1
vote
0answers
20 views

Method of PDE solution by Fourier transform

In Rudin's Functional Analysis (Chapter 7, exercise 17), Rudin claims that for $n=1$ or $2$, if $u$ is a distribution on $R^n$ with compact support $K$, whose Fourier transform $\hat{u}$ is a bounded ...
0
votes
0answers
33 views

Alternative proof of Heisenberg uncertainty principle - step 1

I'm student of physics having problem with pdes. I have rapidly decreasing function in $\mathbb{R}^d$ st $\int{|u|^2}dx=1$ and function $v=e^{i\langle\psi\rangle x}u(x+\langle x\rangle)$, where ...
1
vote
0answers
27 views

Fourier transform for pde

I solved the following PDE: $u_{t}-(u_{t})_{x}+(b+1)uu_{x}=bu_{x}u_{xx}+uu_{xxx}$ numerically, using Fourier Transform method. For this i wrote it in the following way: ...
0
votes
1answer
21 views

$f, \hat{f} \in L^{1}(\mathbb R) \implies \widehat {\text{Re}(f)}, \widehat{\text{Im}(f)} \in L^{1}(\mathbb R)$?

Let $f:\mathbb R \to \mathbb C$ such that $f(x)= (f_{1}(x), f_{2}(x))$; where $f_{1}(x)=\text{Re}(f(x))=\text{the real part of} \ f $ and $f_{2}(x)=\text{Im}(f(x))= \text{ the imaginary part of} \ ...
3
votes
0answers
27 views

Anyone help me with this PDE using Fourier Transform?

I have this: $$\frac{\partial c}{\partial t} + p\frac{\partial c}{\partial z}+\lambda p\frac{\partial^{2} c}{\partial z\partial t}-\frac{\partial^{2} c}{\partial z^2}=0\quad(1)$$ $$c(z,0)=\delta(z)$$ ...
0
votes
0answers
20 views

properties of frequency- decomostion operator $\square_{k}^{\sigma}=\sum_{|\ell|_{\infty}\leq 1}\square_{k}^{\sigma}\square_{k+\ell}^{\phi}$?

Let $\rho \in S(\mathbb R^{n})= \text{Schwartz space}, \ \rho:\mathbb R^{n}\to [0,1]$ be smooth radial function verifying $\rho(\xi)=1$ for $|\xi|_{\infty}\leq \frac{1}{2}$ and $\rho(\xi)=0$ for ...
0
votes
1answer
19 views

$\int_{|x|<t} |\mathcal{F}^{-1}f(x) |dx \leq C t^{\frac{n}{2}}\|f\|_{L^{2}}$?

Let $f\in L^{2}(\mathbb R^{n}).$ Fix $t>0,$ My Question:How to show, $\int_{|x|<t} |\mathcal{F}^{-1}f(x)| dx \leq C t^{\frac{n}{2}}\|f\|_{L^{2}}$ ? [We note $\mathcal{F}$ denotes the ...
1
vote
1answer
39 views

Solve PDE with the Fourier transform

I have a problem with solving PDEs with the fourier trasform method when the function not depends only on x and t but also on the y variable. In particular, when I have to solve this equation ...
0
votes
0answers
8 views

von Neumann analysis

When performing a von Neumann analysis on a PDE, we end up getting an expression in Fourier-space on the form (k is a "wave number", as in $\exp(ikr)$) $$ A_{n+1}(k) = G(k)A_{n}(k) $$ where $G$ is our ...
1
vote
1answer
54 views

Forced wave equation question?

I'm studying for my PDEs midterm and trying to do practice problems. I'm really not sure how to do this question - I've never seen anything like it. Thanks in advance for your help. Solve the ...
0
votes
0answers
11 views

How to use $(|u|^{2}u - |v|^{2}v)(s)= (u-v)|u|^{2}(s)+ v(|u|^{2}-|v|^{2}) (s)$; to prove contraction in Banach sapace $C([0, T];M^{p,1})$?

