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

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18 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
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1answer
25 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 ...
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0answers
38 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 ...
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0answers
10 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
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1answer
33 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 ...
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1answer
17 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} ...
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1answer
24 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 ...
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0answers
36 views

Solving a boundary-value problem where the function is not differentiable at the boundary?

Let us say we have a initial-boundary value problem $$ \frac{\partial u}{\partial t} = Lu $$ on $(0, T]\times [0, \infty)$ with initial condition $u(0, x)=h(x)$. I don't specify $L$ here in the hope ...
2
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1answer
15 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 ...
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1answer
50 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 ...
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0answers
28 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
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1answer
36 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 $u(0,t)$?
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1answer
25 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 ...
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1answer
24 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 ...
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0answers
30 views
+50

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

I am trying to implement a Finite Element Solution for the Stabilized Eikonal Equation : $$ |\nabla u| = 1 + \Gamma \Delta u, \quad \text{ where } \quad u = \text{ distance function }$$ $$ ...
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1answer
19 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 ...
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0answers
25 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
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0answers
16 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) ...
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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 ...
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2answers
40 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 ...
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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
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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 ...
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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 ...
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0answers
21 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
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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 ...
0
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0answers
12 views

Connection between two same form equation but with different independent variables [closed]

I have two different nonlinear equations. Both of them are reducible to heat equation with different independent variables. Then all the analysis which are true for 1st equation will work out for the ...
1
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1answer
36 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. ...
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0answers
16 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 ...
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1answer
27 views

Compact Imbedding into $L^{1} (\Omega)$ [closed]

Let $\Omega$ be a bounded subset of $\mathbb R^{d}$ with a Lipschitz continuous boundary.Prove that: the canonical imbedding of $BV(\Omega)$ into $L^{1}(\Omega)$ is COMPACT .
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0answers
23 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 ...
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0answers
24 views

Existence and uniqueness of a pde solution

I have the PDE system: $\frac{\delta}{\delta t}u(t,r)=-\int_0^1 H(|r-r'|)v(t,r')dr'u(t,r)$ $\frac{\delta}{\delta t}v(t,r)=\int_0^1 H(|r-r'|)v(t,r')dr'u(t,r)-v(t,r)$ $x(0,r)=\rho(r), ...
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0answers
29 views

Trace zero not needed for $H^2$ regularity if $V_N\subset H^2$ is finite dim?

Reading Evans and this note after asking this question, I have been thinking about the estimates for interior/global regularity in Evans, 6.3.1, theorem 1, and thoerem 4 in 6.3.2, of the form ...
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0answers
49 views

Orthonormal basis of $L^2$ and it's impact on the solution to the heat equation.

If we consider the homogeneous Dirichlet eigenvalue problem on a bounded domain $\Omega\subset\Bbb R^n$ - (one part of my question is if we can assume $\Omega$ to be a Lipschitz domain and still ...
0
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1answer
26 views

Existence of the first weak eigenvalue of the Laplacian in a bounded domain

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain and $$H:=W_0^{1,2}(\Omega):=\left\{u\in L^2(\Omega):\nabla u\in L^2(\Omega)\right\}$$ be the Sobolev space, where $\nabla u$ denotes the weak ...
6
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0answers
207 views

How to classify/ solve this PDE?

I am searching how to solve the PDE below but I can not seem to find a decent example online. My major did not focus much in solving PDEs so I feel very deficient. I know how to solve for the steady ...
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0answers
24 views

Stability in partial differential equations

I have the following PDE, with parameters $a$ and $b$: $$ \frac{\partial c}{\partial t} = \frac{\partial}{\partial z} \left( a c + b \frac{\partial c}{\partial z}\right) $$ with, for now, just one ...
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1answer
35 views

Uniqueness of harmonic function with Mixed Dirichlet Neumann condition

Let $u \colon \{\mbox{Im } z>0\}\subset\mathbb{C}\to \mathbb{R}$ be a positive harmonic function in the upper half plane, i.e $$ \Delta u=0,\,\, \mbox{for}\,\mbox{ Im } z>0. $$ Consider now the ...
2
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0answers
30 views

