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

learn more… | top users | synonyms (2)

3
votes
1answer
67 views

Is it possible to solve this PDE

It would be pretty sweet if I could solve this for $A$. Is it possible? $$\frac{dA}{dx}+\frac{dA}{d\tau}=wx\tau$$ where $w$ is a constant and $x$ is a function of $\tau$. It might help that it is ...
3
votes
1answer
89 views

Differential equation on $\Bbb R$

We have a differential equation on $\Bbb R$ of the form $$\frac {d^2}{dx^2}u = \chi_{[0,1]},$$ where $\chi_{[0,1]}$ is the characteristic function of the interval $[0, 1] ⊂ \Bbb R$. I want to find a ...
0
votes
0answers
107 views

check coercive in Lax-Milgram

Today, I try to solve a PDEs problem, but I have a obstacles: Let $u\in H^1(0,1)$ such that $u'(1)=-u(1), u'(0)=u(0)$, and $$a(u,u)=u^2(1)+u^2(0)+\int_0^1 (u'(x))^2dx-\int_0^1 (u(x))^2dx$$ define ...
2
votes
1answer
59 views

Poisson's equation under translation and scaling

Let $u\in C^\infty (\Omega,\mathbb R)$ be a solution of $$\begin{array}{cccl} -\Delta u & = & 0 & \mathrm{in}\ \Omega \\ u & = & g & \mathrm{on}\ \partial\Omega\end{array}$$ ...
6
votes
1answer
363 views

Regularity of a solution of Laplace equation

Assume $\Omega$ is some open, bounded domain with smooth boundary - say $\Omega = B(0,1) \subset \mathbb{R}^3$. Let $v$ be a solution of the Laplace equation \begin{equation} \begin{cases} \Delta v =0 ...
3
votes
2answers
210 views

about weak derivative of Bochner integrable function

Hi I am studying the following theorem ( the theorem can be found in the classical Evans PDE book in the apendix 5.9 ) Theorem : Suppose $ u \in L^{2}(0 , T ; H^{1}_{0}(U))$ with $u^{'} \in L^{2}(0 ...
6
votes
1answer
232 views

Is there an integral form of Newton's method?

Warning : This seems like a silly sort of question, not the kind I'd ask out loud. The contraction mapping theorem is a basic tool for proving existence of, and finding solutions to, equations. Given ...
10
votes
3answers
13k 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?
3
votes
2answers
213 views

Eigenvalues of the laplacian on a compact manifold without boundary

Let $ M $ be a compact manifold WITHOUT boundary. It is clear that the first eigenvalue of the Laplace operator $ -\Delta $ is $ \lambda_0=0 $. Now we suppose that M has constant sectional curvature ...
0
votes
1answer
51 views

General solution of a simple PDE

What is the general solution of this equation: $Q \frac{\partial C}{\partial V} + r = \frac{\partial C}{\partial t}$ where Q and r are constants? I tried to use the method of separation of ...
2
votes
1answer
997 views

What is convection-dominated pde problems?

Can you explain for me what is convection-dominated problems? Definition and examples if possible. Why don't we can apply standard discretization methods (finite difference, finite element, finite ...
2
votes
1answer
165 views

Method of Moving Planes and Method of Moving Spheres

Under what conditions should I use the methods of moving spheres instead of the method of mobile plans? Under what conditions should I use the method of moving planes, but I can not use the method of ...
3
votes
0answers
80 views

Integration by Parts for PDEs

I'm reading a paper on PDEs in preparation for some research. In it, integrals like this appear repreatedly: $$ I(x,t) = \int_{|y|>1} e^{-i\alpha xy} \frac{e^{i\beta y^2}}{(1+y^2)^m} dy. $$ Here ...
4
votes
1answer
283 views

Where can I find this “standard elliptical argument”?

I recently came across the following problem: Assume $x_0 \in \mathbb{R}^N$ and \begin{equation} \Vert w \Vert_{L^\infty (B(x_0,1))} \leq C \end{equation} for some real constant $C>0$, where ...
1
vote
0answers
73 views

Is this a valid application of “separation of variables”?

