# Fast Fourier Solutions

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== Solutions ==
[[FT1|solution]] Find the [[Fourier transform]] of $f(t) = e^{-| t|}\,$<br><br>

[[FT2|solution]] Find the Fourier transform of $f(t) = \begin 1&|t|<1\0&|t|>1\end\,$<br><br>

<table border=0 cellspacing=0 width=70%><tr><td valign=top>[[PDEFT1|
solution]] $u_t=ku_\,$</td><td valign=top
width=50%>$u(0,t) = 0\,$<br>$u(x,0) = f(x)\,</ math><br>[itex]t>0,\,\,0<x<\infty\,$</td></tr></table><br><br>
<table border=0 cellspacing=0 width=70%><tr><td valign=top>[[PDEFT2|
solution]] $u_+u_=0\,$</td><td valign=top
width=50%>
$u(0,y) = 0\,$<br>
$u(1,y) = 0\,$<br>
$u(x,0) = 0\,$<br>
$u(x,1) = B x(1-x)\,$<br>$t>0,\,\,0<x<\infty\,$</
td></tr></table><br><br>

<table border=0 cellspacing=0 width=70%><tr><td valign=top>[[PDEFT3|
solution]] $u_t=-u_\,$</td><td valign=top
width=50%>$u(x,0) = f(x)\,$<br>$t>0,\,\,x\isin\mathbb ,$</td></tr></table><br><br>

<table border=0 cellspacing=0 width=70%><tr><td valign=top>[[PDEFT4|
solution]] $u_=c^2\,u_\,$</td><td valign=top
width=50%>$u(x,0) = f(x)\,$<br>$u_t(x,0)=g(x)\,</ math><br>[itex]t>0,\,\,x\isin\mathbb,$</td></tr></
table><br><br>

<table border=0 cellspacing=0 width=70%><tr><td valign=top>[[PDEFT5|
solution]] $u_+u_+u_=0\,$</td><td valign=top
width=50%>$u(x,y,0) = f(x,y)\,$<br>Auxiliary condition:
$u$ is bounded.<br>
$t>0,\,\,x,y\isin\mathbb,\,\,\,z>0\,$<br>

</td></tr></table><br><br>
*u(x,y,z) = \int\!\!\!\int_\Re e^{i \lambda x + i \mu y} B (\lambda,\mu) e^\,z}\,d\lambda d\mu\,</ math><br><br> <table border=0 cellspacing=0 width=70%><tr><td valign=top> <b>[Quick Answer]</b> Write the form of the solution:<br><br> [itex]u_ =c^2(u_+u_)\,</td><td valign=top width=50%>

$u(x,0,t) = g(x)\,$<br>
$u(0,y,t) = h(y)\,$<br>
<br>
$u(x,y,0) = 0\,$<br>
$u_t(x,y,0) = f(x,y)\,$<br>
<br>
$0<x<\infty,\,\,\,\,\, 0<y<\infty,\,\,\,\,\, t>0\,$<br>
</td></tr></table><br><br>

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

<table border=0 cellspacing=0 width=70%><tr><td valign=top>
<b>[Quick Answer]</b> Write the form of the solution:<br><br> $u_ =c^2(u_+u_)\,$</td><td valign=top width=50%>

$u_y(x,0,t) = g(x)\,$<br>
$u(0,y,t) = h(y)\,$<br>
<br>
$u(x,y,0) = 0\,$<br>
$u_t(x,y,0) = f(x,y)\,$<br>
<br>
$0<x<\infty,\,\,\,\,\, 0<y<\infty,\,\,\,\,\, t>0\,$<br>
</td></tr></table><br><br>

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

<table border=0 cellspacing=0 width=70%><tr><td valign=top>
[[PDEFT6|solution]]$u_=c^2(u_+u_)\,$</td><td
valign=top width=50%>

$u_y(x,0,t) = g(x)\,$<br>
$u(0,y,t) = h(y)\,$<br>
<br>
$u(x,y,0) = 0\,$<br>
$u_t(x,y,0) = f(x,y)\,$<br>
<br>
$0<x<\infty,\,\,\,\,\, 0<y<\infty,\,\,\,\,\, t>0\,$<br>
</td></tr></table><br><br>

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

[[Partial Differential Equations]]
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M. Michael Musatov