operatorname{cap}(A\cup B\cup C)>0$

$\newcommand{\Cap}{\operatorname{cap}}$ For compact disjoint sets A,B,D each with positive Newtonian capacity do we have $$\Cap(A\cup B)+\Cap(A\cup C)-\Cap(A)-\Cap(A\cup B\cup C)>0\text{ ?}$$ ...
3
votes
2answers
62 views

Good Textbook in Numerical PDEs?

I am currently taking a course on Numerical PDE. The course covers the following topics listed below. Chapter 1: Solutions to Partial Dierential Equations: Chapter 2: Introduction to Finite ...
0
votes
0answers
29 views

Finite difference scheme for the continuity equation

I am currently trying to solve a system of PDE's numerically, one being the equation; $$ (1)\quad \partial\rho/\partial t + \partial(\rho v)/\partial x = 0 $$ I have been reading up on ...
3
votes
0answers
54 views

Deriving a formula for an initial boundary-value problem

Given $g : [0,\infty) \to \mathbb{R}$, with $g(0)=0$, derive the formula $$u(x,t)=\frac{x}{\sqrt{4\pi}}\int_0^t \frac 1{(t-s)^{3/2}}e^{-\frac{x^2}{4(t-s)}}g(s)\,ds$$ for a solution of the ...
0
votes
1answer
57 views

An linear elliptic PDE, why it has these properties?

See this image (from this work) The existence is done through Lax-Milgram (at least for $\sigma = \frac{1}{2}$), I think. However, why the author only includes the gradient in defining $H^1$? Is it ...
3
votes
1answer
52 views

The Cauchy problem for Laplace equation in unit cube

Here is the question: We are given a laplace equation $\Delta u=0$ in $Q:=(0,1)\times(0,1)$. Q1: What are some conditions you can put on this to get uniqueness/existence? Q2:What if you wanted to ...
0
votes
1answer
52 views

Property of a transport equation

Let $\Omega\subset\mathbb{R}^N, N=2$ or $N=3$ be a bounded smooth domain, $T>0$ and $y=y(x,t)\in L^{\infty}(\Omega\times (0,T))^N$ is a given vector function such that $\nabla\cdot y=0$. Consider ...
2
votes
1answer
24 views

Find the Eigenfunctions and Eigenvalues for $u_{xx} + \frac{2}{x}u_x + u_{yy} + \theta u = 0$

From an old qual exam: Find the eigenfunctions and eigenvalues associated with the BVP \begin{align} &u_{xx} + \frac{2}{x}u_x + u_{yy} + \theta u = 0,\quad 0 < x < a, \quad 0 < y < ...
0
votes
0answers
20 views

Parallel Programming the 1-D dam breakage problem

I am to write a parallel program to simulate the 1D dam break problem by using the Galerkin Equations with WENO limiter. The equations are on domain [0,2000]. At the beginning a dam divides the ...
2
votes
3answers
73 views

Existence of first order PDEs?

It seems the follow theorem is classical, but I don't know how to proof it: For $x\in\Omega\subset R^n$, where $\Omega$ is a domain with smooth boundary, consider the system of PDEs: ...
1
vote
1answer
37 views

Do there exist nontrivial global solutions of the PDE $ u_x - 2xy^2 u_y = 0 $?

Consider the following PDE, $$ u_x - 2xy^2 u_y = 0 $$ Does there exist a non-trivial solution $u\in \mathcal{C}^1(\mathbb{R}^2,\mathbb{R})$? It is clear that all solutions for $u\in \mathcal{C}^1( ...
2
votes
1answer
93 views

Reasons for convergence

I am interested to know if anyone can see the reasoning behind the convergence $$\int_{\Omega}c(u_{k},\nabla u_{k})(u_{k}-u)dx \rightarrow 0$$ in equation (2.82), page 50 in the following book ...
0
votes
0answers
21 views

Is this really a solution to the PDE?

Suppose we have a function $C(x,\nu,\tau)$ and the PDE $$-C_{\tau}+\frac{1}{2}\nu C_{xx}-\frac{1}{2}\nu C_x+\frac{1}{2}\eta^2\nu C_{\nu\nu}+\rho \eta \nu C_{x\nu}-\lambda(\nu-m)C_{\nu}=0\tag{1}$$ ...
3
votes
1answer
54 views

A question on deriving d'Alembert's formula from change of variables

Let's assume $x \in \mathbb{R}$ and $t \ge 0$. I am asked to find d'Alembert's formula to the wave equation $$\begin{cases}u_{tt}-u_{xx}=0 & \text{in } \mathbb{R} \times (0,\infty) \\ u = g, u_t = ...
2
votes
0answers
40 views

Equipartition of energy

Let $u$ solve the initial-value problem or the wave equation in one dimension: $$\begin{cases}u_{tt}-u_{xx}=0 & \text{in } \mathbb{R} \times (0,\infty) \\ u = g, u_t = h & \text{on } ...
3
votes
0answers
65 views

mononotone and pseudomonotone operators in current research

I know that the following question is quite broad for this forum. But I am interested in references or any other ideas. Can anyone provide some examples of applications of monotone or pseudomonotone ...
1
vote
1answer
21 views

Stokes' Rule for an initial-value problem

Assume $u$ solves the initial-value problem $$\begin{cases}u_{tt}-\Delta u = 0 & \text{in } \mathbb{R}^n \times (0,\infty) \\ u = 0, u_t = h & \text{on }\mathbb{R}^n \times \{t=0\}. ...
4
votes
1answer
108 views

The well-posedness of Laplace equation on half-space

In $2$ dimension, take $-\Delta u=0$ on $\{(x,y\},y\geq 0\}$ with $u(x,y=0)=f(x)$, $u_y(x,y=0)=g(x)$ where $f$ and $g$ are smooth function. I want to justify whether this problem is well posed. My ...
2
votes
1answer
66 views

The Gauss-Green theorem for unbounded domain

This question comes to me when I deal with the following PDE problem. Suppose we have \begin{cases} -\Delta u=0 & x\in \mathbb R^N\setminus B(0,1)\\ u=0 & x\in\partial B(0,1)\\ u\to 0 & ...
1
vote
0answers
19 views

Discretization of the Anisotropic Diffusion Operator for Finite Difference Method

I have to derive and apply a Finite Difference scheme to solve a steady state, anisotropic, diffusion equation. So I have to find a discretization of the following equation $$\nabla \cdot ( ...