I asked this question over at Physics SE. I am not satisfied with the answer. At the heart of the question is this mathematical concern: Can I invoke "separation of variables" to go from this: ...
1
vote
0answers
73 views

Method of characteristics - a general doubt.

This is a continuation of my previous unanswered question Best method to solve this PDE I understand that the method of characteristics, cannot be used for more than one dependent variable from here ...
5
votes
1answer
350 views

Conditions for Unique Solution for this PDE

$$ U_{xy}+\frac{2}{x+y}\left(U_{x}-U_{y}\right)=0 $$ with the boundary conditions $$ U(x_{0},y)=k(x_{0}-y)^{3}\\ U(x,y_{0})=k(x-y_{0})^{3} $$ where $k$ is a constant given by ...
1
vote
0answers
64 views

Fast Poisson Solver Question (PDEs)

I'm working on this question: $(2)$ Solve the two-dimensional Poisson problem $$\Delta u+\lambda u=f\quad\text{in}\quad\Omega=(0,1)^2$$ subject to homogeneous Dirichlet boundary conditions ...
9
votes
2answers
378 views

Show harmonic function is constant on $\mathbb{R}^n$

I'm trying to solve the following question (this is just for practice): If $u$ is harmonic within $\mathbb{R}^n$ with $\int_{\mathbb{R}^n}|Du|^2 dx \leq C$ for some $C > 0$, then show that u is ...
3
votes
0answers
45 views

Classical Parabolic theory (PDE)

I am reading an article and I almost done but I don't understand an argument on page 8: It is known from the classical parabolic theory that in the case when the nonlinearity grows no faster than ...
2
votes
0answers
118 views

Best method to solve this PDE

I need to solve this Partial Differential Equation for $\lambda(x,y)$, $$\frac{\partial \lambda}{\partial x} + h(x,y)\frac{\partial \lambda}{\partial y} - \lambda \frac{\partial h}{\partial y} = 0$$ ...
2
votes
1answer
69 views

Partial differential equation - regularity question

suppose $\Omega \subset \mathbb{R}^n$ open and bounded and $\partial\Omega\in C^{4,\gamma}$. I consider a boundary value problem in the form $\begin{cases} \Delta^2 u(x)=f(x)-u|u|^{p-1} ...
5
votes
1answer
280 views

Definition of smoothness “up to boundary”

Let $U\subseteq \mathbb{R}^n$ be an open set and let $f\in\mathcal{C}^k(U)$ for some positive integer $k$. Are the following definitions of $\mathcal{C}^k$ regularity "up to boundary" equivalent? ...
2
votes
0answers
64 views

Does this PDE have analytic solutions?

I would like to know whether this P.D.E is solvable and if yes, for which values of $a$. $$ 2 (y \cos a + x \sin a) - 4 (y \sin a - x \cos a) f - 2 (y \cos a + x \sin a) f^2 + (x^2 \cos a - 2 x y ...
1
vote
1answer
171 views

Smooth solutions of the Laplace equation with Neumann data

Let $D$ a connected domain, $\Delta u = 0$ on $D$ and $\partial_n u = 0$ on $\partial D$. Using energy methods I can show, that two solutions of this problem are unique up to a constant. Now, how can ...
0
votes
0answers
65 views

How to solve this PDE? A 2-D wave equation.

$\partial_{tt}u=\partial_{xx}u +\partial_{yy}u,r<1,0<\theta<\pi,t>0$ $u(r=1)=t,u(\theta=0,\theta=\pi)=r^{2}t$ $u(t=0)=0,\partial_{t}u(t=0)=f(r)$ Thanks!
4
votes
1answer
1k views

What kinds of PDE can't be solved by separation of variables?

What kinds of PDE can't be solved by separation of variables? Except those without boundary and which are non-linear? Does it matters with the shape of solution domain? When should I use addition or ...
3
votes
0answers
85 views

Is this Stokes problem well-posed?

