14
Special Transformations
In functional analysis, the Laplace transform is a powerful technique for ana-
lyzing linear time-invariant systems. In actual, physical systems, the Laplace
transform is often interpreted as a transformation from the time-domain point
of view, in which inputs and outputs are understood as functions of time, to
the frequency-domain point of view, where the same inputs and outputs are
seen as functions of complex angular frequency, or radians per unit time.
This transformation not only provides a fundamentally different way to un-
derstand the behavior of the system, but it also drastically reduces the com-
plexity of the mathematical calculations required to analyze the system. The
Laplace transform has many important Operations Research applications as
well as applications in control engineering, physics, optics, signal processing
and probability theory. The Laplace transform is used to analyze continuous-
time systems whereas its discrete-time counterpart is the Z transform. The
Z transform among other applications is used frequently in discrete probabil-
ity theory and stochastic processes, combinatorics and optimization. In this
chapter, we will present an overview of these transformations from differen-
tial/difference equation systems' viewpoint.14.1 Differential Equations
Definition 14.1.1 An (ordinary) differential equation is an equation that can
be written as:
#(t,j/,y',...)j/(n)) = 0.
A solution of above is a continuous function y : I H> R where I is a real
interval such that $(t, y, y',..., j/")) = 0, W E I. A differential equation is a
linear differential equation of order n if
j/"> + an^(t)y^-^ + ••• + ai(t)y' + a 0 (t)y = b(t)
where an_i, • • • ,a\, ao,b are continuous functions on I to K. Ificti — Ci, the
above has constant coefficients. If b(t) — 0, Wt 6 /, then the above is called