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

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Is there such a thing as a “partial differential”, brother to “total differential”?

I am familiar with total differentials in the form $$ f = f(x,y,z) $$ $$ df = \frac {\partial f} {\partial x} dx + \frac {\partial f} {\partial y} dy + \frac {\partial f} {\partial z} dz $$ however, ...
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25 views

Solution of a PDE using Bessel Functions of the first an second kind $J_0(z), Y_0(x)$

Solution of a PDE using Bessel Functions of the first an second kind $J_0(z), Y_0(x)$. I want to p}{\partial rurobe that the solution of the PDE: $$g\big(\dfrac{\partial^2u}{\partial ...
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1answer
59 views

Momentum is quantised in compact spaces?

Background One of the first examples given when studying quantum mechanics is the particle on a cylinder, or particle on a ring. One finds that because of the periodic boundary conditions, ...
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1answer
33 views

Which assumptions on $Ω\subseteq\mathbb R^d$ do we need in order to show density of $C_c^∞(Ω)$ in $(L^p(Ω),\left\|\;\cdot\;\right\|_{L^p(Ω)})$?

Let $\Omega\subseteq\mathbb R^d$, $u\in\mathcal L^1(\Omega)$ and $$u_\varepsilon(x):=\frac 1{\varepsilon^d}\int_\Omega\rho\left(\frac{x-y}\varepsilon\right)u(y)\;{\rm d}\lambda(y)\;\;\;\text{for ...
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37 views

Differential Equation Simplification on American Put Option paper

I am currently reading "An Exact and Explicit Solution for the Valuation of American Put Options" by Song-Ping Zhu. The articel is available at ...
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11 views

Does a pde solution completely depend on the parameters

I was wondering whether pde solutions are completely determined by the pde parameters. For example let $f_{a,b}(x,y)$ denote a solution of a pde with UNKNOWN parameters $a,b$. Then, can we always ...
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12 views

Classification of second order linear PDE

I have been looking at the classification of the second order linear PDEs and came across two different definition If a PDE is defined as following: $$Au_{xx}+Bu_{xy}+Cu_{yy}+Du_x+Eu_y+Fu = G$$ ...
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50 views

Why can the solution of a SPDE $\partial_tu(t,x)=\cdots$ be viewed as a stochastic process indexed by $t$ with values in a space of functions of $x$?

Please consider a stochastic partial differential equation of the form $$\partial_tu(t,x)=F(t,x,u(t,x),{\rm D}u(t,x),{\rm D}^2u(t,x))+G(t,x,u(t,x),{\rm D}u(t,x))\partial_tB(t,x)\tag 1$$ where ...
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2answers
23 views

An example for a PDE's

Let $\delta(x^1,x^2)$ and $\beta(x^1,x^2)$ be two functions. Is there any example of $\delta$ and $\beta$ which satisfy in the following PDE's? \begin{align} \frac{\partial \beta}{\partial ...
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1answer
33 views

What if we change one of Fourier's law of heat conduction

I'm studying PDE heat diffusion on 1-D rod using the textbook. It states four intuitions leading to Fourier's law of heat conduction $\phi=-K_0\frac{\partial u}{\partial x}$, where $\phi$ is the heat ...
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35 views

Numerical solution of $k\nabla^2 p(\vec{x}) = \nabla(\vec{f}(\vec{x})p(\vec{x}))$

I have the following equation: $$k\nabla^2 p(\vec{x}) = \nabla(\vec{f}(\vec{x})p(\vec{x}))$$ where the constant $k>0$ and vector field $\vec{f}(\vec{x})$ are both known. I wish to numerically ...
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1answer
17 views

What is the difference between a first order compartment and diffusion

Biologists use Compartment models to represent the flow and storage of fluids in an animals body. In tissue (like muscle) the diffusion of blood is more accurately represented by the diffusion ...
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1answer
194 views

Biharmonic Equation in a Rectangle with Some Uncommon Boundary Conditions

Consider the following boundary value problem (BVP) $$\matrix{ {{\Delta ^2}H = 0,} \hfill & {} \hfill & {{\rm{in}}\,} \hfill & \Omega \hfill \cr {\partial _y^2H = 0} \hfill & ...
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1answer
34 views

Minimizing point for $L^2$ distance.

