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

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Wavelet reference for PDE

Can anyone recommend a very readable introduction to wavelets for use in theoretical PDE/harmonic analysis? I'm frustrated with the account in Lemarie-Rieusset's Navier-Stokes book since he provides ...
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38 views

Weak solution for equation $-\Delta u = f$.

Suppose $f \in L^{2}$. I know that $$\left\{\begin{array}{c} −\Delta u = f(x) & \text{on }\Omega \\ u(x)=0, \partial\Omega \end{array}\right.,$$ where $\Omega \subset \mathbb{R}^{N}$ is open ...
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2answers
155 views
+50

Help with a nonlinear partial differential equation

let : $$\frac{\partial f}{\partial x}=f _{x}\,,\qquad\frac{\partial f}{\partial t}=f _{t}\,,\qquad\frac{\partial}{\partial t}\frac{\partial f}{\partial x}=f_{tx}\,,\qquad\frac{\partial}{\partial x}\...
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Prove this space is a Hilbert space

Let us consider a convex polygonal and bounded domain $\Omega$ in $\mathbb{R}^2$ containing two subdomains $\Omega_1, \Omega_2$ which satisfy $\overline{\Omega}=\overline{\Omega}_1\cup\overline{\Omega}...
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1answer
25 views

Finding $A$ such that $\nabla \times A = B$ for given $B$.

Let $B:U \rightarrow \mathbb{R}^3$ be a $C^\infty$ vector field, where $U = \mathbb{R}^3 \backslash \{(0,0,z):z \in \mathbb{R}\}$, defined by $$B(x,y,z)=\frac1{\rho^l} (-y,x,0)$$ where $\rho = \sqrt{...
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1answer
44 views

A non-strict inequality on skew symmetric matrices

As we know that skew-symmetricity means $A=-A^\top$ where $A\in\mathbb{R}^{n\times n }$. But recently I came across an inequality that states, $A+A^\top\preceq0$ can also be considered as an ...
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+100

Functions with constant divergence of gradient-like field $\phi\nabla \phi/|\nabla \phi|$

I would like to classify functions $\phi : \mathbb{R}^2 \rightarrow \mathbb{R}$ such that $$ \nabla \cdot \left( \phi \frac{\nabla\phi}{|\nabla\phi|} \right) = \text{const}. $$ The only examples I ...
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1answer
22 views

Solving quasilinear PDE - 1D, time-dependant, convection

I have a task to solve the following quasilinear PDE (find $c(x,t)$): $$ c_x v + c_t = - v_x c $$ $c \in (0,20) , t \in (0, \infty)$ where I know function $v(x)$ to be $v(x) = \frac{3}{40}(1+\cos(\...
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29 views

Maple: Symmetries of strange modified heat-equation

I thought about the following PDE (the $u(x=0,t)$ is not a boundary condition! It really is a part of the PDE): $$\dfrac{\partial u(x,t)}{\partial t}=\alpha \dfrac{\partial^2 u(x,t)}{\partial x^2}+u(...
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20 views

Solution route to solve coupled nonlinear PDEs?!

When I have a nonlinear coupled non-steady PDEs, what are the steps to get a solution. My background is for single linear PDE in which I distretize space and time terms, get system of linear equations,...
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1answer
60 views

Is $\Delta C_c^\infty$ a dense subset of $L^p(\mathbb{R}^d)$?

I'm struggling to obtain some density result. It is well known that $C^\infty_c(\mathbb{R}^d)$ is dense in $L^p (\mathbb{R}^d)$ for $1\leq p<\infty$. It is well known that for $\lambda>0$, $(\...
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1answer
705 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} + \...
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22 views

Solving the KGE equation with a potential by changinging into ode? Kink and Antikink, graph just looks wrong..

I'm trying to find solutions to a klien gordon equation with a chosen potential. Particularly kink and antikink solutions. I'm doing this on Maple so will not give the actual results as these tend to ...
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1answer
23 views

Question about trace operator

From the general trace theorem we know for instance that if $f\in W^{1-\frac{1}{p},p}(\partial\Omega)$, then there exists a function $f\in W^{1,p}(\Omega)$ such that $f|_{\partial\Omega}=f$. But is it ...
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1answer
26 views

How to solve this equation: $v_1\partial_xu+v_2\partial_y u=0$

I have the following equation: $$ v_1\dfrac{\partial u}{\partial x}+v_2\dfrac{\partial u}{\partial y}=0 \tag{1}$$ $u(x,y)$ is the unknown function (a scalar-valued function), $v_1$ and $v_2$ are two ...
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1answer
26 views

Solve Kolmogorov differential equations for birth-death process with constant rates

