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

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-6
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0answers
17 views

Neumann problem [on hold]

solve the neumann problem for the upper half plane
0
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1answer
14 views

Prove $\sup_x|u_x(x,t)|\le Ct^{-\frac{3}{4}}\|f\|_2$ for all $t>0$.

Let $u$ be a bounded solution to the heat equation $u_t-u_{xx}=0$ in $-\infty<x<\infty,t>0$ with $u(x,0)=f(x)$ with $f\in L^2(\Bbb R)$. Prove that there is a constant $C>0$, independent ...
-1
votes
0answers
16 views

analytical solution of non-linear least square problem

I am implementing a trust region optimization algorithm and I would like to compare it against already done similar work, where authors measures performance on this problem. $$ \min_{u,\gamma}\Bigg\{ ...
2
votes
2answers
33 views

$\sup_{x\in B_r(z)}|\nabla u(x)|\geq cr$ for $\Delta u=1$?

Let $U\subset\mathbb R^n$ be open and bounded. Let $\Delta u=1$ in $U$. Can you follow $\sup_{x\in B_r(z)}|\nabla u(x)|\geq cr$ provided $B_r(z)\subset\subset U$? I thought about using the ...
1
vote
0answers
16 views

PDE system for two composite functions

Can someone help? I have a PDE system for two unknown functions, $f(x,y,t)$ and $g(x,y,t)$ $$a_1(t)\frac{\partial f(x,y,t)}{\partial x} + a_2(t)\frac{\partial f(x,y,t)}{\partial y} + ...
0
votes
1answer
13 views

integral of 2 functions satisfying Poisson's equation

Let $u,v:\Omega\subset\mathbb{R}^n\rightarrow\mathbb{R}$ be such that $u=v=0$ on $\partial\Omega$, $-\Delta u=\lambda u$ and $-\Delta v=\mu v$ in $\Omega$ with $\lambda\neq\mu$, and neither $u$ nor ...
2
votes
0answers
13 views

Probabilistic interpretation for Fokker-Planck equation

It is well known that if $X_t$ is a stochastic process that solves the SDE $$dX_t = \mu(X_t,t)\,\mathrm{d}t + \sigma(X_t,t)\,\mathrm{d}W_t,$$ with $W_t$ a Wiener process, then the associated ...
0
votes
0answers
17 views

Hyperbolic equations with time dependent coefficients associated with the time derivatives

I'm concerned with evolution equations (of second order) and am hoping for some literature hints regarding a special situation. The equations I'm working with basically look like (complemented with ...
0
votes
0answers
16 views

A pde that cannot solves by Lax-Milgram theorem

Consider the following pde: $-u''(x)+au'(x)+bu(x)=f(x) \qquad\text{in}\; (0,1)\\$ $u'(0)=\alpha\\$ $u'(1)+u(1)=\beta$ How could I prove that it has a nontrivial solution? The bilinear form ...
8
votes
1answer
52 views

Show that $\lim_{t\to \infty}u(x,t)=\frac{A+B}{2}$, for each $x\in\Bbb R$.

Let $u(x,t)$ be a $C^2$ bounded solution of $$u_t(x,t)-u_{xx}(x,t)=0,x\in \Bbb R, u(x,0)=f(x)$$ where $f\in C(\Bbb R)$ satisfies: $\lim_{x\to+\infty}f(x)=A,\lim_{x\to-\infty}f(x)=B$. Show that ...
0
votes
1answer
42 views

Solving for distance in terms of time in inverse squared law [on hold]

I am trying to derive an expression for the distance travelled by an object that obeys an inverse squared law, $\frac{d^2y}{dx^2}=\frac{k}{y^2}$. However, I could not find an expression for distance ...
2
votes
1answer
25 views

Question about fundamental solution to the biharmonic operator $\Delta^2=\Delta(\Delta)$?