I am trying to study the following equation: $$ F(t):=\|u_t-u^*\|_{L^2(\Omega)}^2 $$ where $u^*\in H^1(\Omega)\cap L^\infty(\Omega)$ is fixed and $\Omega\subset \mathbb R^2$ is open bounded with ...
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0answers
16 views

Calculating partial derivative of transformed formula

I am asked to give the partial derivative $\frac{\delta U}{\delta t}$ of the following (Black-Scholes PDE) function: $\frac{\delta V}{\delta t} +rS\frac{\delta V}{\delta S} ...
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28 views

For the wave equation:$u_{tt}=c^2 u_{xx}$, how do I show $E_{kin}=E_{pot}$ for large t

My question has already been asked and partly answered here: LINK I am having difficulties getting the correct term for $u_t^2$ In the link I provided the hint is to differentiate D'Alamberts ...
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1answer
26 views

Intuition for Sobolev spaces $H^{\frac{1}{2}}$ and $H^{-\frac{1}{2}}$

I'm looking for some intuition for Sobolev spaces $H^{\frac{1}{2}}$ and $H^{-\frac{1}{2}}$. Any explanation I've seen is very technical, I'm looking for the most simple explanation possible that gives ...
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15 views

Reference for Inverse Scatering Transform

I am looking for a good introductory text to learn inverse scatering transformation and related topics (Lax pairs, nonlinear FT). Any pointer is much appreciated.
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1answer
35 views

How can I solve this PDE: $u_{tt}(\mathbf{x},t)+ku_t(\mathbf{x},t)-c^2 \Delta u(\mathbf{x},t)=0$

Consider the wave equation: $$u_{tt}(\mathbf{x},t)+ku_t(\mathbf{x},t)-c^2 \Delta u(\mathbf{x},t)=0$$ $\mathbf{x}\in\Bbb R^2, t>0 \\ \space \\ u(\mathbf{x},0)=0 \\ ...
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1answer
77 views

Laplace Equation in a Cylinder with Some Uncommon Boundary Conditions

While I was working on some theorems in PDEs, I encountered the following axisymmetric boundary value problem $$\matrix{ {{\nabla ^2}H = 0} \hfill & {{\rm{in}}} \hfill & \Omega \hfill ...
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3answers
40 views

How to solve the following system of partial differential equations?

I have a system of partial differential equations: \begin{align} & u(a,b,c) \frac{\partial y}{\partial c} = \frac{4}{3} ab, \\ & u(a,b,c) \frac{\partial y}{\partial b} = \frac{2}{3} ac + 2 ...
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0answers
97 views

How can we desribe a particle whose motion is perturbed by a random forcing using a stochastic partial differential equation?

Let $d\in\left\{2,3\right\}$ and $\mathcal V_t$ be the bounded set occupied by a fluid at time $t\ge 0$. Let $x_0\in\mathcal V_0$ be a particle and $$[0,\infty)\to\mathbb R^d\;,\;\;\;t\mapsto ...
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2answers
49 views

cutoff function vs mollifiers

Q1. What are cutoff function? What are mollifiers? I cannot distinguish the two. Could anyone give some concrete/simple examples of cutoff functions? And how they differ from mollifiers I did check ...
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45 views

Laplace Equation in a Cylinder where the Laplacian is Prescribed Over the Boundary

While I was working on some theorems in PDEs, I encountered the following boundary value problem $$\matrix{ {{\nabla ^2}H = 0} \hfill & {{\rm{in}}} \hfill & \Omega \hfill \cr ...
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13 views

integral of solution between two characteristic curves

Suppose we are given a pde: $\frac{\partial u}{\partial t} +\frac{\partial {(b(x)u)}}{\partial x}=0$. Let $u(t,x)\in C^1(\mathbb{R}^2)$ be a solution and $x=X_1(t)$ and $x=X_2(t)$ be two ...
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1answer
31 views