I am solving Stokes problem: $$\begin{gathered} \nabla p = \mu \Delta \vec{u} + \vec{f} \\ \nabla \cdot \vec{u} = 0 \end{gathered}$$ in a domain that's bounded by two surfaces - a cuboid and a small ...
5
votes
1answer
91 views

$\Delta u = \operatorname{div}f \ \ \mbox{in} \ \ B_1, f \in L^2 \Rightarrow \nabla u \in L^2$

I'm looking for results like, If $f \in L^p$ and $$ \begin{array}{rclcl} \Delta u & = & \operatorname{div}f & \mbox{in} & B_1\\ u&=&0& \mbox{on}& \partial B_1 ...
4
votes
1answer
65 views

How can we pick $f \in C(0,T;H)$ with $f(T) =0$ and $f(0) = h$, where $h$ is arbitrary?

Let $C(0,T;H)$ be the space of continuous functions $f:[0,T]\to H$ where $H$ is Hilbert. For every $h \in H$, why is it possible to pick a function $f \in C(0,T;H)$ such that $f(0) = h$ and $f(T) = ...
5
votes
1answer
75 views

Is there a version of mean value property for nonharmonic funcions?

We know by mean value property that harmonic functions satisfies the equalities \begin{equation} u(x) = \dfrac{1}{|B_r|}\int_{B_r}f dx = \dfrac{1}{|\partial B_r|}\int_{ \partial B_r}udS. ...
2
votes
0answers
248 views

Integral of Poisson Kernel for upper halfspace

Let $\beta$ be the surface area of the unit sphere. The Poisson Kernel for the upper half-space $\mathbb{R}_n^+=\left\{x\in\mathbb{R}^n \mid x_n > 0\right\}$ is $$ ...
3
votes
1answer
211 views

Harnack's inequality Evans' PDE book

This is on page 33 of my edition, under the proof of Harnack's inequality. Let $V\subset \overline{V} \subset U$ with $\overline{V}$ compact. Let $r=\frac{1}{4}\text{dist}\left(V,\partial U\right)$. ...
6
votes
1answer
109 views

Euler Darboux PDE solution

Consider the Euler-Darboux PDE $$ u_{xy}+\frac{k}{x-y}(u_{x}-u_{y})=0 $$ What is the solution when $k>0$? All text books I have looked at give solutions for $k<0$ and I don't seem to see how I ...
4
votes
2answers
140 views

Need explanation of passage about Lebesgue/Bochner space

From a book: Let $V$ be Banach and $g \in L^2(0,T;V')$. For every $v \in V$, it holds that $$\langle g(t), v \rangle_{V',V} = 0\tag{1}$$ for almost every $t \in [0,T]$. What I don't ...
4
votes
1answer
148 views

An equality in $L^2(0,T;V')$!? Weak solution to PDE via Galerkin approximations

I have the heat equation $$u' - \Delta u = f$$ as equality in $L^2(0,T;V')$,i.e., $$(u',v) + (\nabla u, \nabla v) = (f,v)$$ for all $v \in L^2(0,T;V)$, where I used the same brackets for duality ...
3
votes
3answers
91 views

$\Delta u = f, f \in L^q \Rightarrow u \in W^{2,q}$ References

I'm looking for references for the following theorem. I will very grateful Theorem: [Calderón Zigmund] If $u$ is a solution of \begin{equation} \Delta u = f \quad \mbox{in} \quad B_2 ...
3
votes
1answer
101 views

Yang–Mills theory

We define the energy as $$E = I_F + I_K + I_V,$$ where, $$I_F [A]= \frac{1}{2} \int d^Dx \operatorname{tr} F^2_{ij},$$ $F_{ij}$ represents the electromagnetic force. $$I_K [\phi,A]= \frac{1}{2} \int ...
4
votes
1answer
71 views

Uniqueness of weight function.

