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

learn more… | top users | synonyms (2)

0
votes
2answers
67 views

Can we show this inequality? (PDE question)

I am attempting to show that $$\int^t_0 \left[e^{-\lambda \kappa (t-s)} a(s)\right] ds \leq \int^t_0 \frac{\lambda}{2\kappa} a(s)^2 ds \tag{1}\label{1}$$ for any $t$, where $\lambda, \kappa >0$, ...
0
votes
1answer
45 views

Understanding Distributional Meanings and Test Functions for PDEs

My question is about distributional meanings in PDEs. My specific question is at the bottom, but I'd be interested in a bit of general theory (even if this is a link to a particularly good set of ...
4
votes
0answers
15 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
1answer
576 views

Numerical techniques for solving systems of first-order semi-linear hyperbolic PDEs in two variables

I need to solve such systems of PDEs numerically and I'm wondering what the "standard" method (if there is one) is. The systems I'm interested in have the form: $\frac{\partial F_i}{\partial t} + ...
-2
votes
1answer
35 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 $u(0,t)$?
3
votes
2answers
107 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 $g \in H^{\frac 12}(\partial\Omega)$. ...
1
vote
1answer
48 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 ...
1
vote
0answers
32 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
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 ...
1
vote
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} ...
0
votes
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~ ...
1
vote
0answers
50 views

Show inequality for modified bilinear form

Let $\Omega_h$ denote to the domain that is bounded by a polygon, and $V_h$ denote to the space of all $v\in C^0(\Omega)$ such that $v_{|T}$ is linear on any (curved) triangle T and $v=0$ in the ...
1
vote
1answer
214 views

How to find if a bilinear form is coercive .

Consider $\Delta u =f(x) , x \in \Omega $ and $\nabla u\cdot n +\alpha u = g(x) , x\in \partial\Omega $, where $n$ is outward normal. Can anyone help me to define a bilinear form for this PDE and ...
1
vote
0answers
28 views
+50

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

I am trying to implement a Finite Element Solution for the Stabilized Eikonal Equation : $$ |\nabla u| = 1 + \Gamma \Delta u, \quad \text{ where } \quad u = \text{ distance function }$$ $$ ...
4
votes
1answer
30 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 ...
2
votes
2answers
31 views

The effect of the CFL number in the numerical solution in this conservation law

I've been studying the very basics of numerical methods applied to conservation laws, and I'm having trouble understanding the role of the CFL number in the upwind scheme. I want to understand it (if ...
0
votes
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 ...
4
votes
2answers
103 views

Is the following PDE boundary value problem well-posed?

My Question Is the following Poisson boundary value problem well-posed, as stated? If so, how could I go about solving it? If not, what would it need to be well-posed? Does it satisfy the ...
2
votes
0answers
27 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 ...
0
votes
2answers
90 views

The space $\{ u \in H^1(\Omega) \mid \Delta u \in L^2(\Omega)\}$

For a open Lipschitz domain $\Omega$, consider the space $$A =\{ u \in H^1(\Omega) \mid \Delta u \in L^2(\Omega)\}.$$ Now I heard somewhere that all the second derivatives of a function $u$ are ...
0
votes
1answer
32 views

I might need some help on this Complex Fourier Series Problem

Here is the problem: Use the complex Fourier Series on $[-L,L] $ with complex coefficients to find a representation of $\frac{1}{2L} \int_{-L}^{L} |f(x)|^{2} dx$ Here is my attempt: The ...
0
votes
2answers
40 views

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

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 ...
1
vote
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 ...
2
votes
1answer
2k views

question 9 - chap 5 evans PDE

The question is : Integrate by parts to prove : $$\int_{U} |Du|^p \ dx \leq C \left(\int_{U} |u|^p \ dx\right)^{1/2} \left(\int_{U} |D^2 u|^p \ dx\right)^{1/2}$$ for $ 2 \leq p < \infty$ ...
2
votes
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 ...
2
votes
0answers
24 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 ...
1
vote
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 ...
1
vote
1answer
46 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 ...
3
votes
2answers
438 views

Well-posedness of the Poisson problem with mixed boundary conditions

Let $\Omega \subset \mathbb R^n$ be a subdomain with Lipschitz boundary, i.e. locally any part of the boundary looks like the graph of a Lipschitz continuous function, after some affine coordinate ...
0
votes
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 ...
6
votes
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 ...
3
votes
1answer
27 views

Convolution with standard mollifier

Let $\Omega \subset \mathbb{R}$ open and $f \in L^p(\Omega).$ Now, we define $$\eta(x):=\chi_{[-1,1]}(x) e^{\frac{-1}{1-x^2}}.$$ Then we define $$\eta_h(x):=\frac{1}{h} \eta\left( ...
7
votes
0answers
121 views
+50

Question about boundedness of a sequence in $ W^{3,q} $ for any $ 1\leq q < \frac{N}{N-1} $

I have asked this question several months ago, I have understood every thing and there are good comments and they have helped me , but only I have a question about tomas comment, how can ...
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 ...
2
votes
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) ...
1
vote
1answer
435 views

Relationship between integral equations and partial differential equations

In my functional analysis class, we did a lot of problems involving integral equations such as proving existence & uniqueness using spectral theory and the Banach fixed point theorem. I've never ...
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 ...
2
votes
2answers
70 views

Proof of an inequality involving gradient function

I'm reading ahead in my course, and I've encountered the following problem; Let $\Omega \subset \mathbb{R}^n$ be a bounded domain such that the divergence theorem holds. Assume that $u \in ...
4
votes
0answers
119 views

Question 13 in Taylor's PDE vol III section 16.1.

My question comes from Taylor's PDE textbook, volume III. Consider a semilinear hyperbolic system, $u_t=Lu+g(u)$, $u(0)=f$, where $Lu=\sum_j A_j \partial_{x_j}u$, $g(0)=0, \ |g'(u)| \le C$, take ...
1
vote
1answer
58 views

How to prove the inequality in Burgers equation and what is the relationship to energy dissipation

Consider Burgers equation: \begin{equation} u_t+uu_x-\epsilon u_{xx}=0 \quad , \quad u(x,0)=g(x) \quad , \quad u = 0 \mbox{ when } |x| \mbox{ large}, \end{equation} where ...
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
19 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 ...
8
votes
1answer
299 views

Heat Kernel Property

Let $\phi$ be the Heat Kernel in $\mathbb{R}^n$, i. e. $$\phi (x,t)={(4\pi t)}^{-n/2}\exp\left( - \frac{\mid x \mid ^2}{4t}\right)$$ and let $u$ satisfy Heat equation. Show that: ...
0
votes
3answers
74 views

Simple Partial Differential Equations

Solving Partial Differential Equation $x f_{y}+y f_{x} =0$ ? $f=f(x,y)$ is a real-valued function of two variables $x,y$. Thanks.
1
vote
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
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
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. ...
2
votes
2answers
76 views

Analytical Solution of a PDE

I need to solve a PDE which seems to be quite simple and to have an analytical solution. I tried the method of separation of variables, but could not complete the solution. Could you please let me ...
0
votes
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 ...
0
votes
0answers
75 views

How do I solve first order non-linear system of PDE: $\partial f^i(x,y)/\partial z = F^i(f^1,f^2,…,f^n)$?

Suppose that I have a system of PDEs of the following form: \begin{eqnarray} \frac{\partial f^i(x,y)}{\partial z} = F^i(f^1,f^2,...,f^n), \qquad i = 1,..,n \end{eqnarray} Where $z = x + iy$, ...