  A wave view of the universe is proposed in which each natural phenomenon is
equipped with its own unique natural viewing lens. A self-sameness modeling
principle and its systematic application in Fourier-Laplace transform space is
proposed as a novel, universal discrete modeling paradigm for
advection-diffusion-reaction equations (ADREs) across non-integer derivatives,
time scales, and wave spectral signatures. Its implementation is a novel exact
spectral derivative discretization finite difference method (ESDDFD), a way for
crafting wave viewing lenses by obtaining discrete wave models from ADRE
models. The template for building these lenses come in the form of natural
derivative representations obtained from the wave signature probability
distribution function and its harmonic oscillation in FL transform space; use
of the ESDDFD method in the discrete numerical modeling of wave equations
requires no a-priori theory of any mathematical derivative. A major
mathematical consequence of this viewpoint is that all notions of the
mathematical integer or non-integer derivatives have representation as limits
of such natural derivative representations; this and other consequences are
discussed and a discretization of a simple integer derivative
diffusion-reaction equation is presented to illustrate the method. The
resulting view lenses, in the form of ESDDFD models, work well in detecting
both local and non-local Debye or Kohlrausch-Williams-Watts exponential
patterns; only Brownian motion and sub-diffusion are discussed in the present
article.