Show that when $n=2$, the function $u(x)=-\frac{1}{8\pi}|x|^2log|x|$ is a fundamental solution to the biharmonic operator $\Delta^2=\Delta(\Delta)$. That is, show that $$\varphi(0)=\int_{\Bbb ...
2
votes
2answers
59 views

Fourier transform to find an harmonic function (Strauss)

I am trying to solve one of the problems of section 12.4 of the book "Partial Differential Equations" by Strauss. The problem says: Use the Fourier transfor in the $x$ variable to find the harmonic ...
3
votes
0answers
37 views

looking for some exercise to test understanding of covector

I am trying to understand the concept of "covector" so that I can work with it for problems I come across in PDE. My knowledge on differential geometry is very little. And my topological background ...
4
votes
4answers
295 views

Basic of Partial Differential Equation

I pretty new to calculus and I am trying to understand the following transformations: $2uu_{t} = \frac{\partial }{\partial t}u^{2} $ $2u_{t}u_{tt} = \frac{\partial }{\partial t}u_{t}^{2} $ $2uu_{xx} ...
0
votes
0answers
14 views

PDE reduced to ODE Uniqueness??

Could you please help me with the following problem. As a first help, I know the solution of the following ODE: \begin{align} j_1(t)[r \log(j_1(t)) + \beta] &= j_1'(t) \\ \nonumber j_1(T) ...
2
votes
1answer
29 views

How to solve the following PDE

What steps should be taken in order to get a solution (that only depends on v) for the following?: $\dfrac{\partial ^2f}{\partial v^2}+\dfrac{1}{v}\dfrac{\partial f}{\partial ...
2
votes
0answers
32 views

find the total differential of this equation $ xyz + \sqrt{ x^2 + y^2 + z^2} = \sqrt 2 $

How to calculate the total differential of $ z= z(x,y)$, which is $ xyz + \sqrt{ x^2 + y^2 + z^2} = \sqrt 2 $ at point (1, 0, -1)? The evaluation of mine seems wrong, $ dz= \frac{\partial ...
-2
votes
0answers
31 views

ROAD MAP FOR PDE [on hold]

I am not a specialist in PDE, but I am looking for a "ROAD MAP FOR PDE" : I mean, a general picture about main ideas during the last 100 years until today, which allow to solve all kind of PDE. The ...
0
votes
0answers
10 views

Convergence to the entropy solution a.e.

Consider the equation: $u^{\epsilon}_{t} + f(u^{\epsilon})_{x} - \epsilon u^{\epsilon}_{xx} = 0$ in $\mathbb R$ X $(0,\infty)$ ; $u^{\epsilon}(x,0) = u^{\epsilon}_{0}$ To show the existence of ...
1
vote
0answers
13 views

Basic examples of functions in Hörmander class

The Hörmander class $S_{\rho,\delta}^m$ (with $\rho,\delta\in[0,1]$) consists of smooth functions $p(x,\xi)$ with $$|D_x^\beta D_\xi^\alpha p(x,\xi)|\leq ...
0
votes
0answers
18 views

Time advance in Adaptive Mesh Refinement method

I am working on solving complex system of 2D PDEs governing the behaviour of plasma in a gas lamp during discharge. Recent tests have shown that because of steep gradients in temperature field and ...
1
vote
1answer
42 views

A question regarding harmonic function.

Can any one provide some hint on the following question? I have being thinking about this for a while but cannot figure out where to start. I have been thinking about Taylor expansion but it seems not ...
0
votes
0answers
28 views

integration by parts in 2 dimensions with heat equation pde

I'm working on some PDE problems and my biggest issue is vector calculus facts. Let $u \in C^2(\Omega)$, where $\Omega$ is some bounded subset of $R^2$ with smooth boundary such that $u_{t}-\Delta ...
4
votes
1answer
28 views

Estimates for parabolic vs elliptic PDE

Elliptic and parabolic PDE share many properties. They each, for example, have an associated maximum principle and their value at any point depends on the entirety of the boundary data. I have been ...
2
votes
0answers
43 views