I need to solve the Kolmogorov forward equations for a birth-death process whose birth/death rates $\lambda_k,k=0,\ldots$ and $\mu_k,k=1,\ldots $ are constant, i.e., $\lambda_k=\lambda$ and $\mu_k=\mu$...
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1answer
32 views

Vainberg Theorem in measure theory

In a lecture notes about Variational Methdos, I found the following theorem: THEOREM: Let $(f_n)$ a sequence in $L^{p}(\Omega)$ and $f \in L^{p}(\Omega)$, such that $f_{n} \rightarrow f$ in $L^{p}(\...
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1answer
26 views

difference between second order quasi linear and semi linear PDE

I am studying the second order PDE's and I am a bit confused with classification of quasi linear and semi linear PDEs. Could anybody explain on examples what is a difference between them please?
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Expressing spherical harmonics as a combination of other spherical harmonics

Spherical harmonics are a useful tool in physics, particularly in classic electrostatics and electrodynamics. Given an integer $l$, the spherical harmonic $Y_{l,m}$, where $-l\leq m\leq l$, solves the ...
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Wave equation in a cube

Is it possible find a computable solution to the following homogeneous wave equation problem: Let $\mathcal{C}=\{(x,y,z)\in \mathbb{R}^3, 0< x,y,z < 1\} $ be the open unit cube. Find $u$ such ...
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1answer
18 views

“Hessian” differential equation

In my homework, I'm given the following problem: Let $f: \mathbb{R}^n \to \mathbb{R}$ be a twice differentiable function. For an $\alpha \geq 2$, let: $$f(\lambda x) = \lambda^a f(x)$$ for ...
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1answer
12 views

Derivative of volume of given set

As picture below ,how to compute the $\partial_t |\Omega_t|$ ? The picture below is from the 32 page of Maximum principles and the method of moving planes. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...
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1answer
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Rellich-Kondrachov

I read an article about the Rellich-Kondrachov embedding theorem in Sobolev spaces: https://en.wikipedia.org/wiki/Rellich%E2%80%93Kondrachov_theorem. Nevertheless, when I checked the refererence in ...
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Reference about the surgery of Ricci flow

I roughly read the Topping's LECTURES ON THE RICCI FLOW. Seemly, there is not introduction about surgery. Seemly,it is enough to deal singularity by blow up. Then, for knowing surgery I read the ...
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1answer
32 views

what is the definition of the space $C([0,T];H^s)$?

What is the definition of the space $C([0,T];H^s)$? Here, we are considering the solutions of a PDE, and $H^s$ is the Sobolev space. My book says we are assuming that a solution lies in this space, ...
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1answer
25 views

What is wrong about this proof for the mean-value theorem for harmonic functions?

Let $\Omega\subset\mathbb{R}^n$ be an open connected domain, and let $u\in C^2(\Omega)$ be a harmonic function on $\Omega$. Then for every ball $B_R(x)=\{y\in\Omega:|x-y|<R\}$ in $\Omega$ we have $...
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21 views

Classification of higher order partial differential equations

For second order linear PDEs we have the classifications parabolic (e.g. heat equation), hyperbolic (e.g. wave equation), elliptic (e.g. laplace equation) and ultrahyperbolic (at least two positive ...
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An algorithm to find the general classical solution to a linear gradient system in partial derivatives

I'm looking for a book where the algorithm to construct the general solution for system $$\nabla u(x,y) = \vec a(x,y)\cdot \nabla v(x,y)$$ is given. Could ypu please advice me some source?
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finite difference time domain grid question

The finite difference time domain method is a finite difference method for solving maxwell equations numerically. There are several pieces to it, but this is the root of my question $H_{i +1/2 , j+1/...
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1answer
486 views

PDE Evans Chapter 7 problem 16

Problem 16 of chapter 7 states Use problem 15 to prove that if $u$ is the semigroup solution in $X=L^2(U)$ of $$ \left\{ \begin{array}{rl} u_t - \Delta u =0 & \text{in } U_T \\ u=0 & \...
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0answers
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Poincaré's inequality proof for $u \in W_0^{1,p}(\Omega)$.

I am trying to prove Poincare's inequality for $u \in W_0^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^n$ is an open bounded set and $1 \leq p < \infty$. This is Poincare's inequality: $||...
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1answer
35 views

solving a partial differential equation (Damped heat equation)

I am trying to solve the below-mentioned PDE that represents a damped heat diffusion in one-dimensional space. I am using the separation of variables to solve it, however when I try to find the ...
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What is necessity for integral to be well-defined in defining solutions?

