What are the differences between laplace & fourier transform

[serimc]

What are the differences?
I mean when we will make a decision "hmm now i must use laplace transform or now i must use Fourier transform".

What are the absences in laplace transform so Fourier design a new transfom?

I want to know these transforms' main idea, differences.

I am looking for your answers.

Thanks .
Sincerely;

 

[JohanL]

in applications you usually use Laplace transforms when you have initial conditions for a p.d.e. and u start at t=0. Thats because when you Laplace transform derivatives you get the initial conditions into the p.d.e.

Often you Fourier-Laplace transforms differential equations too. If you have a p.d.e. for u(r,t) you can Fourier transform the position and Laplace transform the time. Then you get an easier equation for
u(k,z), where k is the wave-vector. This is very useful when you want to derive the Greenfunction for a particular problem.