Using fourier/laplace transform and green's function

[captain]

i am having trouble distinguishing when to use Fourier or laplace transform to solve any linear differential equation (it can be an ODE or PDE). What are the advantages and disadvantages of using each? Also for a green's function (take it to be a function of x, x') when solving for it, is it true that for an ODE you take a case when x>x' and x<x' so that you get two homogenous equations when the dirac delta function equals 0 and for the PDE you use Fourier transform? Can you used a laplace transform to evaluate a greens function (although i haven't seen it done in any text i have read)? sorry about all these questions but i feel confused. thanks in advance to whoever answers some or all these questions.

 

[gtoguy97]

Fourier transforms are used to analyze functions of a continuous variable, whereas Laplace transforms are used to analyze functions of a discrete variable. The advantages of using Fourier transforms over Laplace transforms are that Fourier transforms are more efficient and can be used to solve certain types of problems that cannot be solved with Laplace transforms. Additionally, Fourier transforms are better able to represent certain types of signals, such as periodic or non-periodic signals. Finally, in some cases, Fourier transforms can be used to simplify the solution of differential equations.The advantages of using Laplace transforms over Fourier transforms are that they can be used to solve linear differential equations in a more efficient manner than using Fourier transforms and they can also be used to solve nonlinear differential equations. Additionally, Laplace transforms can be used to calculate the inverse of a function and can be used to solve initial value problems. Finally, Laplace transforms can be used to obtain analytic solutions of systems of ordinary differential equations.For a Green's function, you can use both Fourier and Laplace transforms to evaluate it. For an ODE, you would typically take the cases of x>x' and x<x' so that you get two homogenous equations when the Dirac delta function equals 0. For a PDE, you would use Fourier transforms to calculate a Green's function. Laplace transforms can also be used to evaluate a Green's function, although it is not as common as using Fourier transforms.