ABSTRACT
In Chapter 1 we considered integral transforms, which can be used for the solution of
linear differential equations with respect to a certain differential operator. For example,
in section 1.4 we considered the Laplace transformation, which is fit for the solution of
linear differential equations with respect to the operator of differentiation D = ddx . The
disadvantage of the use of integral transforms is that some integral has to be convergent.
So one should look for a pure algebraic version of operational rules with respect to the
operator D for the application to the solution of differential equations. This was first done
by D. Heaviside [H.1] through [H.3] in a quite formal manner. In the 1950s the problem was
solved by J. Mikusin´ski, see [Mi.7], who used elements of algebra to develop an operational
calculus for the operator D in an elementary but perfect manner; see also [DP] and [Be.1]
for similar representations of the same topic. Meanwhile, operational calculi for many other
differential operators were developed; see, for example, [Di]. In this book we deal only with
the classical one, i.e., operational calculus for the operator D.
