Operational Calculus

a method of mathematical analysis that in a number of cases permits the solution of complicated mathematical problems by means of simple rules. It is particularly important in such fields as mechanics, automation, and electrical engineering.

Operational calculus essentially involves the replacement of functions under study by other functions called transforms, which are obtained from the original functions by certain rules. Usually the transform is obtained from the original function through a Laplace transformation. In the substitution, the differential operator p = d/dt is interpreted as an algebraic quantity, with the result that the integration of certain classes of linear differential equations and the solution of a number of other problems of mathematical analysis reduce to the solution of simpler algebraic problems. Thus, the solution of a linear differential equation reduces to the generally simpler problem of solving an algebraic equation; from the algebraic equation one finds the transform of the solution of the original equation, and then the solution itself is recovered from the transform. The operation of finding the transform of the original function and the reverse operation are facilitated by the existence of extensive tables of transforms.

The work of the British scientist O. Heaviside was very important for the development of operational calculus. Heaviside set forth formal rules for dealing with the operator p = d/dt and certain functions of this operator. By means of operational calculus he solved a number of very important problems in electrodynamics. He did not, however, supply a mathematical grounding for operational calculus, and many of his results remained unproved.

A rigorous foundation for operational calculus was provided with the aid of the Laplace integral transform. If the function f(t), 0 ≤ t < + ∞, is transformed into the function F(z), z = x+ iy.

f(t)→ F(z)

then the derivative

(*) fʹ →zF(z)–f(0)

and the integral

Thus, the differential operator p is transformed into an operator of multiplication by the variable z, and integration reduces to division by z. Table 1 gives some examples, for t ≥ 0, of transforms.

Example. With the aid of operational calculus we shall find the solution y = f(t) of the linear differential equation

yʹʹ – yʹ–6y = 2e^4t

given with the initial conditions

y₀ = f(0) = 0 and yʹ₀ = fʹ(0)= 0

Using Table 1, we move from the unknown function f(t) and the given function 2e^4t to the transforms F(z) and 2/(z —4). Applying formula (*) for the transforms of the derivatives, we obtain

or

Hence, again using Table 1,

In 1953 the Polish mathematician J. Mikusiński suggested that operational calculus could be based on the concept of a

function ring. The theory of generalized functions can also be used as a foundation for the methods of operational calculus. There are various generalizations of operational calculus, such as multidimensional operational calculus, which is based on the theory of multiple integrals. An operational calculus of differential operators other than the operator p = d/dt has been created, for example, B = (d/dt)t(d/dt). This theory is also based on the study of function rings, in which the concept of the product of functions is defined in a suitable manner.