How do I solve the pde:
$\ -s_x(x,t) -p(x,t)s_t(x,t)=p(x,t)$
for s(x,t) when p(x,t)=2x, subject to the condition s(0,t)=0?
Generally p(x,t) may not be analytical so I would like to use finite differences to transform this into a linear algebra problem:
$\ (-D_x-pD_t)s=p $ But how do I incorporate my "boundary" condition s(0,t)=0 into this?
Thanks in advance for any answers!