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.

 

[tacman]

The Laplace and Fourier transforms are both mathematical tools used to analyze signals and systems in various fields such as engineering, physics, and mathematics. While they share some similarities, they have distinct differences in their main purpose and applications.

The main difference between the Laplace and Fourier transforms lies in the type of signals they can analyze. The Laplace transform is used for analyzing signals that are time-continuous and have an exponential decay or growth. On the other hand, the Fourier transform is used for analyzing signals that are time-limited and have periodic components.

Another key difference is in the domain in which these transforms operate. The Laplace transform operates in the complex frequency domain, while the Fourier transform operates in the frequency domain. This means that the Laplace transform can handle both real and imaginary components of a signal, while the Fourier transform only deals with the real component.

In terms of their main idea, the Laplace transform is primarily used for solving differential equations and analyzing control systems, while the Fourier transform is used for analyzing signals in the frequency domain and solving problems involving periodic functions.

There are also differences in the way these transforms are applied. The Laplace transform is usually used to find the transfer function of a system, which represents the relationship between the input and output signals. In contrast, the Fourier transform is used to decompose a signal into its frequency components, which can then be analyzed separately.

The absence of imaginary components in the Fourier transform was a limitation in its application to certain types of signals, leading to the development of the Laplace transform. The Laplace transform allows for a more comprehensive analysis of signals with exponential components, which are common in many real-world systems.

In summary, the Laplace and Fourier transforms have distinct differences in their main purpose, domain, and applications. The decision to use one over the other depends on the type of signal and the specific problem being analyzed. Both transforms have their strengths and weaknesses and are valuable tools in various fields of study.