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

learn more… | top users | synonyms (2)

0
votes
0answers
5 views

Applying different boundary conditions in a quasi Helmholtz problem.

I have tried to solve this exercise from Applied Partial Differential Equations-Richard Haberman, Consider the two-dimensional eigenvalue problem $$ \nabla^2\phi+\lambda\sigma(x,y)\phi=0 $$ ...
1
vote
0answers
10 views

Proving identity of a $C_0$ semigroup

We have a strongly continuous semigroup $\{T(t)\}_{t\ge 0}$ on a Banach space X with a generator $ A:D(A)\subset X\to X$. I need to show that $\displaystyle T(t)x = \sum\limits_{n=0}^{\infty} ...
0
votes
0answers
31 views

Exercise to Heat equation

I am dealing with the following exercise on a generalized heat equation Let $U$ be a open bounded subset of $\mathbb{R}^n$ and $T \in (0, \infty).$ Now, let $u$ be a solution to $$(\partial_t - ...
1
vote
2answers
29 views

Counter example in Sobolev spaces: Is there a standard example for showing that $H^1(\mathbb{R}^2)\not\subset L^\infty(\mathbb R^2)$?

One way to define the Sobolev space $H^s(\mathbb{R}^n)$ is as the completion of the $C^\infty$ functions on $\mathbb{R}^n$ under the norm $$\lVert f \rVert_{H^s(\mathbb{R}^n)}:= \left( ...
2
votes
0answers
15 views

The adjoint operator of the second order partial differential operator.

I'm studying the second order elliptic partial differential equations in the 'Partial Differential Equations, EVANS'. The section 6.2.3 begins with defining the adjoint operator $L^*$ of the operator ...
1
vote
0answers
15 views

Fisher's equation

In question 2)c)i) here https://www.maths.ox.ac.uk/system/files/legacy/3333/b08_10_0.pdf then $$\frac{\partial u}{\partial t} = u(1-u-\beta v) + \frac{\partial^2 u}{\partial \xi^2}$$ and ...
0
votes
0answers
18 views

Mollifiers and smooth path connetction

Generaly it is about smooth path connection. I have a two smooth paths f,g and a points x,y,z of open subset of $R^n$ such that f is smooth path from a to y and g is smooth path from y to z. I define ...
1
vote
1answer
23 views

Solve the following PDE: $(1+\sqrt{z-y-x})z'_x+z'_y=2$

Solve the following PDE: $(1+\sqrt{z-y-x})z'_x+z'_y=2$ given that $z(x,2x)=2x$. I want to explain to you how we were taught to solve these at class, and this method seemed to work with other ...
0
votes
1answer
31 views

Solving the heat equation

I have the heat equation $$\frac{\partial u}{\partial t}=\frac{\partial^2u}{\partial x^2}$$ where $0\leq t<\infty$ and $0\leq x\leq1$ The Boundary conditions are $u(0,t)=0$ and $u(1,t)=1$ with ...
0
votes
0answers
16 views

Bounds for the solution of heat equation using convolutions [on hold]

We know that the solution of the heat equation $$u_t=u_{xx}$$ with initial condition $ u(0,x)=u_0(x)$ is given by $$u(t,x)=(u_o * H_t)(x)$$ where the heat kernel is given by ...
0
votes
0answers
57 views
+100

Behavior of a Solution to Heat equation Compactly Supported in Time

We say that a function $g\in C^{\infty}(\mathbb{R}^{n})$ is of Gevrey class $G^{\sigma}(\mathbb{R}^{n})$ if for each compact $K\subset\mathbb{R}^{n}$, there exist $C,R>0$ such that $$\sup_{x\in ...
0
votes
1answer
30 views

Equivalence of Dirichlet problems. Gilbarg & Trudinger

I do not understand the proof of theorem 11.4 in the book "Elliptic Partial Differential Equations of Second Order" by Gilbarg & Trudinger. The reason is that I do not understand the text right ...
0
votes
1answer
33 views

If $u \in H^1(0,\infty)$, does $u_x(\infty) = 0$?

