For the sake of brevity, consider the 1D Equation 2-54. In order to derive the PML formulation in the time domain, the following steps are taken (
Ref. 39). First, consider a special form of the mapping
Equation 2-53:
Then, taking Equation 2-55 into account, multiply
Equation 2-54 by
1+σ(x)/iω.
Equation 2-54 transforms to the following form:
Equation 2-56 and
Equation 2-57 yield a system of partial differential equations in the time domain equivalent to the frequency domain
Equation 2-54: