Operational calculus

Operational calculus

Operational calculus, also known as operational analysis, is a technique by which problems in analysis, in particular differential equations, are transformed into algebraic problems, usually the problem of solving a polynomial equation.

Contents

History

The idea of representing the processes of calculus, derivation and integration, as operators has a long history that goes back to Gottfried Leibniz. The mathematician Louis François Antoine Arbogast was one of the first to manipulate these symbols independently of the function to which they were applied. This approach was further developed by Servois who developed convenient notations. Servois was followed by a school of British mathematicians including Heargrave, Boole, Bownin, Carmichael, Doukin, Graves, Murphy, William Spottiswoode and Sylvester. Treatises describing the application of operator methods to ordinary and partial differential equations were written by George Boole in 1859 and by Robert Bell Carmichael in 1855. This technique was fully developed by the physicist Oliver Heaviside in 1893, in connection with his work on electromagnetism. At the time, Heaviside's methods were not rigorous, and his work was not further developed by mathematicians. Operational calculus first found applications in electrical engineering problems, for the calculation of transients in linear circuits after 1910, under the impulse of Ernst Julius Berg, John Renshaw Carson and Vannevar Bush. A rigorous mathematical justification of Heaviside's operational methods came only after the work of Bromwich that related operational calculus with Laplace transformation methods (see the books by Jeffreys, by Carslaw or by MacLachlan for a detailed exposition). Other ways of justifying the operational methods of Heaviside were introduced in the mid 1920's using integral equation techniques (as done by Carson) or Fourier transformation (as done by Norbert Wiener).

A different approach to operational calculus was developed in the 1930s by Polish mathematician Jan Mikusinski, using algebraic reasoning.

Principle

The key element of the operational calculus is to consider differentiation as an operator p = d / dt acting on functions. Linear differential equations can then be recast in the form of an operator valued function F(p) of the operator p acting on the unknown function equals the known function. Solutions are then obtained by making the inverse operator of F act on the known function.

In electrical circuit theory, one is trying to determine the response of an electrical circuit to an impulse. Due to linearity, it is enough to consider a unit step, i. e. the function H(t) such that H(t < 0) = 0 and H(t > 0) = 1. The simplest example of application of the operational calculus is to solve: py = H(t), which gives:

 y=p^{-1} H = \int_0^t H(u) du= t H(t) .

from this example, one sees that p − 1 represents integration, and p n represent n iterated integrations. In particular, one has that p^{-n} H(t)=\frac{t^n}{n!} H(t). It is then possible to make sense of \frac{p}{p-a}H(t)=\frac{1}{1-\frac{a}{p}}H(t) by using a series expansion. One finds that:

\frac{1}{1-\frac{a}{p}}H(t)=\sum_{n=0}^\infty a^n p^{-n} H(t)=\sum_{n=0}^\infty \frac{a^n t^n}{n!} H(t)=e^{at} H(t)

Using [partial fraction] decomposition, it becomes possible to define any fraction in the operator p and compute its action on H(t). Moreover, if the function \frac{1}{F(p)} has a series expansion of the form:

\frac{1}{F(p)}=\sum_{n=0}^\infty a_n p^{-n},

it is straightforward to find that:

\frac{1}{F(p)}H(t)=\sum_{n=0}^\infty a_n \frac{t^n}{n!} H(t)

Applying the above rule, solving any linear differential equation is thus reduced to a purely algebraic problem.

Heaviside went farther, and defined fractional power of p, thus establishing a connection between operational calculus and fractional calculus.

Using the Taylor expansion, one can also see that eapf(t) = f(t + a), so that operational calculus is also applicable to finite difference equations and to electrical engineering problems with delayed signals.

References

  • Terquem and Gerono, Nouvelles Annales de Mathematiques: journal des candidats aux écoles polytechnique et normale 14 , 83 (1855) [Some historical references on the precursor work till Carmichael].
  • V Bush Operational Circuit analysis (J. Wiley & Sons, 1929). with an appendix by N. Wiener.
  • B van der Pol, H Bremmer Operational calculus (Cambridge University Press, 1950)
  • RV Churchill Operational Mathematics (McGraw-Hill, 1958).
  • J Mikusinski Operational Calculus (Elsevier, Netherlands, 1960).

External links


Wikimedia Foundation. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • operational calculus — operacinis skaičiavimas statusas T sritis automatika atitikmenys: angl. operational calculus vok. Operatorenrechnung, f rus. операционное исчисление, n pranc. calcul opérationnel, m …   Automatikos terminų žodynas

  • operational calculus — Math. a method for solving a differential equation by treating differential operators as ordinary algebraic quantities, thus obtaining a simpler problem. * * * …   Universalium

  • operational calculus — noun : a branch of mathematics that subjects to algebraic operations symbols of operation as well as of magnitude * * * Math. a method for solving a differential equation by treating differential operators as ordinary algebraic quantities, thus… …   Useful english dictionary

  • Operational semantics — In computer science, operational semantics is a way to give meaning to computer programs in a mathematically rigorous way. Operational semantics are classified into two categories: structural operational semantics (or small step semantics)… …   Wikipedia

  • Simply typed lambda calculus — The simply typed lambda calculus (lambda^ o) is a typed interpretation of the lambda calculus with only one type combinator: o (function type). It is the canonical and simplest example of a typed lambda calculus. The simply typed lambda calculus… …   Wikipedia

  • Process calculus — In computer science, the process calculi (or process algebras) are a diverse family of related approaches to formally modelling concurrent systems. Process calculi provide a tool for the high level description of interactions, communications, and …   Wikipedia

  • Oliver Heaviside — Heaviside redirects here. For other uses, see Heaviside (disambiguation). Oliver Heaviside Portrait by Francis Edwin Hodge …   Wikipedia

  • Jan Mikusinski — Prof. Jan Mikusiński (April 3 , 1913 Stanisławów July 27, 1987 Katowice) was a Polish mathematician known for his pioneering work in mathematical analysis. Mikusiński developed an operational calculus 44A40 Calculus of Mikusiński , which is… …   Wikipedia

  • Generalized function — In mathematics, generalized functions are objects generalizing the notion of functions. There is more than one recognised theory. Generalized functions are especially useful in making discontinuous functions more like smooth functions, and (going …   Wikipedia

  • John Renshaw Carson — (June 28, 1886 October 31, 1940), who published as J. R. Carson, was a noted transmission theorist for early communications systems. He invented single sideband modulation.Carson was born in Pittsburgh, Pennsylvania, graduated from Princeton… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”