Let $u \in H^1(0,\infty)$. Then is it true that $u'(x) \to 0$ as $x \to \infty$? I am wondering by Green's formula/IBP in this setting; do I get something like $$\int_0^\infty u_{xx}v = ...
3
votes
2answers
43 views

How to address multiple cases in this BVP? (Laplace equation in quarter-annulus)

The original problem: $$\nabla^2 u =0 \ \ \ \ for \ \ \ 0<a<r<b\ \ \ ,\ \ \ 0<\theta <\frac \pi 2$$ $$u(r,0)=0,\ \ u(r,\frac \pi 2)=f(r),\ \ u(a,\theta)=u(b,\theta)=0$$ My ...
0
votes
1answer
45 views

Exponential decay estimate involving $C_{0}$-semigroup and the principle eigenvalue

Consider the heat equation defined as $$\begin{cases} \partial_{t}u(x,t)=\Delta_{x}u(x,t) &\mbox{ } t>0\mbox{, } u\in U \\ u(x,t)=0 &\mbox{ } t\ge 0\mbox{, } x\in\partial U \\ u(x,0)=g(x) ...
1
vote
1answer
17 views

Solving PDE using normal mode

Given a linearised PDE $u_t=u_{xx}+\mu u$ where $x\in[0,1]$. A hint given is $u=V(x)\exp(st+ikx)$, where $s$ can be complex and $k$ is real. When I substituted into the PDE, I get ...
1
vote
0answers
90 views

How to find an ODE with prescribed terminal values? [on hold]

Let us consider an ODE $$\frac{dx_t^y}{dt}=g(x_t^y),$$ where y is the initial condition i.e. $x_0^y=y$. Now, given a function $f$ (increasing and smooth) is it possible to find $g$ (i.e. an ODE) ...
0
votes
1answer
32 views

How to solve a particular PDE (which reminds of heat equation)

I suddenly ran into this equation: Let $u:[a,b]\times \mathbb{R} \rightarrow \mathbb{R}$ be a function satisfying: $$\partial_t u = -u' + \frac{1}{2}u''$$ with bountary conditions $u(0,x)=g(x)$ where ...
4
votes
1answer
48 views

Good sources to learn about Geometric Analysis

So far the topics that have captured my interest the most are analysis and geometry. I want to learn a bit more about geometric and global analysis. There seems to be no shortage of geometry books but ...
2
votes
0answers
22 views

Resolvent estimate of hyperbolic Laplacian

Consider the Laplacian $-\Delta$ on the hyperbolic space $\mathbb{H}^n$. For $\lambda \in \mathbb{C} \setminus [0, \infty)$, do we have resolvent estimates of the form $$\Vert (-\Delta - \lambda ...
2
votes
3answers
22 views

inhomogeneous pdes by separation of variables

This is the problem: $u_t=c^2 u_{xx}+g(x,t),0<x<l,\text{ and } t>0$ $u(0,t)=0=u(l,t)$, $t\ge 0$ $u(x,0)=f(x)$ I have trouble passing this problem to homogeneous form
3
votes
1answer
48 views

Maximum principle-estimation

Let $S=\{x \in \mathbb{R}^2 \mid |x| <1\}$. Using the maximum principle I have to show that the solution of the problem $$-\Delta u(x)=f(x), x \in S \\ u(x)=0, x \in \partial{S}$$ satisfies the ...
0
votes
1answer
21 views

A problem with a simple PDE

My task is to find a general solution to such a PDE: $xu_x+yu_y=0$. My approach: Such an equation is constant on its characteristics. So at first I want to find out what they look like. ...
0
votes
0answers
16 views

Poisson equation with nonlinear Neumann conditions

Let $\beta:\mathbb{R}\to\mathbb{R}$ be a Lipschitz function such that $0<a\leq \beta^\prime\leq b$, for some constants $a,b$. Give the weak formulation of the problem \begin{equation} ...
1
vote
1answer
57 views

heat conduction problem

Find the solution of the heat conduction problem $U_{xx} =4U_t , 0 < x < 2, t>0;$ $U(0,t)=0, U(2,t)=0, t>0$; $U(x,0)=2\sin(\frac\pi2x)-\sin(\pi x) + 4\sin(3\pi x), 0 \le x \le 2 $ ok ...
0
votes
1answer
24 views

Laplacian operator on $L^2(\Omega)$?

