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Question 1

0votes

0answers

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Sufficient condition at infinity for uniqueness of a non-linear parabolic PDE

Let me consider the harmonic map heat flow from $\mathbb R^2$ onto $S^2 \subset \mathbb R^3$, given by\begin{equation}\begin{cases}\partial_t u = \Delta u + |\nabla u|^2 u & \text{in } \mathbb ...

Falcon

4,536

asked15 hours ago

Question 2

1vote

2answers

109views

Problem with solving $U_t=U_{xx}+\sum_{n=1}^\infty \frac{e^{-nt}}{n!} \sin (nx)$

Here is the problem:Solve the following P.D.E with the given initial and boundary conditions:$$U_t=U_{xx}+\sum_{n=1}^\infty \frac{e^{-nt}}{n!} \sin (nx), \quad U(0,t)=U(\pi,t)=U(x,0)=0$$So, I ...

User

8,489

asked2 days ago

Question 3

0votes

0answers

47views

Can we apply the principle of superposition on the initial condition for a P.D.E.?

The problem is:Solve the P.D.E. $U_t=U_{xx}$ with the following initial and boundary conditions:$$U(0,t)=U(\pi,t)=0,\quad\text{and}\quad U(x,0)=u_0-u_0\sin x$$For the given fixed boundary ...

User

8,489

askedNov 25 at 19:22

Question 4

0votes

2answers

71views

Having problem understanding the answer to the $1$D Heat Equation with the initial condition $U(x,0)=u_0-u_0\sin x$

Here is a problem and answer from my notes, where I have issues understanding parts of the provided answer.Problem:Solve the P.D.E. $U_t=U_{xx}$ with the following initial and boundary conditions:$...

User

8,489

askedNov 25 at 16:47

Question 5

0votes

1answer

44views

How to show that the solution of wave equation uniformly convergent to the initial data?

Picture below is from Evans' PDE. I want to show the red line.First, $g$ is smooth, with compact support. Besides, I treat$$\lim_{t\rightarrow 0} \frac{e^{-s^2/4t}}{(4\pi t)^{1/2}} = \delta(s)\tag{...

Enhao Lan

6,956

askedNov 25 at 11:50

Question 6

0votes

0answers

37views

How to perform the singular perturbation analysis for this reaction-diffusion PDE?

Suppose I want to determine the transient solution of the following IBVP involving a nonlinear reaction-diffusion PDE:$\partial u(x,t)/\partial t = D \partial ^2 u(x,t)/\partial x^2 - kR(u(x,t)) $$u(...

Leszek

121

askedNov 25 at 11:41

Question 7

1vote

0answers

16views

Boundary conditions of weak solutions in $H^{2}_{0}$

I'm currently working through Chapter 6 of Partial Differential Equations by Evans. I'm now attempting some exercises at the end of the chapter but I'm a little confused about the definition of a weak ...

mmga

53

askedNov 24 at 18:15

Question 8

0votes

0answers

31views

Fourier transform and the acoustic tensor

Let $C$ the fourth-order tensor of elastic constants which can be seen as as a linear transformation from Sym into Sym (matrices). We denote the action of $C$ on a symmetric matrix $A$ as $C[A]$. ...

Guillermo García Sáez

8,158

askedNov 24 at 12:49

Question 9

1vote

1answer

116views

+50

Huygens' principle in Evans' PDE

Pictures below are taken from paragraph 4.3.2 of Evans'Partial Differential Equations. I want to prove the result underlined by the second red line.Since $\text{spt} \, {\bf g} \subset B(0, R)$, by ...

Enhao Lan

6,956

askedNov 24 at 11:25

Question 10

0votes

1answer

113views

How should I find the general solution of the partial differential equation $u_{xx}+4u_{yy}=0$ and what's the correct answer to this given PDE?

Find the general solution of the partial differential equation $u_{xx}+4u_{yy}=0$.Here's my work:When it comes to finding the general solutions of this type of partial differential equations, does ...

Enigma

61

askedNov 24 at 0:26

Question 11

0votes

0answers

47views

Proving the solution of heat equation is $C^2_1(\mathbb R^n \times (0,\infty))$

I'm trying to prove part 1 in the following theorem about heat equationwhere (2.28) and $\mathcal H$ are given byIn the following proof, why can we differentiate the integral after a change of ...

HIH

663

askedNov 23 at 16:35

Question 12

0votes

0answers

37views

Spectral decomposition of the Neumann-Laplacian

I'm currently trying to find a spectral decomposition for the Laplace Operatorsuch that the eigenfunctions are a part of $H^2_N$ := $\{u \in H^2 \,|\,\frac{\partial u}{\partial \nu} = 0 \,\,\text{on} ...

Zaza

1

askedNov 23 at 13:26

Question 13

3votes

2answers

93views

Problem with understanding the technique used to calculate $B_n=\frac{2u_0}{\pi}\int_{x=0}^\pi\sin(nx)(\sin x+\sin 3x)dx$ for heat equation.

This is a problem and answer from my notes:Solve heat equation for $l=\pi$ and with the initial and boundary conditions:$U(0,t)=U(\pi,t)=0;\;\;U(x,0)=u_0(\sin x+\sin 3x)$The answer to the above ...

User

8,489

askedNov 22 at 20:10

Question 14

1vote

1answer

91views

The Gateaux derivative and the Jacobian

QuestionIs it accurate to claim that the Gateaux derivative is the continuous linear operator represented by the Jacobian matrix? That is, the Jacobian is the finite dimensional equivalent of the ...

Jared

267

askedNov 21 at 13:58

Question 15

1vote

1answer

73views

A question concerning the duality bracket of $W^{-1,p'}(\Omega)$ with $W_0^{1,p}(\Omega)$ relevant to $L^2(\Omega)$ inner product

Let $\Omega$ be an bounded open set in $\Bbb R^d$, $1<p<\infty$, $p'$ be the Holder conjugate of $p$,let $u\in L^2(\Omega) \cap W^{-1,p'}(\Omega)$ (the intersection is undertood by embedding ...

Enrui Zhang

13

askedNov 20 at 15:43

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Questions tagged [partial-differential-equations]

Sufficient condition at infinity for uniqueness of a non-linear parabolic PDE

Problem with solving $U_t=U_{xx}+\sum_{n=1}^\infty \frac{e^{-nt}}{n!} \sin (nx)$

Can we apply the principle of superposition on the initial condition for a P.D.E.?

Having problem understanding the answer to the $1$D Heat Equation with the initial condition $U(x,0)=u_0-u_0\sin x$

How to show that the solution of wave equation uniformly convergent to the initial data?

How to perform the singular perturbation analysis for this reaction-diffusion PDE?

Boundary conditions of weak solutions in $H^{2}_{0}$

Fourier transform and the acoustic tensor

Huygens' principle in Evans' PDE

How should I find the general solution of the partial differential equation $u_{xx}+4u_{yy}=0$ and what's the correct answer to this given PDE?

Proving the solution of heat equation is $C^2_1(\mathbb R^n \times (0,\infty))$

Spectral decomposition of the Neumann-Laplacian

Problem with understanding the technique used to calculate $B_n=\frac{2u_0}{\pi}\int_{x=0}^\pi\sin(nx)(\sin x+\sin 3x)dx$ for heat equation.

The Gateaux derivative and the Jacobian

A question concerning the duality bracket of $W^{-1,p'}(\Omega)$ with $W_0^{1,p}(\Omega)$ relevant to $L^2(\Omega)$ inner product

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