Operational calculus

The Heaviside operational calculus according describes an empirical named after Oliver Heaviside operational calculus, which was published in 1887 in his famous work "Electromagnetic Theory". (English operator calculus or operational calculus )

General

The purpose of the operational calculus, it is in the solution of differential equations with an operator to replace the operation of differentiation by the algebraic operation of multiplying and thus " transform " a usually relatively difficult separable differential equation in a more easily separable algebraic equation. Heaviside built, based on previous work of Leibnitz and Cauchy, as first the operational calculus to a calculus of which triggered many of the at that time pending theoretical problems of electrical engineering.

To this end, he generalized the complex AC circuit analysis, in which replaces the differentiation by introducing the differential operator and uses it without any deeper reasoning as a multiplicative factor:

Finally, it separates from the time functions and the resulting expressions are a " very existence " as an operator. Intuitively, he concludes that the reciprocal of logically represents the operator for integration.

The Heaviside operational calculus (also called the Heaviside calculus ) is a generalization of the extended symbolic method to non- stationary signals and is thus a forerunner of the modern Operator bills, such as the Laplace and the operational calculus by Mikusiński.

Example

It is the characteristic of the voltage of Ua (t ) are calculated at the capacitor of an RC element, if at the input of the DC voltage Vin at time t = 0 is "on" is. With the notation of Heaviside unit step as for the "fat one" so ue (t ) = true Ue · 1

From the circuit we obtain the relationship

It follows the inhomogeneous differential equation for t ≥ 0:

Heaviside is now using the differential operator p:

He excluded from

... And solving for the expected size on

So although is " operator for the result " found, but what does this phrase mean? Heaviside tried the solution by series expansion:

By an integration operator ( applied to the unit step ) interpreted, we obtain the series elements ( but vanish for t < 0):

And generally

So

Here you have now - as Heaviside - identify " with a trained eye " that the exponential function (with a negative argument) hidden behind this series and thus receives the self-contained solution ( for t ≥ 0):

The transfer function

Heaviside now defined as a characteristic of the system is independent of the signal transmission function as an operator. For the above-mentioned Example, one obtains:

This definition is identical to the transfer function in the sense of extended symbolic method of alternating current technology and other Operators bills and still has outstanding importance.

Interpretation by decomposition

Heaviside interpreted the results obtained in operator form by partial fractions or series expansion (as in the example above). Heaviside developed a reliable method that is still used even in modern Operators bills as Heavisidescher development kit for the partial fractions. Practically, there are however problems with the determination of the roots if the degree of the denominator polynomial to be fractionated is greater than 4 when a root is 0 or occurs more than once.

In contrast, the decomposition is by series expansion in principle quite difficult and, depending on the approach, different possible. The result is ambiguous, and thus this method is " for practical engineering activity not suitable". If we decompose the above-mentioned For example the operator as follows in a power series

And interprets p as a differential operator, then you get a false or meaningless result. To be sure, mathematically extensive studies on the convergence or divergence of the power series would be required.

Criticism

Heaviside considered mathematics as experimental science and thought that the success justifies the procedure. He made ​​no distinction between the operators and the objects to which he applied them. For this purpose, a mathematical field theory would have been necessary, but it was not worked out at that time. Heaviside always sat (implicitly) vanishing initial conditions of the differential equations ahead, thus " discharged power memory " at time 0 Although Heaviside solved many of then current problems with his operational calculus, she could not sit down and was many " attacks " the mathematician exposed. Only through the interpretation of the operators using the Laplace transformation, the operational calculus was able to establish on the basis of the integral transformation and the theory of functions in theory and practice. Finally, in 1950 by the mathematician January Mikusiński an operational calculus " without Laplace " justified mathematically exact algebraic methods.