Decay with source

                The data d(t) satisfies the equation

(1.1)

                               

Introducing the Fourer transforms

(1.2)     

Equation (1.1)becomes

              (1.3)       

The homogeneous solution U(f), u(t)

                The equation is

                  (1.1)

This is possible only for f values such that the square bracket is equal to zero

                       (1.2)

In the time domain this is

(1.3)       

The value of A is determined from the boundary conditions.  Frequently A = 0.

Green's function

For the case

          (2.1)

Then  (1.3) becomes

         (2.2)       

So that

                    (2.3)

In the time domain

                (2.4)

The poles of the integrand are at the locations where .  This is in the negative f plane.  For t > 0 , f needs to go to -jR to converge as a contour integral, while for t < 0, it goes to +jR.  Since the value of the integral is -2pj times the sum of the residues at the enclosed poles (for a contour in the negative plane), g(t) = 0 for t < 0.  For t  > 0

          (2.5)

                This is a particular solution, to which there may need to be added the homogeneous solution to the differential equation in order to satisfy any given set of boundary conditions.  Note that this solution already satisfies the conditions that g(t) is zero in the distant past and distant future.

Solution for an arbitrary function of time

A solution to equation 1 is

      (3.1)

Or

        (3.2)

The homogeneous solution may need to be added to satisfy the boundary condition.