Definitions

Operational calculus

Operational calculus 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. It is also known as operational analysis.

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 $F\left(p\right)$ 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 impulse, i. e. the function $H\left(t\right)$ such that $H\left(t<0\right)=0$ and $H\left(t>0\right)=1$. The simplest example of application of the operational calculus is to solve: $py=H\left(t\right)$, which gives:

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

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

$frac\left\{1\right\}\left\{1-frac\left\{a\right\}\left\{p\right\}\right\}H\left(t\right)=sum_\left\{n=0\right\}^infty a^n p^\left\{-n\right\} H\left(t\right)=sum_\left\{n=0\right\}^infty frac\left\{a^n t^n\right\}\left\{n!\right\} H\left(t\right)=e^\left\{at\right\} H\left(t\right)$ Using [partial fraction] decomposition, it becomes possible to define any fraction in the operator $p$ and compute its action on $H\left(t\right)$. Moreover, if the function $frac\left\{1\right\}\left\{F\left(p\right)\right\}$ has a series expansion of the form:

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

it is straightforward to find that:

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

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 $e^\left\{ap\right\} f\left(t\right)=f\left(t+a\right)$, so that operational calculus is also applicable to finite difference equations and to electrical engineering problems with delayed signals.

An example of this calculus is given below: Problem: L(2n) = L2(n) + 2*(-1)n+1
Solution:
(eD+e-D)F(n)=enD-(-1)ne-nD
For function F(a) we have:
[F(a+1)+F(a-1)]F(n)=F(a+n)-(-1)nF(a-n)
or
L(a)F(a)=F(a+n)-(-1)nF(a-n)
for a=2n and F(3n)=[L2(n)+(-1)n+1]F(n)
L(2n)=L2(n)+2*(-1)n+1

References

LF Arbogast Du calcul des dérivations (Levrault, Strasbourg, 1800).

Servois Annales de Gergonne 5, 93 (1814).

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].

G Boole A treatise on differential equations Chapters 16 and 17 (Mc Millan, 1859).

RB Carmichael A treatise on the calculus of operations (Longman, 1855). O Heaviside Proc. Roy. Soc. (London) 52. 504-529 (1893), 54 105-143 (1894). [Original articles]

JR Carson Bull. Amer. Math. Soc. 32, 43 (1926).

JR Carson Electric Circuit Theory and the Operational Calculus (Mc Graw Hill, 1926).

N Wiener Math. Ann. 95, 557 (1926).

H Jeffreys Operational Methods In Mathematical Physics (Cambridge University Press, 1927). also at | Internet Archive

HW March Bull. Amer. Math. Soc. 33, 311 (1927), 33, 492 (1927).

EJ Berg Heaviside's Operational Calculus (McGrawHill, 1929).

V Bush Operational Circuit analysis (J. Wiley & Sons, 1929). with an appendix by N. Wiener.

HT Davis The theory of linear operators (Principia Press, Bloomington, 1936).

NW Mc Lachlan Modern operational calculus (Macmillan, 1941).

HS Carslaw Operational Methods in Applied Mathematics (Oxford University Press, 1941).

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).

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