Nonlinear Solution to PDE (sine-Gordon Equation)

So I have this nonlinear PDE, the sine-Gordon Equation, $u_{tt}-c^{2}u_{xx}+\omega_{p}^{2}\sin u=0$ whose linearized solution is given by $u_0$. ($c$ and $\omega_p$ are constant.) My reference tells ...
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0answers
15 views

Limit of the solution of viscosity problem associated to Conservation Laws

If $u^{\epsilon}$ be a solution of approximating viscous problem: $u^{\epsilon}_{t} + F(u^{\epsilon})_{x}-\epsilon u^{\epsilon}_{xx}=0$ in $\mathbb R$ X $(0,\infty)$ $u^{\epsilon} = g$ on $\mathbb ...
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0answers
14 views

Doubt regarding proof of Uniqueness of Entropy solution In Evans

At page $610$ of the PDE book by Evans I have read this: $\int_{0}^{\infty} \int_{0}^{\infty}G(\bar y, \bar s) \eta_{\epsilon}(\bar y) \eta_{\epsilon}(\bar s)d \bar y d \bar s \geq 0$ ; where: ...
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0answers
18 views

Weak harnack type inequality

I have reached a lemma which I do not have any reference and hint for it. Assume $ \Omega \subset \mathbb{R^N} $ is smooth bounded domain and let $u$ be a positive distributional supersolution ...
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0answers
19 views

PDEs, Monte-Carlo methods and hyperbolic problems

I often hear that Monte-Carlo methods provide good solutions to elliptic and parabolic type PDE problems. The main apparent reason being that the Feynman-Kac formulae, modernly derived from the Ito ...
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0answers
10 views

strong minima and maxima condition in calculus of variation

I am going through the topic CALCULUS OF VARIATION. There are not many examples on the topic strong/weak maxima minima. Can anybody provide the link of the source or book name where this topic is ...
2
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1answer
26 views

Couple stress tensor reference.

Can someone give me a good mathematical reference for couple stress tensor in its most basic form. Thank you.
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0answers
154 views

Regularity of a weak solution

The problem is formulated as follows: Let $\Omega\subset\mathbb{R}^2$ a bounded Lipschitz domain. For $u\in L_2(\Omega)$ one can define the one dimensional weak dirivative $\partial_1 u$. Now define ...
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30 views

Uniqueness of the solution of a PDE system

If I have the following PDE system: $\frac{\delta}{\delta t}x(t,r)=-\int_0^1 G(|r-r'|)y(t,r')dr'x(t,r)$ $\frac{\delta}{\delta t}y(t,r)=\int_0^1 G(|r-r'|)y(t,r')dr'x(t,r)-y(t,r)$ $x(0,r)=a(r), ...
0
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1answer
24 views

Prove that each $W_0^{1,2}$-function is weakly differentiable

Let $\Omega\subseteq\mathbb R^n$ be open. $u\in\mathcal L^1_\text{loc}(\Omega)$ is called weakly differentiable $:\Leftrightarrow$ $\exists v\in\mathcal L^1_\text{loc}(\Omega;\mathbb R^n)$ with ...
1
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1answer
47 views

How to numerically solve the Poisson equation given Neumann boundary conditions?

I want to solve the Poisson equation on a 2D domain given Neumann-type boundary conditions: The PDE: \begin{equation} \nabla^2 \; u(r,\theta) \;=\; f(r,\theta) \end{equation} The boundary ...
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1answer
30 views

Help with a proof using energy method for PDE

The question is Prove the uniqueness of solution of the initial value problem for $\left\{ \begin{array}{l} \Delta u - {u_{tt}} - q(x)u = f(x,t)\\ u(x,0) = g(x)\\ {u_t}(x,0) = h(x) \end{array} ...
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0answers
14 views

Heat equation with $x\in [0,+\infty[$ and non-homogeneous initial and boundary condition

The IVBP that i need to solve is the follow: \begin{equation} \begin{cases} u_t=au_{xx} & x>0,t>0,a\in\mathbb{R}^+\\ u(x,0)=B_0e^{-kx}\cos(kx) & x\geq ...