It has been proved that the inverse operator is the solution of the related equation of the following form, in which the non-homogeneous term has been substituted by the Dirac's delta function :
It should be noted that the Dirac's delta function, because of its rather complex nature, is formally described in terms of the derivative of the Heaviside unit step function :
Diagrams of the Dirac's delta function and Heaviside unit step function are schematically shown in Fig.
Following the method described in [17,20], it is required to solve the following differential equation, by making use of the Dirac's delta function:
It requires the Dirac's delta function to be expanded in the same way, i.e., in a half-range Fourier sine series expansion, which yields the following expressions:
* In electrical engineering and other disciplines, the output (response) of a (stationary, or time- or space-invariant) linear system is the convolution of the input (excitation) with the system's response to an impulse or Dirac's delta function
To express a very sharp peak in branching rate at a time t = [Tau], we may use Dirac's delta function [Delta](t - [Tau]):
This can be shown by considering the case in which the branching rate has a very sharp peak at a time t = [Tau], expressed in terms of Dirac's delta function [Delta](t - [Tau]):