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This page is under construction. Some items are incomplete.
Operational Methods
There are several kinds of operational calculus methodologies
which have been developed over the years. I have heard of four
approaches, and am more or less familiar with two of them. Two
approaches which I am interested in are the approach via Laplace
transforms, and the approach by Mikusinski operational
calculus.
I have studied Laplace transform methods along with associated
distribution methods (generalized function methods) due to
Schwartz. This study was years ago. Recently I have resumed studies
to "brush up" on knowledge and techniques, to obtain more knowledge
regarding these methods, and to improve my level of mathematical
rigor regarding these methods.
Operational methods are interesting (to me) in their own right.
They are also interesting because they are practical tools: whose
application leads to solution of differential equations and to
partial differential equations.
Laplace Transformation
Naive engineers are taught that there is little difference
between Fourier transforms and Laplace transforms. Electrical
engineers are taught that the Fourier transform is a "trivial"
special case of the Laplace transform in which s=iw.
Mathematically, however, these two transforms are very different,
and the mathematical methods used to discuss each technique is very
different. That is, mathematically, the Fourier transform and the
Laplace transform are very different.
Mikusinski Operational Calculus
When I first started studying Mikusinski calculus, it reminded
me of some techniques Richard Asky taught in my college freshman
math class. My memory is a little rusty, and it would be curious to
try to find my old notes (I am not sure where they are) to try to
recall how much of this was presented in my first semester freshman
year.
Mikusinski operational calculus offers a number of interesting
features as an analytical tool. Some elements are discussed in the
list below.
- In Laplace methods there are two distinct
transform domains: one containing the Laplace variable s and
the other containing the time variable t. These are called
the Laplace domain and the time domain respectively. By comparison,
in Mikusinksi operational calculus, there are not two domains, and
the variables in each domain can be "mixed". That is s and
t can appear in the same expression; and this makes sense.
In fact, the operator s t even makes sense.
- In Mikusinski operational calculus, all
multiplications, when not scalar multiplications, are
multiplications in the Mikusinski algebra: they are convolutions of
generalized functions.
- Transcendental operational operators such as
exp[a s] are defined. That is, the differential operator s is a
logarithm. With this many properties follow, and a function can
also have a Laplace transform-like decomposition in terms of
exp[lambda s]. In form this is similar to the Laplace transform,
but the meanings of the terms and multiplications are different in
the Mikusinksi context.
- In this and other respects, the Mikusinksi
decomposition and Laplace transform are isomorphic. In Mikusinksi,
{exp[at]} = 1/(s-a) where equality is used. Notice that there is
not a Laplace transform "domain" here; instead the time function is
just equal to the expression containing the s
variable. This is a subtle point and a very beautiful point. The
derivative theorem also holds true in Mikusinksi algebra: a'
= s a - a[0]. This is a Laplace transform like result but with no
transform - just equality. This is purely amazing!
- The most striking aspect regarding
Mikusinksi operational calculus, is that the Laplace
transform like decomposition holds for a continuous function f no
matter how fast the continuous function f increases. That is,
Mikusinksi operational calculus will work for the function
exp[exp[exp[t]]] whereas Laplace transform methods assume that
functions are bounded by an exponential of a given order. Thus
Mikusinksi can find solutions which Laplace methods might not.
- Mikusinksi operational calculus can be applied
to solve ordinary differential equations. It is also applied to
find solutions of the classical wave and diffusion equations.
Formulations involving Bessel functions also occur in Mikusinksi
calculus.
References with Annotations
Mikusinski operational calculus is discussed in the
references below.
-
Operational Calculus, Lectures given by A. Erdelyi,
Mathematics Department, California Institute of Technology,
Pasadena, 1955.
This set of lecture notes, written by Erdelyi,
gives an excellent account of the Mikusinski technique.
His presentation is smooth and readable. It is
mathematically tight and clean. He derives all needed
results.
There are applications to ordinary differential
equations, to the diffusion equation, and to the wave
equation. Operational transforms are derived for erf and
related functions, and for Bessel functions.
- Operational Calculus, J. Mikusinski,
Pergamon Press, Oxford, 1959.
This gives a little more detail than the above
Erydeli notes. It appears that the Erdelyi notes and lectures are
based on this.
There are numerous good books on Laplace transformations.
Laplace transforms.
- Churchill's book on
Operational Mathematics discusses some rigorous aspects of
Laplace transforms.
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