Variable coefficient wave equation

Consider the equation $$u_{tt} - f(x)^{2}u_{xx} + u_{t} = 0$$ for $(x,t) \in \mathbb{R} \times [0, \infty)$ with $u(x, 0) = 0$ and $u_{t}(x,0) = 0$ for all $x \in \mathbb{R}$. Furthermore, suppose ...
0
votes
0answers
15 views

Impose initial condition on partial differential equation

After solving a Fokker-Planck equation (using expansion in eigenfunctions) I have obtained the following, general solution for the probability density: \begin{equation} p(x,t) = \int_0^\infty dk~ ...
4
votes
1answer
44 views

PDE: Fokker-Planck equation with time-dependent boundary conditions

We have the following PDE: \begin{equation} \frac{\partial p(x,t)}{\partial t}= - a\frac{\partial p(x,t)}{\partial x} + \frac{D}{2} \frac{ \partial^2 p(x,t) }{\partial x^2}, \quad0<x<L, \quad ...
1
vote
2answers
23 views

Finite difference : relationship involving gamma

Given the following PDE, $$ \frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2S^2\frac{\partial^2V}{\partial S^2}=0 $$ and its finite difference approximation, $$ \frac{V_n^{m+1}-V_n^m}{\Delta t} ...
0
votes
1answer
25 views

Continuously differentiable functions are weakly differentiable

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain and $u\in C^1(\Omega)$. I want to show, that $u$ is weakly differentiable, i.e. $$\int_\Omega\psi\frac{\partial u}{\partial ...
2
votes
1answer
16 views

Implicit finite differences: Sufficient conditions for non-negativity

Given the finite difference approximation for black scholes with zero interest rate, $$ \frac{V_n^{m+1}-V_n^m}{\Delta t} + \frac{1}{2}\sigma^2S^2 \frac{V_{n+1}^{m}-2V_n^m+V_{n-1}^{m}}{\Delta ...
1
vote
1answer
51 views

Using Feynman-Kac, compute the following:

Let $B(t)$ be Brownian Motion and let $\alpha$ be a constant and $T>0$. Compute $\mathbb{E}_{B_{0} = x}\left[\exp\left(-\alpha \int_0^T B(s)^2 ds\right)\right]$. I'm just having a hard time with ...
2
votes
0answers
30 views

Are smooth solutions to a PDE dense in the space of $L^2$ solutions to the PDE?

Let's say I have a linear differential operator $P$ with smooth coefficients between bundles $E$ and $F$ over a smooth compact manifold $X$ with smooth boundary. Let's consider $P$ as an operator ...
-2
votes
1answer
38 views

Partial differential equation

PDE: $u(x,t)$ satisfies $u_t=u_{xx}+x+t-2$ IC: $u(x,0)=2x$ BC: $u_x(0,t)=-2(u(0,t)-t)$ and $u_x(1,t)=2+t$ Question: How shall I find $u(0,t)$?
1
vote
1answer
27 views

About the boundary conditions of the Black-Scholes-Merton PDE

I have a question about the solution of the Black-Scholes PDE for the European call option when I read the book Stochastic Calculus for Finance II of Steven E.Shreve. Let $c(t,x)$ be the value of the ...
1
vote
1answer
25 views

Weak convergence in the Sobolev space and compact embeddedness

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain $H:=W_0^{1,2}(\Omega)$ be the Sopolev space $u\in C^0\left(\overline\Omega\times [0,\infty)\right)\cap C^{2,1}\left(\Omega\times ...
3
votes
0answers
54 views
+50

Jacobian of Stabilized Eikonal Equation $| \nabla u| = 1$

I am trying to implement a Finite Element Solution for the Stabilized Eikonal Equation : $$ |\nabla u| = 1 + \Gamma \Delta u, \quad \text{ where } \quad u = \text{ distance function }$$ $$ ...
0
votes
1answer
20 views

Evaluating 'Constant' Term

suppose I have a pde $$u_{xt}(x,t)+u(x,t)u_{xx}(x,t)=h(t),\,\,\,\, x\in[0,\pi],\,\, t>0$$ for some unspecified function $h(t)$. This question is about finding what $h(t)$ is. Please, you may ...
3
votes
0answers
29 views

Reference request: The compactness and compact embedding in Besov Space?