Second order elliptic equation with nonlinearity depending on the gradient

Let us consider the problem $$-\Delta u =f(x,u,\nabla u)\text{ in }\Omega$$ $$u=0 \text{ on }\partial\Omega,$$ where $\Omega\subset \mathbb{R}^n$ is a smooth and bounded domain. I have seen at many ...
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3answers
81 views

how to bring the PDE $u_{tt}-u_{xx} = x^2 -t^2$ to the canonical form

How to bring to the canonical form and solve the below PDE? $$u_{tt}-u_{xx} = x^2 -t^2$$ I recognize that it is a hyperbolic PDE, as the $b^2-4ac=(-4(1)(-1))=4 > 0$. I don't know how to proceed ...
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0answers
22 views

second order PDE's - change of variables - how to choose variables $\xi$ and $\eta$

second order PDE's how do you choose the alternative $\xi$ and $\eta$ variables when you deal with a new problem? I understand that it should be something that simplifies once we substitute the ...
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2answers
28 views

The relation between the discrete Harmonic function and the Harmonic function in PDE

Given a Markov chain with state space $\Omega$ and its transition matrix $P$, a function $h(x):\Omega\to\Bbb R$ is called a harmonic at state $x$ if $h(x)=\sum_{y\in\Omega}P(x,y)h(y)$, and is called ...
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0answers
25 views

Propagation speed for the wave equation

Let $g\in C_c(\mathbb{R}^n,[0,\infty))$ with $\int g(x)dx=1$. Denote $\Phi$ the Heat kernel given by $\Phi(x,t)=\frac{1}{(4\pi t)^{n/2}}e^{-\frac{|x-y|^2}{4t}}$. Let ...
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1answer
28 views

Solving Laplace equation in a square with one insulated border

I keep getting stuck on this problem, so if someone could point out where my method is flawed and how I should approach this problem, that would be extremely useful. We're considering the square ...
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0answers
12 views

poisson equation- decay rate

Given a solution $\phi\in C^2(\mathbb{R})$ to the equation $$ \Delta \phi=u $$ If I know that $u\in C(\mathbb{R})$ and $u\to 0$ at $x\to \pm\infty$ faster than any polynomial can I know something ...
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1answer
50 views

Boundary conditions for vibrating beam

I'm solving the equation for the transverse vibrations of a Euler-Bernoulli beam fixed at both ends and subject to axial loading (as per this diagram). It's a similar problem to that described by Rao ...
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0answers
25 views

LTE for the Cahn Hilliard Equation

I'm trying to Calculate the LTE of a first order finite difference (central difference) scheme for the following 1D equation: $u_t = \frac{\partial^2}{\partial x^2}(u^3 - u - \frac{\partial^2 ...
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1answer
47 views

Show that PDE does not have a solution

I need to show, that the following PDE does not have a solution: \begin{align} u_x + u_t &= 0 \\ u(x,t) &= x , \forall x,t: x^2 + t^2 = 1 \end{align} My attempt: It's first-order linear PDE ...
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1answer
40 views

Some computation about Laplace operator

Let $$y=\frac{x}{|x|^2}$$ in $\mathbb{R}^n$. Let $u(x)$ be a smooth function, and let $$v(y)=|x|^{n-2}u(x)$$ Then we have $$\Delta_y v(y)=|x|^{n+2}\Delta_x u(x)$$ How to prove this? I know some ...
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1answer
31 views

Transform PDE to ODE (3 variable case) with given boundaries

How can I transform the following PDE into an ODE? I tried using three different functions $H(x),G(y)$ and $F(t)$ but that didn't help hence I did not post it here. I really hope someone can help me ...
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0answers
18 views

Poisson equation and Green's function: consider three dimensions

Consider in three dimensions: (1)$\nabla^2u=f(x)$, with on boundary $\nabla u \dot\ \hat n=0$ Now show that $\phi_h=1$ is a homogeneous solution, satisfying homogeneous boundary conditions. I ...
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1answer
26 views

Construct an example where x(t, x_0) is bounded but limt→+∞ x(t, x_0) does not exist.