When trying to give a notion of solutions for differential equations with non-local terms, e.g., integral of unknown functions, to guarantee that the integral is well-defined, i.e., finitely ...
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Showing that ODE is not of Sturm-Liouville form

The PDE $$\frac{\partial u}{\partial t} = D\frac{\partial^2 u}{\partial x^2}-V_0\frac{\partial u}{\partial x}$$ can be separated into two ODEs by the method of separation of variables, and the ODE ...
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Eigenfunctions of $x^2M''+xM'+\lambda M=0$ with $M'(1)=0$ and $M'(L)=0$

If we make the substitution of variables by $z=\ln(x)$ in $$x^2M''+xM'+\lambda M=0$$ then we will get $$M''(z)=-\lambda M(z)$$ We can consider different cases for $\lambda$: Case 1: $\lambda>0$ ...
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second order linear PDE classifications

I am trying to systematize the second order linear PDE's. Does the three types of solutions (hyperbolic, parabolic, elliptic) apply to the all of the three types (linear, semi-linear, quasi-linear) ...
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2answers
23 views

Particular integral of PDE.

The PDE $$\frac{\partial ^{2}u}{\partial x^{2}}+2\frac{\partial^{2}u}{\partial x\partial y}+\frac{\partial^{2}u}{\partial y^{2}}=x$$ Has $1.$ Only one particular integral. $2.$ a particular integral ...
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1answer
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Proof of strong maximum principle for harmonic functions

Let $u\in\mathscr C^2(U)\cap\mathscr C(\bar U)$ be harmonic in the non-empty open and connected set $U\subset\mathbb R^n$. If there exists a Point $x_0\in U$, so that $u$ has a local Maximum at $x_0$, ...
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PDE with Robin like boundary conditions

Solve for $\phi:[0,L]\times [0,\infty) \to \mathbb R$: \begin{align*} \frac{\partial^2}{\partial t^2} \phi(x,t) &= c^2\frac{\partial^2}{\partial x^2} \phi(x,t) \\ \frac{\partial^2}{\partial t^2} \...
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What are complexiton solutions of PDEs?

I have seen written in some article that the solution of PDEs containing more than one transcendental functions is called complexiton solution, is it correct ? what are properties of complexition ...
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461 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: $$\frac{d}{dt}\...
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PDEs - prove continuity of operator

Consider the following nonlinear problem $$ \begin{cases} -div(a(u)\nabla u ) =0 & \text{in $\Omega$} \\ u=0, & \text{on $\partial \Omega$ } \end{cases} $$ We can assume $\Omega$ to be a ...
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2answers
37 views

Numerical analysis of wave equation in polar coordinates:

Is there a simple solution to deal with the problem of radial symmetry when solving a pde numerically. If so can someone provide some references/resources that explain this. Any help would be greatly ...
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70 views

Name of a Sobolev/Poincare type inequality

It is a basic analysis exercise to verify that for $u \colon [0,1] \times [0,\infty) \to \mathbb R$, if $u(0,t) = 0$ for all $t$, then $$ ||u(\cdot,t)||_\infty \leq ||u_x(\cdot,t)||_2,$$ where the ...
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2answers
53 views

Rewriting ODE in terms of a different variable ($z=e^x$)

Given the ODE $$x^2M''+xM'+\lambda M = 0$$ where $1<x<L$, with boundary conditions $M(1) = 0$, $M(L)=0$, we can rewrite it in the Sturm-Liouville form and get $$\left[M'\exp\left(\int\limits_0^L{...
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332 views

A Question about the strong maximum principle in Evans Partial differential equation

Evans stated the strong maximum principle as follows: $U\subset\mathbb{R}^n$ a bounded and open set. If $u\in C^2(U)\cap C(\overline{U})$ is harmonic within $U$. Then, $\max_{\overline{U}}u=\max_{\...
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1answer
49 views

Analytical solution for a Non-linear differential equation $\frac{d^2y}{dt^2} = A\left(\frac{dy}{dt}\right)+B \sin(2Cy)$

Analytical solution for a non-linear differential equation: $\frac{d^2y}{dt^2} = A \left(\frac{dy}{dt}\right)+ B \sin(2Cy)$ A,B are non-zero constants and y (position) is a scalar-value parameter ...
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16 views

Almost the minimal surface equation

I came across the following quasilinear PDE: $$ \nabla \cdot \left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}} \right) = 0. $$ This is almost the minimal surface equation, except that there is a minus sign ...
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1answer
67 views

Explanation of spaces of functions in PDE

Let's consider following equation: The problem $$ \begin{cases} -\operatorname{div}\left( p\left(x\right) \nabla{u} \right) + q(x)u = f \quad\text{... on } \Omega \\ u = h(x) \quad\text{... on } \...
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25 views

Solving n-dimensional first order linear pde

While working on a problem in game theory, I'm stuck at a problem which requires me to solve the following linear first order PDE on $K$ independent variable: $\sum_{k=1}^K(\frac{\partial u}{\partial ...