(For details of this question you may see the paper,(proof of theorem 1, page.9), MR2506839, Local well-posedness of nonlinear dispersive equations on modulation spaces; Bull. Lond. Math. Soc. 41 ...
0
votes
1answer
19 views

composition and commutators of Fourier multiplier operators

I am working with some Fourier multiplier operators arising in study of a PDE. I have a general question: Suppose $S$ and $T$ be two Fourier multiplier operators (on some space) with multipliers $m_1$ ...
0
votes
1answer
34 views

A Specific Example about Parabolic PDE

I am solving a PDE numerical problem. And I have already had a algorithm. However, it seems to be hard to find a specific example to test my solution. Could you please give me one? The equation ...
0
votes
0answers
10 views

decay of coefficients in the expansion into Bessel functions

Let us consider the generalized Fourier expansion into Bessel functions, as illustrated in the Wikipedia page: http://en.wikipedia.org/wiki/Generalized_Fourier_series. Let $J_0 (r)$ be the 0th ...
3
votes
0answers
47 views

$f, f|f| \in L^{1} (\mathbb R) \cap C_{0} (\mathbb R) \implies |f| \in H_{1} (\mathbb R)$?

Suppose $f \ \text{and} \ f|f|\in L^{1}(\mathbb R).$ Then, clearly, $|f|\in L^{2}(\mathbb R)$ and therefore by Plancheral theorem, we get, $\widehat{|f|} \in L^{2}(\mathbb R).$ Also, assume, $f, ...
1
vote
0answers
33 views

When it is possible to integrate an oscillatory integral?

Let $\phi(x,t;\theta) = \theta f(x,t)$, $\theta \in \mathbb R$, $x \in \Omega \subset \mathbb R^k$ ($\Omega$ is a domain), $t \in (0,+\infty)$, be a phase function and define an oscillatory integral ...
0
votes
1answer
44 views

Solving the heat equation using Fourier series; specific questions

Like this previous question, Solving the heat equation using Fourier series, I too am reading the same wikipedia article, ...
0
votes
0answers
12 views

2D Wave propagating in duct with height change

Suppose that we have a two - dimensional rigid wall duct cosisting of two semi - infinite regions $x<0,\ 0\leq y\leq a\ $ and $x>0,\ a<y\leq b$ (this means exactly that there is a height ...
0
votes
0answers
28 views

Fourier Sine Transform Identity Relation through Integration by Parts

This is purely for my own recreational interest. I've spent the last few days trying to demonstrate to myself that the Fourier Sine Transform and the inverse Fourier Sine Transform return their ...
1
vote
1answer
53 views

Inhomogeneous diffusion equation and initial conditions inversion

While working on a physical diffusion process, I encountered the following Fokker-Planck equation $$ \frac{\partial F}{\partial t} = D (x) \frac{\partial^2 F}{\partial x^2} \tag1$$ where $D(x) > ...
1
vote
1answer
150 views

Inhomogenous Heat equation using fourier transform

Is it possible to transform the inhomogenous heat equation: $ u_t = u_{xx} + h(x,t)$ for $ - \infty < x< \infty , t > 0$ and $u(x,0) = 0$ to the integral equation: $$\int_0^t ...
0
votes
2answers
76 views

Using fourier analysis in order to solve differential equations.

http://www.enm.bris.ac.uk/admin/courses/EMa2/Lecture%20Notes%2009-10/LSPDE5.pdf The above PDF teaches us the separation of variables method. However, there are some things I dont understand, that I ...
0
votes
2answers
57 views

Laplace's equation in polar coords

Question: Suppose that the function u(r, $\phi$) satisfies Laplace’s equation for plane polar co-ordinates (r, $\phi$) i.e. $$ ∇^2u = \frac{1}{r} \frac{∂}{∂r}(\frac{r∂u}{∂r}) + ...
3
votes
1answer
100 views

Does anyone know a solution to this PDE?

I ran into this PDE: $$\frac{\partial y(x,t)}{\partial t} = A\,x^{\gamma-1} \left(\frac{\partial y(x,t)}{\partial x} + x \, \frac{\partial^2 y(x,t)}{\partial x^2}\right)$$ If it helps in any way, ...
0
votes
1answer
19 views

von Neumann stability analysis for irregular meshes

All the litterature I have come across about the von Neumann stability analysis is performned on regular grids. Can the analysis be performed analytically on irregular grids, or does it have to be ...
0
votes
1answer
40 views

Why is $\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}\cos\left[ a\xi \right]\hat{f}(\xi)d \xi = f(a)$?