Suppose we are given an IVP $x = f(x), x(0) = x_0 $, x ∈ R^n for which we know that the unique solution x(t, x_0) exists globally in time. Construct an example where x(t, x_0) is bounded but limt→+∞ ...
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19 views

Reduction of a quadratic form to a canonical form

I'm supposed to reduce following polynomial to its canonical form. But my result differs from the one given in my book, so I'm not sure if it's correct too. $$ q = u_{xx} - u_{xy} - 2 u_{yy} + u_x + ...
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14 views

Pde Problem with Delta initiative condition.

The problem is to solve the heat equation $\frac{\partial T}{\partial t} = ~\frac{\partial^2 T}{\partial x^2}$ $T=T(x,t)$ inside a disc of radius $R$ with initial condition $T(x,0)=\delta^{(2)}(x)$ ...
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0answers
11 views

Solution curves to characteristic equations

I had a question about solution curves of characteristic equations given by dx/a=dy/b=du/c My teacher says that if you can solve two of these equations at a time, each solution will satisfy the whole ...
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1answer
19 views

Why $H^1(\Omega)\cap H^2(\Omega_-)\cap H^2(\Omega_+) \subset W^1_p(\Omega),\forall p>2$?

I'm reading a book and it says that: Let $\Omega=\Omega_-\cup\Omega_+$. Define $X(\Omega)=H^1(\Omega)\cap H^2(\Omega_-)\cap H^2(\Omega_+)$. Then by the Sobolev embedding theorem, $X(\Omega)\subset ...
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1answer
26 views

Confusion on taking the second partial derivative:

From this previous question I gained understanding of why the chain rule for a function $u(x,y)=f(p)$ is expressed as $$\frac{\partial u}{\partial x}=\frac{\mathrm{d}f(p)}{\mathrm{d}p}\times ...
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2answers
24 views

Need to change variables in equations with cosh.

i have these five functions: $x=\tau \cosh(s)$ $q=\tau \sinh(s)$ $y= \sinh(s)$ $p= \cosh(s)$ $u= 1/2*\tau*\cosh(2s)+1/2*\tau$ I need to write $u$ in terms of $x$ and $y$ I know the answer is ...
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2answers
40 views

Solving a PDE $\frac{\partial H(x,t)}{\partial t} = H(x,t) + f(x)$

I have an equation of the general form: $$ \frac{\partial H(x,t)}{\partial t} = H(x,t) + f(x) $$ The actual $f(x)$ is a bit complex but is purely a function of x. I'd like to get a general ...
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1answer
117 views

Is a general smooth rescaling of a complete vector field itself complete?

$\newcommand{\Ga}{\Gamma}$ $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\til}{\tilde}$ $\newcommand{\M}{M}$ $\newcommand{\ep}{\epsilon}$ $\newcommand{\brk}[1]{\left(#1\right)} $ ...
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0answers
35 views

Brownian motion, harmonic functions and the Dirichlet problem

I am having trouble understanding one detail of the standard use of Brownian motion to solve the Dirichlet problem, I will write the statement and proof and then point to the detail I don't ...
2
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0answers
22 views

smoothness up to the boundary and the compatibility conditions

Let us say $\Omega$ is a smooth bounded domain. I have a baby quesiton: when we can say a solution $u$ of a heat equation is in $C^{2, 1}(\overline\Omega\times (0, T])\cap C(\overline\Omega\times ...
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0answers
45 views

Solving a simple Schrodinger equation with Fast Fourier Transforms

While trying to solve a stochastic Gross-Piaevskii equation I have found a problem that can be tracked down to something buggy occuring in the simplest Schrodinger equation possible: $\partial_t \psi ...