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

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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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0answers
18 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 ...
1
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1answer
17 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 $\mu_k=\mu$...
1
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1answer
23 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
23 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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12 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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0answers
11 views

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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16 views

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
16 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
10 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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9 views

Wave equation, existence and solution for inital values in $C^{\infty}$ using partition of unity

here is the problem: http://prntscr.com/bm25rb I need to proof existance and uniqueness for wave equation with inital values in $C^\infty(\mathbb{R^d})$. This problem is from Jeffrey Rauch, PDE, 3. ...
2
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1answer
15 views

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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19 views

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
23 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 $...
2
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1answer
113 views
+50

Help with a nonlinear partial differential equation

let : $$\frac{\partial f}{\partial x}=f _{x}\;,\;\;\frac{\partial f}{\partial t}=f _{t}\;,\;\;\frac{\partial}{\partial t}\frac{\partial f}{\partial x}=f_{tx}\;, \;\;\ \frac{\partial}{\partial x}\frac{\...
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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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11 views

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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55 views
+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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12 views

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/...
2
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1answer
485 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 & \...
2
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0answers
18 views

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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13 views

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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20 views

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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0answers
29 views

Find solution of Laplace equation

Hey I need help with these example: Solve boundary problem on $\mathbb{R}^{+} \times \mathbb{R}^{+}$ \begin{equation*} \left\{ \begin{array}{l} \Delta u = 0 \\ u(0,.)=0 \\ u(.,0)= f\end{array}\right....
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0answers
19 views

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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9 views

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 ...
2
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1answer
1k views

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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0answers
36 views

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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0answers
20 views

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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1answer
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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14 views

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
36 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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69 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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2answers
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_{\...
0
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1answer
48 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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0answers
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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 ...
1
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1answer
28 views

Laplace Operator Times Function

I'm just going through some proofs of a PDE book and have a question about one of them. It is stated that: $$ \int_U w \Delta w \text{ d}x = -2 \int_U |Dw|^2 \text{ d}x $$ Where $w$ is a solution of ...
3
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2answers
66 views

Temperature/heat equation

I solved this problem $$\left\{\begin{array}{ll} u_{t}=ku_{xx}, & x\in(0,1), t>0 \\ u(0,t)=2, u(1,t)=3, & t>0 \\ u(x,0)=x^{2}+x+2, & x\in(0,1) \end{array}\right.$$ and I got this $$u(...
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Enquiry regarding a step in the proof of Theorem 13 on page 35 of Evans' PDE second edition.

In the proof of symmetry of Green's function on page 36, I have: On the other hand, $v(z) = \Phi(z-x)-\phi^x(z)$, where $\phi^x$ is smooth in $U$. Thus $$\lim_{\epsilon \to 0} \int_{\partial B(x,\...
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10 views

Bounds on flux integrals

What are some handy upper bounds for surface integrals (and their proofs)? Specifically, suppose $f$ is a bounded function on a surface $S$. Do we have $$ \int_{\partial S} F \cdot n \; \mathrm{d}S \...
0
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1answer
28 views

Nonhomogeneous heat equation [closed]

I really don't know how to start to solve it: $$\left\{\begin{array}{ll} u_{t}=ku_{xx}-\lambda^{2}u, & x\in(0,\ell), t>0 \\ u(0,t)=u(\ell,t)=0, & t>0 \\ u(x,0)=h(x), & x\in(0,\ell) \...
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0answers
25 views

PDE with analytical solution in some cases

I studied PDE, if a classification of them means studying. I haven't studied solving methods for PDE so I'd like to have an elementary answer if it is possible. I got this one: $$ \frac{\partial \...
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0answers
36 views

Affine projection of polynomial

Please i need your help to solve the following problem: Let $V(Y)$ belong to $\mathbb{R}[Y_1,...,Y_d]$. Prove that one can find an affine change of coordinate $Y=AX+B, (X_1,..,X_d)$ on $\mathbb{R}^d$...
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26 views

Textbook/monograph for microlocal analysis

I want to grasp the theory of microlocal analysis and apply this theory to some PDEs in $R^n$. But most textbooks I found put much priority on manifolds. Sadly, I know little about them and don't ...