This post has been on MathOverflow for couple of days but receive no response. So I put it here hoping for more attentions. Thank you guys! Let $\Omega\subset \mathbb R^N$ be open bounded with ...
2
votes
0answers
18 views

A stochastic variant of the heat equation modulo $2\pi$ has weird unstable particle-antiparticle solutions. Does this equation have a name?

I implemented a discretization of a weird 2D heat equation "mod $2\pi$", $$\dot{f}(\mathbf{x},t)=\Delta^*f(\mathbf{x},t)$$ where (WARNING: handwavy, I'm not sure I understand it) ...
2
votes
1answer
39 views

Fundamental solution for a parabolic PDE with costant coefficents

as it is well known, the fundamental solution of the heat equation is the function $G(t,x)=\frac{1}{(4\pi t)^{n/2}}e^{\frac{|x|^2}{4t}}$, for all $t>0,x\in\mathbb{R}^n$. I wonder if exists (and ...
0
votes
2answers
43 views

how to solve the system of partial differential equations? [closed]

I need to solve the following system: $$ \begin{cases} \frac {\partial K(x,y)}{\partial x}x + \frac {\partial K(x,y)}{\partial y}y+2k(x,y) = 0 \\ \frac {\partial K(x,y)}{\partial x}y - \frac {\partial ...
0
votes
1answer
15 views

Higher accuracy of numerical derivative in 2D case

Recently, I face a problem about solving a PDE (2D in spatial direction) and I stuck on the discretization of the 1st order derivative. My stencil is as follow There are five points in my stencil. ...
1
vote
1answer
21 views

Numerical method for nonlinear wave equation

I need to solve the following nonlinear wave equation numerically $U_{tt}=(1+\epsilon U_{x}^2)U_{xx}$ with Initial conditions. what is the best method for solving it? I tried the finite elements ...
0
votes
0answers
109 views

Rearranging a spherical harmonics expansion

Referring to this article (click to enlarge): and How is it that they get from equation 2 to equation 3? Whenever I do it, I can't cancel the imaginary terms Is there some spherical harmonics ...
1
vote
0answers
23 views

Under some regularity assumptions to the boundary $\partial\Omega$, the first weak eigenfunction of $-\Delta$ in $\Omega$ is also a strong one

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain, $H:=W_0^{1,2}(\Omega)$ be the Sobolev space, $C^{2,\alpha}(\Omega)$ be the Hölder space for some $\alpha\in (0,1]$ and ...
1
vote
1answer
18 views

Convergence of a sequence of Hölder continuous functions with respect to the Sobolev norm

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain, $W_0^{1,2}(\Omega)$ be the Sobolev space and $C^{2,\alpha}$ be the Hölder space for some $\alpha\in (0,1]$. Suppose $(u_k)_{k\in\mathbb ...
1
vote
1answer
39 views

What theorem is this? (in PDE)

I'm confused because it's titled as "Gauss's theorem about heat flux" (not in English though, I'm translating), but instead of the heat equation there's Laplace's equation written above the theorem. ...
0
votes
0answers
17 views

finite differences nonlinear least squares

I am facing to following nonlinear least-squares problem: $$\min_{u,\gamma} \frac{1}{1000} \int_{ \gamma(x,y)^2} + \int_{ [u(x,y)−u (x,y)]^2} + \int_{ [∆u(x,y)−\gamma(x,y)u(x,y)]^2}$$ where the ...
1
vote
0answers
24 views

Regularity of solutions of Parabolic PDE

Consider the equation: $u_{t} - a \Delta u + \lambda u = g $ in $\mathbb R$ X $(0,T)$ $u(x, 0) = u_{0}(x)$ Where $a \gt 0$ & $\lambda \in \mathbb R$ are given constants!! Now, assume that for ...