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

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Reformulation of Goldbach's Conjecture as optimization problem correct?

Question I think I managed to reformulate a stronger version of Goldbach's conjecture as an optimization problem: $$ \frac{\partial F_n}{\partial a_n} = \frac{\partial F_n}{\partial \overline a_n} ...
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24 views

Fundamental solution of the Poisson equation with variable exponent

Let the variable exponent $p(x)$, where $p(x) \in C(\overline{\Omega})$, I want to know the fundamental solution of $$-(\Delta u)^{p(x)}=\delta_0.$$
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1answer
28 views

Solution to piecewise heat equation

We have piecewise diffusion equation $u_{t}(x,t)=\left\{\begin{matrix} D_{1}u_{xx}(x,t) &x\in(0,1) \\ D_{2}u_{xx}(x,t) & x\in (1,2)\end{matrix}\right.$, with IC: $u(x,0)=2x+1$ and BC : ...
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0answers
19 views

Bounding a Subsolution of the Heat Equation

As the title suggests, I'd like to bound a subsolution of the heat equation. I have \begin{align*} u_t - \Delta u &\le 0 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text{ in } \mathbb R^n \times (0,\infty) \\ ...
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1answer
18 views

$L^1(0,T;(L^1_{loc}(\mathbb{R^N}))^N)$ - Confused about notations used for space and time dependent vector fields

I found this notation - $L^1(0,T;(L^1_{loc}(\mathbb{R^N}))^N)$ - in a paper of DiPerna and Lions concerning vector fields space and time dependent, "Ordinary differential equations, transport theory ...
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40 views

Barenblatt solution for diffusion on the whole line

Question It is my second course on PDEs and the teacher asked us to find solutions like: $$u(x,t) = t^\alpha · f\left(\frac{|x|^2}{t^\beta}\right)$$ for the diffusion (with $k=1$): $$u_t - u_{xx} = ...
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11 views

Greens function for Poisson equation with Neumann boundary condition

Consider Poisson's equation with Neumann boundary condition: \begin{eqnarray*} -\triangle u & = & f,\quad x\in\Omega\\ \frac{\partial u}{\partial n} & = & g,\quad x\in\partial\Omega ...
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1answer
13 views

How do you choose basis functions in finite element analysis?

I'm having some trouble understanding the underlying mathematics in finite elements. I know it's widely used to solve PDEs especially in structural applications. From what I understand you discretize ...
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37 views

Reference request: about inverse Laplacian operator

I am currently studying some problems about inverse Laplacian and the Yosida approximation and wishing to learn more about it. Here is a post about one of the problems that I am interested in. ...
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1answer
51 views

The Burgers equation $u_y + u u_x = 1$ with $u=0$ on the parabola $y^2=2x$

For the PDE $u_y + u u_x = 1$, sketch a plot of $\Gamma$ and a few representative curves, including the envelope curve. Conditions: $u=0$ on the curve $y^2=2x$, and $y,x>0$. Express $u$ as a ...
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2answers
21 views

A general solution for a 2d pdf (ode)

I have the following 2 dimensional PDE: $$ \partial_{x_1}^2 u(x_1,x_2)+\frac{1}{x_1^2}\partial_{x_2}^2 u(x_1,x_2)+\frac{1}{x_1}\partial_{x_1}u(x_1,x_2)=k $$ where $k>0$ is a constant, and ...
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0answers
34 views

The inverse of Laplacian for different orders.

This question is related to my previous question here Let $u,v\in C_c^\infty(\Omega)$ and $\Omega\subset \mathbb R^N$ is open bounded with smooth boundary. Let $\Delta$ denote the Laplacian operator, ...
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1answer
39 views

Solving a system of ODE that arose in solving Burgers' equation

Consider the Burgers' equation $$\partial_t u = \alpha u\partial_xu$$ Intend to solve this using Fourier Galerkin method. So When I convert this into $N$th Fourier partial sum, I get a system of ODE's ...
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5 views

Looking for references of sobolev spaces involving time

I am looking for references which introduce the the sobolev spaces involving time. What I have at the moment is only a short chapter from Evan's PDE-book. Is there any other similar literature but ...
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34 views

Verify function is a Solution to Wave Equation in 3D Spherical Coordinates

The wave-equation is given by $$\nabla^2E=\frac{1}{c^2}\frac{\partial ^2E}{\partial t^2} $$ And I'm trying to prove that this wave $ E(r,\theta,\phi)= \frac{A_o}{r}sin(\theta) cos(\omega t - ...
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1answer
42 views

Interior estimate for derivatives of harmonic function

I'm learning PDE from the book of Trudinger and Gilbarg and I'm attempting to prove the following theorem: Let $\hspace{0.1ex}u\hspace{0.1ex}$ be harmonic in $\hspace{0.1ex}\varOmega \subset ...
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1answer
47 views

Computing the inverse of Laplacian operator.

