How can I write poisson's equation
$\partial_{xx} u = f$
solution in 1d explicitly?
I have seen somewhere I can write $u(x) = \int^{x}_{0}\int^{y}_{0} f(z) dz dy - \int^{1}_{0}\int^{x}_{0}\int^{y}_{0} f(z) dz dy dx$
in $[0,1]$ ?
and say u is comparable?
How do you get this explicit form?some integration?what regularity does $f$ need?
A complete solution is $$ u(x) = \int_0^x \int_0^y -f(z) ~dz ~dy + ax + b, $$ where $a$ and $b$ are chosen to satisfy the boundary conditions. E.g., if you know that $u(0) = 0$ and $u'(0) = 1$, then $a = 1$ and $b = 0$. On the other hand, if you know that $u(0) = 3$ and $u(1) = 7$, then you have $$ b = 3\\ a = -\left(\int_0^1 \int_0^y -f(z) ~dz ~dy + 3 \right) + 7. $$