Background: We are looking at the wave equation on $\mathbb{R}^n$ via the Fourier transform. If $u(x,t)$ solves $\Delta u = u_{tt}$ in $\mathbb{R}^n$, with $u(x,t) = f(x)$ at $t=0$ and $u_t(x,t) = ...
1
vote
0answers
35 views

Estimative in Hs spaces

Consider $W(t)$ defined by $\widehat{W(t)} = (2 \pi)^{\frac{-n}{2}} \cos (t|\xi|)$. Consider $g \in H^{s-1} (R^{n-1})$. Show that: $$ \| W(t ) *g\|_{H^s} \leq c \sqrt2 (1 + |t|)\| g\|_{H^{s-1}}$$ ...
0
votes
2answers
92 views

Differential equation for heat equation

Consider the equation $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2} + a\frac{\partial u}{\partial x}$$ for a function $u(x,t)$ with initial value $$u(x,0)=f(x).$$ Let ...
0
votes
0answers
20 views

estimate for a function in the H^s space

Consiser $f \in H^s(R^n)$ $(s> n/2)$. Show that : $$ |f(x) - f(y)| \leq \gamma_s(|x-y|) || f||_s , \ \ \forall x,y \in R^n .$$ Where $\gamma_s(|x-y|) = 2 (2 ...
3
votes
1answer
35 views

Compare mixed derivatives to laplacian

Suppose $u,f$ periodic and smooth in $Q=[0,1]^n$ such that $\Delta u=f$. Show that for each $i,j$, $$\int_Q \left| \frac{\partial^2 u}{\partial x_i \, \partial x_j} \right|^2 \leq C \int_Q |f|^2.$$ ...
2
votes
2answers
51 views

$C$ such that $\sum_{k\in \mathbb{Z}^n} k_i^2k_j^2|a_{ij}|^2 \leq C \sum_{k\in \mathbb{Z}^n} \|k\|^4|a_{ij}|^2$

More generally, can we find $C_n>0$ such that $$\sum_{k\in \mathbb{Z}^n} k_i^2k_j^2|a_{ij}|^2 \leq C \sum_{k\in \mathbb{Z}^n} \|k\|^2|a_{ij}|^4$$ for all $\{a_k\}_{k\in \mathbb{Z}^n} \in ...
2
votes
1answer
93 views

Wave equation 1D inhomogeneous Laplace/Fourier Transforms vs Green's Function

I am trying to solve the following 1D inhomogeneous wave equation. Forgive me if I a miss any rigorous mathematical concept. $$ \frac{\partial^2 u}{\partial x^2} - \frac{1}{c^2}\frac{\partial^2 ...
3
votes
0answers
60 views

An estimate For the Laplacian semi-group

Let $S(t)$ be the semi-group generated by the Dirichlet Laplacian in $L^2(0,1)$, which is given, for $y\in L^2(0,1)$, by $$S(t)y=\displaystyle\sum_{n=1}^\infty e^{-n^2\pi^2 t} \langle y,\sin(n\pi x) ...
1
vote
2answers
65 views

Comparison between Bessel's coefficients

The spatial solution is written as $$\Phi_k(r) = r^{1-\frac{d}{2}} \left(c_1 J_{1-\frac{d}{2}}(k r) + c_2 Y_{-1+\frac{d}{2}}(kr)\right).$$ In the case $d=3$, the solutions can be written as ...
2
votes
1answer
32 views

The conservation of a critical non-linear dispersion equation.

Consider the non-linear problem $$ \frac{1}{i}\frac{\partial{u}}{\partial{t}}-\frac{d^2u}{dx^2}=\sigma|u|^{\lambda-1}u$$ $$u(x.0)=f(x)$$ Suppose that $u$ is a smooth solution that decays ...
3
votes
1answer
84 views

Schwartz kernel theorem for induced distributions…

I'm studying periodic pseudo-differential operators on torus and I have a question concearning the Schwartz kernel theorem: If $A:C^\infty(\mathbb T^n)\rightarrow \mathcal{D}^{'}(\mathbb T^n)$ is a ...
3
votes
2answers
154 views

Laplace equation Fourier transform

I tried to solve but I didn't. Please help me. $$u_{xx} + u_{yy} = 0$$ $$-\infty < x < \infty$$ $y>0$ , $u_{y}(x,0)=f(x)$. Show that $$u(x,y)= \frac{1}{2 \pi} ...
10
votes
1answer
289 views

Solving Poisson's equation for $\varrho(\mathbf{r}) = \sigma \cos\left(\frac{2 \pi}{L} x\right) \, \delta(y)$

Problem statement I took an exam, where I had the following task: Determine the electrostatic potential for the charge distribution $$\varrho(\mathbf{r}) = \sigma \cos\left(\frac{2 \pi}{L} x\right) ...
2
votes
2answers
95 views

Prove $\varphi\in\mathcal{S}(\mathbb R^n)$ if and only if the following inequality holds..