Let $\Omega\subseteq \mathbb R^n$ be an open subset and $\displaystyle \Delta:=-\sum_{j=1}^n D_j^2$ be the Laplacian operator. I have some questions concearning this operator: $(i)$ Does it map ...
1
vote
0answers
10 views

Nonlinear Schrodinger equation - modified

I have a nonlinear Schrodinger equation: $ia_1\dfrac{\partial A}{\partial x}-a_2\dfrac{\partial^2 A}{\partial t^2}+|A|^2A=0$, $A$ is the amplitude and the above equation governs the slow modulation of ...
3
votes
0answers
33 views

$\Box u= | u |^2 u$ global solution in $C^\infty$

Let $u_0, u_1 \in C^\infty( \mathbb{R},\mathbb{R}^3)$. Consider the cubic defocusing NLW $$(\ast)\begin{cases} \Box u= |u|^2 u \\ (u,\partial_t u) \restriction_{t=0} = (u_0,u_1) ,\end{cases}$$ where ...
0
votes
1answer
15 views

Mean Value Property/Bulk Mean Value Property: conceptual issues (PDEs)

So my lecturer proves the mean value property which poses that under a few conditions on $u$ and $\overline{B_r(x_0)} \subset \Omega \subset \mathbb{R}^n$: $$u(x_0) = \dfrac{1}{\text{Vol}(\partial ...
-1
votes
0answers
15 views

PDE with Robin boundary condition [on hold]

I need any advice (sugesseted book or a method etc..) to solve it
4
votes
1answer
75 views

Can I combine the wave and heat equations?

I have this equation $$\frac{\partial^2u}{\partial x^2} = 2\frac{\partial u}{\partial t} + \frac{ \partial^2u}{\partial t^2}$$ Is it possible for me to use both the wave and heat equations to solve ...
0
votes
1answer
12 views

Transforming the diffusion equation.

We need to transform the diffusion equation : $u_t = k(u_{xx} + u_{yy})$ into axisymmetric form : $u_t = k(u_{rr} + \frac{u_{r}}{r})$ , I first converted the laplace equation $u_{xx} + u_{yy}$ = 0 ...
0
votes
1answer
25 views

Proof of the Harnack inequality

Let $\Omega\subseteq\mathbb{R}^n$ be a domain, $\Omega'\subset\subset\Omega$ be a domain and $u\in C^0(\overline{\Omega})$. Suppose we know $$\sup_{\Omega'}u\le 3^n\inf_{\Omega'}u\tag{1}$$ if ...
1
vote
1answer
17 views

Deciding when to use $\mu^2$ or $-\mu^2$ in separation of variable (PDE'S)

I have the following question Consider the two-dimensional PDE on $u = u(x,y)$ $$u_{xx}-u_{yy}=0$$ $$u(x,0)=\phi(x)$$ $$u_y(x,0)=0$$ where $\phi:\mathbb{R}\rightarrow \mathbb{R}$ is a function ...
0
votes
2answers
20 views

Show that lamda is greater than or equal to zero for a sturm liouville problem

To show that this problem can be put into S-L form for an eigenvalue problem, Observe that The S-L form is of $$\text{p'(x)}\phi _x\text{+p(x)}\phi _{\text{xx}}\text{+q(x)$\phi $+$\lambda \phi ...
0
votes
1answer
32 views

Need help with this S-L problem

Consider the problem: $$\frac{\text{$\delta $u}}{\text{$\delta $t}}=\frac{\delta ^2u}{\text{$\delta $x}^2}\text{-u+x(1-x)}$$ The IC are given as: $$u(0,t)=1$$ $$u_x\text{(1,t)=0}$$ ...
2
votes
1answer
25 views

Partial Differential equation with laplacian and gradient.