I am considering the following equation: $$ f(t):=\int_\Omega[(I-t\Delta)^{-1}\Delta(I-t\Delta)^{-1}u]\cdot u\,dx $$ where $u\in C_c^\infty(\Omega)$ and $t\geq 0$ a real number. $I$ is the identity ...
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1answer
14 views

Is L^2-norm of Laplace operator equivalent to 2-seminorm?

Let's say $\Omega$ is a bounded domain in $\mathbb{R}^2$. Is it true that $C_1\|\Delta u \|_{0,\Omega} \leq |u|_{2,\Omega} \leq C_2\|\Delta u \|_{0,\Omega}$? The left inclusion is obvious by ...
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16 views

Lax pair for Painleve V equation

Deformations of this linear system (which contains only fuchsian singularities) $$\frac{dY(z,t)}{dz}=(\frac{A_0}{z}+\frac{A_1}{z-1}+\frac{A_t}{z-t})Y(z,t)$$ are isomonodromic if and only if ...
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26 views

The first eigenvalue for Dirichlet boundary condition positive?

Let $M$ be a compact, n dimensional Riemannian manifold with boundary. Then we know that $W^{1,2}(M)=W^{1,2}_0(M)$, the latter is the completion of $C_0^{\infty}(M)$ function w.r.t $W^{1,2}(M)$-norm. ...
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1answer
18 views

Subsequence of $L^{2}(\Omega)$ - bounded sequence weakly * converging to a measure

I was reading a well available article in the internet: "THE COMPENSATED COMPACTNESS METHOD APPLIED TO SYSTEMS OF CONSERVATION LAWS by Tartar"; there at Page-266, it is written: "$L^{1}(\Omega)$ is ...
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1answer
32 views

Solving $u_{xx}+u_{x}^{2}=ku$ for various k

Here is a nonlinear ODE: $u_{xx}+u_{x}^{2}=ku$ for various k Attempts For $k=0$, we have $u=log(x+c_{1})+c_{2}$. For $k\neq 0$, 1)Divide by $u_{x}$ and integrate both sides to get: ...
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24 views

Exact solution of $u_{t}=a_{1}u_{xx}+a_{2}\Phi(t)u_{x}^{2}$, where $\Phi(t)$ is cdf normal

The PDE for $(x,t)\in \mathbb{R}\times (0,1]$ is $$u_{t}=a_{1}u_{xx}+a_{2}\Phi(t)u_{x}^{2},$$ where $a_{1},a_{2}$ are constants and $\Phi(t)=\int_{-\infty}^{t} \phi(\frac{x-0.5}{0.2})$. I picked ...
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1answer
39 views

how to solve a pde whose coefficient is the function itself

I am studying differential geometry, Walker metric in three dimension. I try to find the geodesic equations of a Walker manifold and I need to solve the following PDE. Unfortunately, I didn't take any ...
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20 views

Solving Hyperbolic PDE using MATLAB (finite difference Method)

I am given the PDE $u_t + u_x = 0, x \in (-2,3), t > 0$ with initial condition $ u(x,0) = 1 - |x|$ when $|x| \leq 1$ and $u(x,0) = 0$ when $|x| \geq 1$. With boundary condition $u(-2,t) = 0$. How ...
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1answer
35 views

Existence of a solution to the poisson equation with a Radon measure on the right hand side.

I was trying to guarantee the existence of a solution to the problem $$-\Delta u=\mu,\quad u\lvert_{\partial\Omega}=0$$ where $\mu$ is a signed Radon measure, i.e., $\mu(A)=\int_{A}f\,dx$, $f\in ...
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25 views

HOW to work out mixed partial derivatives

I dont understand how to work out the partial derivative using the chain rule, for exaple, $ u = \psi_{y} $ $v = -\psi_{x}$ $ \psi = (2x)^{1/2} f(\eta) , \eta = (2x)^{-1/2)}y$ so $ \psi_{y} = ...
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A problem about Frechet derivative:

Let A be a $n \times n $ matrix with real entries and eigenvalues with strictly negative real parts. Let $g \in C^1(R^n;R^n)$ with $ g(0) = 0 $ and with Frechet derivative $ Dg(0) $ satisfying ...
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1answer
24 views

mixed partials for PDE

Can someone please help me. if $ u = \frac{\partial\phi}{\partial y} , v = -\frac{\partial\phi}{\partial x}$ and $ \phi = (2x)^{\alpha} f(\eta)$ where $ \eta = (2x)^{\beta}y$ I need, to work out ...
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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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0answers
24 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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11 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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49 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
32 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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12 views

How to determine the right initial and boundary conditions of the nonlinear PDE system

The nonlinear PDE system is from a research paper in 2000. The authors solved the system by using an ordinary differential equation integrator in FortranVariable-coefficient Ordinary Differential ...
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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
156 views
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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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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
25 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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0answers
14 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
34 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
74 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 ...