i have read many of the answers and explanations about the similarities and differences between laplace and fourier transform.
Laplace can be used to analyze unstable systems.
Fourier is a subset of laplace.
Some signals have fourier but laplace is not defined , for instance cosine or sine from -infinity to +infinity.
i have studied signals and systems and basic control theory in my undergrad. My question is related to the solution of differntial equations using these transforms.
Wherever i have seen it is written that fourier is used for steady state analysis (example : in solution of circuits) whereas for transient response we resort to laplace.
What exactly is the thing that enables laplace to incorporate initial conditions (hence unilateral laplace) and solve ODEs ? Similarly what prevents fourier from taking these into account?
Let me explain at least what i understand. when solving for the laplace of the derivative of a function (using integration by parts) we input the initial conditions there and they usually end up appearing as decaying exponentials (at the natural modes/poles of the system) in the final response. The constants are chosen so as to satisfy the initial conditions. On a sidenote, this is also related to linearity and upon initial conditions that are not 0 (rest) the system ends up becoming non-linear because of not satisfying the zeros input zero output property.
Now when the fourier is found for the derivative of a function the limits of the integral go from -infinity to + infinity and consequently the initial conditions can't be absorbed. Why cant we define the integral to be from 0 to infinity and incorporate initial conditions just as in laplace? i know that fourier is closely linked to convolution and changing these limits would wrong that but still can this formulation be used for solving an initial value ODE? if yes , kindly provide an example. here is one that i found . i don't understand one step (second step of finding the general solution'u') in it but other than that i can't find a mistake.
Solving an initial value ODE problem using fourier transform
i know this might be too basic for some or too narrowed down but it has bugged me my whole undergrad and now i really need an answer from an expert. Thank u and regards khurram
