# Solving a System and taking inverse Laplace/Fourier Transforms

I have a set of linear equations for 4 quantities which have been both Fourier transformed and Laplace transformed. The system needs to be solved for the quantities and then each of the quantities has to be inverse Laplace transformed, and then inverse Fourier transformed (where the transforms are all one-dimensional so there is only one frequency).

I have tried to solve the system with Maple and then use the invlaplace function to take the inverse Laplace transforms of the quantities. The code I use is as follows:

e1 := -2*Dikpi + As = 0

e2 := 2*Aik*pi + 2*Cikpi + Bs = a

e3 := s*C + 4/5*Pw3(2*piik*D - 2*1/3*piikD)/w2 = -2(C + 2*K*(2*piik*B - 2*1/3*piik*B))/(w2*K)

e4 := s*D + 2*5/4*P*t4*piik*C/(t2*K) = -5/(2*P)*D/(t2*K)

sys := {s*C + 4/5*Pw3(2*piik*D - 2*1/3*piikD)/w2 = -2(C + 2*K*(2*piik*B - 2*1/3*piik*B))/(w2*K), s*D + 2*5/4*P*t4*piik*C/(t2*K) = -5/(2*P)*D/(t2*K), -2*Dikpi + As = 0, 2*Aik*pi + 2*Cikpi + Bs = a}

solve(sys, [A, B, C, D]),

where everything is a constant apart from the frequency $$k$$, the variable $$s$$ is from the Laplace transform and $$i = \sqrt{-1}$$. The solution which Maple produces for the 4 quantities is a fairly complicated fraction, and when I use invlaplace on each quantitity I obtain the answer as a sum over roots of a quartic polynomial.

I was wondering if it would be possible to use Mathematica to solve the system and obtain the quantities in the simplest possible form, and then if there is a function similar to invlaplace which then enables me to obtain a relatively simple analytic expression for the 1D inverse Laplace transform for each of the quantities, such that they can be passed to a function which gives the 1D inverse Fourier transform?

• Use LaplaceTransform and InverseLaplaceTransform. If this does not resolve your problem, edit your question to show what you have tried (in proper Mathematica syntax) and clarify what problem you are having. – Bob Hanlon Feb 20 '20 at 16:25