Thread: solving partial differential equations using complex numbers

hi there folks,

what i wanted to ask today is the following :

i know that complex numbers can be used to simple differential equations of the second order such as 10y''+y'+1=0

my question is this , lets take a partial differential equation such as the wave equation (for those not familiar )

now , given the proper initial values , or an adequate set of boundary limits i can guess a solution , quantimize(spelling?) it and find a soltuion for the equation dependant on n,x,t
my question is the following , i know that some people will prefer to solve such an equation using complex numbers, so my request is this , could someone provide an example to doing so ? any partial differential equation will do , i only gave the wave equation as an example.

Usually these equations are solved by letting u(x,t)=Aexp(i(kx-wt)).

Every class of equations has its own method of solving it.