Analytical solutions of linear delay-differential equations with Dirac delta function inputs using the Laplace transform

In this paper, we propose a methodology for computing the analytic solutions of linear retarded delay-differential equations and neutral delay-differential equations that include Dirac delta function inputs. In numerous applications, the delta function serves as a convenient and effective surrogate for modeling high voltages, sudden shocks, large forces, impulse vaccinations, etc., applied over a short period of time. The solutions are obtained using the Laplace transform method, in conjunction with the Cauchy residue theorem. The accuracy of these solutions are assessed by comparing them with the ones provided by the method of steps. Numerical examples illustrating the methodology are presented and discussed. These examples show that the Laplace transform solution is very reliable for linear retarded delay-differential equations, because the analytic solution, for a single delta function input, is continuous. However, for linear neutral delay-differential equations with a delta function input the analytic solution is discontinuous. Consequently, the well-known Gibbs phenomenon is observed in the vicinity of the discontinuities. However, for neutral delay differential equations, we show that in some cases, the magnitude of the jumps at the discontinuities decrease, as time increases. Therefore, the Gibbs phenomenon of the Laplace solution dissipates.