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They are similar. In many problems, both methods can be used. You can view Fourier transform is the Laplace transform on the circle, that is |z|=1. When you do Fourier transform, you don't need to worry about the convergence region. However, you need to find the convergence region for each Laplace transform. The discrete version of Fourier transform is discrete Fourier transform, and the discrete version of Laplace transform is Z-transform.
Ans: because of essay calucation in s domine rather than time domine and we take inverse laplace transfom
Laplace is used to write algorithms for various programs. More info is available on wiki .
The most generalized reason would be:"To solve initial-valued differential equations of the 2nd (or higher) order." Laplace is a little powerful for 1st order, but it will solve them as well.There is a limitation here: Laplace will only generate an exact answer if initial conditions are provided. Laplace cannot be used for boundary-valued problems.In terms of electronics engineering, the Laplace transform is used to get your model into the s-domain, so that s-domain analysis may be performed (finding zeroes and poles of your characteristic equation).This is particularly useful if one needs to determine the kind of response an RC, RLC, or LC circuit will provide (i.e. underdamped, overdamped, critically damped).Once in the s-domain, we may begin discussing the components in terms of impedance. Sometimes it is easier to calculate the voltage or current across a capacitor or an inductor in terms of the components' impedances, rather than find it in a t-domain model.The node-voltage and mesh-current methods used to analyze a circuit in the t-domain work in the s-domain as well.