PDE:Fourier Transforms

From Exampleproblems

Jump to: navigation, search

solution Find the Fourier transform of $f(t) = e^{-|t|}\,$

solution Find the Fourier transform of $f(t) = \begin{cases}1&|t|<1\\0&|t|>1\end{cases}\,$

solution $u_t=ku_{xx}\,$

$u(0,t) = 0\,$
$u(x,0) = f(x)\,$
$t>0,\,\,0<x<\infty\,$

solution $u_{xx}+u_{yy}=0\,$

$u(0,y) = 0\,$
$u(1,y) = 0\,$
$u(x,0) = 0\,$

$u(x,1) = B x(1-x)\,$
$t>0,\,\,0<x<\infty\,$

solution $u_t=-u_{xxxx}\,$

$u(x,0) = f(x)\,$
$t>0,\,\,x\isin\mathbb{R},$

solution $u_{tt}=c^2\,u_{xx}\,$

$u(x,0) = f(x)\,$
$u_t(x,0)=g(x)\,$
$t>0,\,\,x\isin\mathbb{R},$

solution $u_{xx}+u_{yy}+u_{zz}=0\,$

$u(x,y,0) = f(x,y)\,$
Auxiliary condition:

u

is bounded.

$t>0,\,\,x,y\isin\mathbb{R},\,\,\,z>0\,$

$u(x,y,z) = \int\!\!\!\int_\Re e^{i \lambda x + i \mu y} B(\lambda,\mu) e^{-\sqrt{\lambda^2+\mu^2}\,z}\,d\lambda d\mu\,$

[Quick Answer] Write the form of the solution:

$u_{tt}=c^2(u_{xx}+u_{yy})\,$

$u(x,0,t) = g(x)\,$
$u(0,y,t) = h(y)\,$

$u(x,y,0) = 0\,$
$u_t(x,y,0) = f(x,y)\,$

$0<x<\infty,\,\,\,\,\, 0<y<\infty,\,\,\,\,\, t>0\,$

$u(x,y,t) = \int_0^\infty \int_0^\infty U(\lambda,\mu,t) \sin(\lambda x) \sin(\mu y)\,d\lambda\,d\mu\,$

[Quick Answer] Write the form of the solution:

$u_{tt}=c^2(u_{xx}+u_{yy})\,$

$u_y(x,0,t) = g(x)\,$
$u(0,y,t) = h(y)\,$

$u(x,y,0) = 0\,$
$u_t(x,y,0) = f(x,y)\,$

$0<x<\infty,\,\,\,\,\, 0<y<\infty,\,\,\,\,\, t>0\,$

$u(x,y,t) = \int_0^\infty \int_0^\infty U(\lambda,\mu,t) \sin(\lambda x) \cos(\mu y)\,d\lambda\,d\mu\,$

solution $u_{tt}=c^2(u_{xx}+u_{yy})\,$

$u_y(x,0,t) = g(x)\,$
$u(0,y,t) = h(y)\,$

$u(x,y,0) = 0\,$
$u_t(x,y,0) = f(x,y)\,$

$0<x<\infty,\,\,\,\,\, 0<y<\infty,\,\,\,\,\, t>0\,$

$u(x,y,t) = \int_0^\infty \int_0^\infty U(\lambda,\mu,t) \sin(\lambda x) \cos(\mu y)\,d\lambda\,d\mu\,$

Partial Differential Equations

Retrieved from "http://www.exampleproblems.com/wiki/index.php/PDE:Fourier_Transforms"

Views

Personal tools

Create an account or log in

Search

Toolbox

Argan Oil
Natural Skin Care
Organic Skin Care