By Vrinda Prabhu.

Both Laplace Transforms and Fourier Transforms can be used to solve differential equations, so a natural question is to ask “where to use what?” or “Which one is better?”.

The Fourier Transform is decent enough to be used for functions (or “signals”) that are not having infinite energy and i guess it can be extended to certain finite power, infinite energy functions, like DC or a sinusoid or a periodic function.

But for some functions, like the unit-step function,Weak Dirichlet Condition for the Fourier Transform [integral |f(t)dt| between limits (-infinity,infinity) < infinity] is not satisfied.In simple words function is not absolutely summable so the Fourier Transform doesn’t converge nicely. To counter this there is a change variable of the transformed function from ω or jω to s=σ+jω by adding a little real part to jω which makes some of these integrals converge.

Laplace transform is a more general form of Fourier transform. Laplace transform can be applied to all signals,and all systems. As said before Fourier transform of a system exists only if a system is stable.Because of the negative exponential term in the Laplace Transform integration, convergence is a lot stronger, and polynomial expressions which could not be handled with the Fourier Transform can be considered. The Laplace Transform picks out precisely the solution to a differential equation that obeys certain initial conditions at t=0. Hence it is ideally suited to initial value problems.

However as we are aware  Fourier Transforms tie in beautifully with many areas of applied mathematics, particularly quantum mechanics.Fourier Transform also has many applications beyond the solution of differential equations.The Fourier Transform picks out precisely the solution to a differential equation which decays to zero at large distances. In many applied problems this is exactly the solution we want, since a function which grows at large distances would be considered unphysical.

On the other hand The Fourier transform and the Laplace transform are closely related; in a sense, the Fourier transform can be seen as a special case of the bilateral Laplace transform, where the complex variable ‘s’ in the integral is restricted to be on the imaginary axis. Because of this, Laplace transforms apply to a larger set of functions than Fourier transforms, which can run into discontinuities along the imaginary axis that cause either the transform or its inverse to be divergent.

Briefing it out one of the main areas where Laplace transform is essential is circuit theory; as you probably know Laplace can be used to convert differential equations into algebraic ones this allows for analyzing circuits in there transit state without the need to use the regular methods of solving differential equations, this greatly simplifies the effort needed to analyze and solve whereas Fourier Transform is probably the most important transform because it ties together two of the most used parameters time and frequency, meaning that if you have a signal or waveform in the time domain and you want to see what frequencies does this signal contain you apply the Fourier transform. Looking at the signal in the frequency domain allows us to preform a lot of manipulation on the signal including filtering, sampling, modulation……….etc.

Often a mixture of the two methods proves the most effective. We are interested in functions that depend on time and space. The most physical solution to an equation is one which obeys initial conditions at t=0, and which tends to zero at large distances. This directly corresponds to Fourier transforming with respect to space and Laplace transforming with respect to time!