I need some help for showing the following result: Let $\varphi\in C^\infty(\mathbb R^n)$. Then $\varphi\in \mathcal{S}(\mathbb R^n)$ if and only if for all $\alpha\in\mathbb N^n$ and $N\geq 0$ there ...
-3
votes
1answer
52 views

How to prove $\mathrm{supp} ~ R(t)\in \{|x|<t\}$

Let $R(t)$ define as $$R(t):=F^{-1} \left(\frac{\sin |\xi|t}{|\xi|} \right),$$ how to show that $$\mathrm{supp} ~ R(t)\in \{|x|<t\},$$ where $F$ is the Fourier transform.
1
vote
1answer
32 views

What is the definition of $H^{-k}(\mathbb{R}^n)$ and its norm?

What is the definition of $H^{-k}(\mathbb{R}^n)$ and its norm? How can I understand the fact $$\|f\|_{H^{-k}(\mathbb{R}^n)}=\|(I-\triangle)^{-k}f\|_{H^{k}(\mathbb{R}^n)}.$$
2
votes
1answer
91 views

Discontinuity of the Laplacian

I'm searching for a proof of the following fact: For every $d$ there is a sequence $\{\phi_n\}_{n\in \mathbb{N}} \subseteq C_c^{\infty}(\mathbb{R}^d)$ that converges to $0$ in $L^2(\mathbb{R}^d)$, ...
3
votes
1answer
151 views

Solution of a differentiation in integral form

How will I get the solution in the form of integration $$ \phi (0,t)=\frac{R^{3}}{2}\frac{A}{\sqrt{\pi }}\int_{0}^{\infty }k^2e^{-R^{2}k^{2}/4}\cos (\sqrt{k^2+2} t)\ dk. $$ from the equation, when ...
1
vote
2answers
70 views

Dirichlet problem on $[0,1] \times [0, \pi]$

Let $\Omega := [0, 1] \times [0,\pi]$. We are searching for a function $u$ on $\Omega$ s.t. $$ \Delta u =0 $$ $$ u(x,0) = f_0(x), \quad u(x,1) = f_1(x), \quad u(0,y) = u(\pi,y) = 0 $$ with $$ f_0(x) ...
8
votes
2answers
167 views

A Differential operator.

What are the fundamental solutions for the operator $$\mathcal D=a{\partial^2\over\partial x_1^2}+b{\partial^2\over\partial x_2^2}$$ on $\Bbb R^2 $ with standard cordinates $(x_1,x_2)$. Here ...
3
votes
1answer
79 views

Differential equation on $\Bbb R$

We have a differential equation on $\Bbb R$ of the form $$\frac {d^2}{dx^2}u = \chi_{[0,1]},$$ where $\chi_{[0,1]}$ is the characteristic function of the interval $[0, 1] ⊂ \Bbb R$. I want to find a ...
2
votes
2answers
331 views

Solve Laplace equation in the upper half plane

I need to solve \begin{eqnarray} u_{xx} + u_{yy} = 0 \quad \quad y>0 \quad -\infty < x< \infty \end{eqnarray} With boundary condition \begin{eqnarray} \frac{\partial u(x,0)}{\partial y} = ...
2
votes
1answer
116 views

How to show $C_p^k([-\ell, \ell])$ is not a Banach space?

I need to show the space $$C_p^k([-\ell, \ell])=\{f\in C^k(\mathbb R; \mathbb C); f(x+2\ell)=f(x), \forall x\in\mathbb R\},$$ is not a banach space with the norms ...
0
votes
1answer
24 views

Convergence of a series of a given metric..

I'm trouble with a metric defined over a given set: Consider $\mathcal{P}=C_p^\infty([-\pi, \pi])$, that is, $\mathcal{P}$ is the set of all infinitely differentiable functions $f:\mathbb R\rightarrow ...
2
votes
1answer
165 views

Change of variables in a convolution..

I'm in trouble with change of variables in a convolution: Definition: The convolution of two $2\pi$-periodic functions $f$ and $g$ is defined as $$(f*g)(x)=\frac{1}{2\pi}\int_{-\pi}^{\pi} ...