Let $L=p(x)\frac{d^2}{dx^2}+q(x)\frac{d}{dx}+r(x).$ Where L stands for differential operator. Now inner product defined $(f,g)=\int_a^bf(x)g(x)w(x)dx$. Where $w(x)$ is a weight function. Now $L$ is ...
1
vote
1answer
1k views

Black-Scholes PDE to heat equation, nonconstant coefficients

Can someone provide me with details or a reference on how to transform the Black-Scholes PDE with nonconstant coefficients (i.e. $r=r\left(S,t\right)$, $\sigma=\sigma\left(S,t\right)$) to the heat ...
7
votes
0answers
284 views

Connection between distributional and renormalized solutions for Boltzmann equation

I have a problem understanding the meaning of a distributional solution. Let me tell you the context the problem appeared: I read through some papers by DiPerna and Lions concerning the Cauchy Problem ...
3
votes
1answer
120 views

Property of the difference quotient in Evans(Partial Differential Equations)

Why holds the property of the difference quotient in Evans(Partial Differential Equations) \begin{equation} \int_{U}v D_k^{-h}dx = -\int_U w D_k^hv dx \end{equation} for $v,w \in H^{1}_0(U)$ (16) in ...
4
votes
1answer
62 views

Change of variables for integral involving curl

So I'm working on a finite element problem. Usually, I have to deal with the integral of a gradient over a triangular region $K$. I map the integral to the reference triangle $\hat{K}$ with vertices ...
2
votes
1answer
47 views

Transferring Time-Derivative Inside Spacial-Norms Inequality

A colleague and I have come across what looks to be a very easy inequality to prove, but we are stumped. Anyway, assuming that $u$ is in all the appropriate spaces, does anyone have any idea how to ...
1
vote
1answer
177 views

Vector analysis: $(\vec v \cdot \vec \nabla) \vec v=(\vec \nabla \cdot \vec v) \vec v$?

If I know that $\vec \nabla \cdot \vec v=0$, can I say that: $$( \vec v \cdot \vec \nabla )\vec v=\underbrace{(\vec \nabla \cdot \vec v)}_{=0} \vec v=0 $$ ? Note: this is a question I asked in ...
5
votes
1answer
237 views

Semilinear Parabolic PDE

I'm considering the following type of PDE: $u_{t}=u_{xx}+u_{x}+u_{x}^2+u_{x}^3+\frac{u_{x}}{x(1-x)}+\left(\frac{u_{x}}{x(1-x)}\right)^3$ with periodic boundary conditions $u_{x}(0)=u_{x}(1)=0$. ...
3
votes
0answers
148 views

Checking initial condition of PDE is satisfied in Galerkin method

The following is an argument I read, it is part of a proof of checking that the solution given by the Galerkin method satisfies the initial conditions. From our PDE $$\langle u', v \rangle_{V',V} + ...
4
votes
0answers
55 views

An average estimate

In the paper here ,which the aim is to prove the Calderor-Zygmund theorem with a new approach in the lemma 7, page 8 we have \begin{equation} \dfrac{1}{|B_4|}\int_{B_4} | D^2u|^2 \le 2^n \ \ ...
1
vote
1answer
117 views

Solving the equation $\nabla u=f$.

Let $\Omega\subset\mathbb{R}^N$ be a open set, $\mathcal{D}(\Omega)=C_0^\infty(\Omega)$ and $D'(\Omega)$ the set of distributions. Suppose that $f_i\in \mathcal{D}'(\Omega)$ for $i=1,\ldots,N$. Define ...
1
vote
1answer
56 views

$\bigcup_{n}V_n$ is dense in $V$ implies $\bigcup_{n}L^2(0,T;V_n)$ is dense in $L^2(0,T;V)$?

Let $V$ be a separable Hilbert space with basis $w_j$ and let $V_n$ denote the linear span of $w_j$ for $j=1,...,n$. Clearly $V_n$ are Hilbert spaces and $V_n \subset V_{n+1}$ for all $n$. We have ...