Suppose that $\Omega\subset \mathbb{R}^n$ is open and bounded. Let $u\in C^2(\Omega)\cap C^0(\bar\Omega)$ is a solution for the equation $\triangle u+\sum_{k=1}^na_ku_{x_k}+c(x)u=0$ where ...
1
vote
1answer
9 views

Vector Identity that appears in the Poisson Kernel of the $B(0,1)$

Given that $y\in \partial B(0,1)$ how do i get that $|x|^2|y-\frac{x}{|x|^2}|=|y-x|$? It may be a silly question but I don't see why? Thanks!
2
votes
0answers
28 views

First order partial differential equation.

We need to solve the given first order partial differential equation : $(y-xu)u_x$ + $(x+yu)u_y$ = $x^{2} + y^{2}$ . I tried this : $\frac{dx}{y-xu}$ = $\frac{dy}{x+yu}$ = $\frac{du}{x^{2} + ...
0
votes
1answer
40 views

Radial Solution to the wave equation in terms of odd functions

Show that if $f \in C^3_c (\mathbb{R})$ is an odd function then for $|x|=r$ define \begin{equation} u(x,t) = \frac{f(r+t)+f(r-t)}{2r} \end{equation} then $u$ extends as a $C^2$ function that solves ...
2
votes
1answer
26 views

Inversion map is a Conformal map

I'm studying PDE by Evans book and I need to show that the inversion map $f:\mathbb{R}^n-\{0\}\to \mathbb{R}^n$, defined by $$f(x)=\frac{x}{\|x\|^2}$$ is conformal. So I have a hint, show that ...
0
votes
0answers
14 views

GODOUNOV scheme

Help me, i can't understand how to decribe godounov scheme with discontinuous initial conditions for nolinear PDE , is it related with Riemann problem ? \begin{equation} u_t+f(u)_x=0 \end{equation} ...
-1
votes
0answers
12 views

classification of first order PDE hyperbolic and parabolic

how i can classify a first PDE \begin{equation} u_t+cu_x=0 , \end{equation} is it hyperbolic or parabolic? . with details .
0
votes
1answer
24 views

Eigenfunction and their orthogonality with respect to the weight function

The Eigenfunction and their orthogonality with respect to the weight function $$\sigma$$ is defined as $$\int _a{}^b\phi _n\text{(x)}\phi _m\text{(x)$\sigma $(x)dx=0}$$. Given that I have some ...
-4
votes
0answers
28 views

sde problem, may require Ito? [closed]

Given $dU_t = -\gamma U_t \, dt + dX_t$; How do I solve this equation for $U_t$?
1
vote
1answer
44 views

Computing Fréchet derivative

I am reading Methods in Nonlinear Analysis by Kung-Ching Chang and having trouble in obataining a Fréchet derivative in the text. For those who has the book, it is on page 37, which concern Euler ...
1
vote
3answers
29 views

Fourier series sketching

Whenever I am asked to draw fourier series, is it correct to first draw the function on the interval first (in this case 0<= x < pi), then extend the the graph to the desired interval ...
0
votes
0answers
35 views

PDE complex boundary condition

My attempt to this question was setting T''-lambda T=0 and try lambda=0, >0 and <0. However, I do not seem to have sufficient information to determent which case have non-trivial solution ( since ...
0
votes
0answers
16 views

How do I solve this system of PDEs numerically?

Suppose that I have a system of PDEs of the following form: \begin{eqnarray} (\frac{\partial}{\partial x} - i\frac{\partial}{\partial y}) f(x,y) = F(f,g,h) \\ (\frac{\partial}{\partial x} - ...
1
vote
2answers
39 views

Heat equation-unicity

We have the folllwing problem: $\begin{cases} & \dfrac{\partial u}{\partial t} = k \dfrac{\partial^2 u}{\partial x^2}, 0 < x < l, t > 0\\ & u(0,t)=0,\\ & \dfrac{\partial ...