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One of the methods of mathematical analysis which in many cases makes it possible to reduce the study of differential operators, pseudo-differential operators and certain types of integral operators (cf. [[Differential operator|Differential operator]]; [[Integral operator|Integral operator]]; [[Pseudo-differential operator|Pseudo-differential operator]]) and the solution of equations containing them, to an examination of simpler algebraic problems. The development and systematic use of operational calculus began with the work of O. Heaviside (1892), who proposed formal rules for dealing with the differentiation operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o0683301.png" /> and solved a number of applied problems. However, he did not give operational calculus a mathematical basis; this was done with the aid of the [[Laplace transform|Laplace transform]]; J. Mikusiński (1953) put operational calculus into algebraic form, using the concept of a function ring. The most general concept of an operational calculus is obtained using generalized functions (cf. [[Generalized function|Generalized function]]).
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The simplest variant of operational calculus is as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o0683302.png" /> be the set of functions (with real or complex values) given in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o0683303.png" /> and absolutely integrable in any finite interval. The integral
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o0683304.png" /></td> </tr></table>
+
One of the methods of mathematical analysis which in many cases makes it possible to reduce the study of differential operators, pseudo-differential operators and certain types of integral operators (cf. [[Differential operator|Differential operator]]; [[Integral operator|Integral operator]]; [[Pseudo-differential operator|Pseudo-differential operator]]) and the solution of equations containing them, to an examination of simpler algebraic problems. The development and systematic use of operational calculus began with the work of O. Heaviside (1892), who proposed formal rules for dealing with the differentiation operator  $  d/dt $
 +
and solved a number of applied problems. However, he did not give operational calculus a mathematical basis; this was done with the aid of the [[Laplace transform|Laplace transform]]; J. Mikusiński (1953) put operational calculus into algebraic form, using the concept of a function ring. The most general concept of an operational calculus is obtained using generalized functions (cf. [[Generalized function|Generalized function]]).
  
is called the convolution of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o0683305.png" />. With the usual addition operation and the operation of convolution, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o0683306.png" /> becomes a ring without zero divisors (Titchmarsh's theorem, 1924). Elements of the quotient field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o0683307.png" /> of this ring are called operators and are written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o0683308.png" />; the fact that division in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o0683309.png" /> is not always possible is precisely the source of a new concept, operators, which generalizes the concept of a function. To indicate the necessary difference in operational calculus between the concepts of a function and of its value at a point, the following notation is used:
+
The simplest variant of operational calculus is as follows. Let  $  K $
 +
be the set of functions (with real or complex values) given in the domain  $  0 \leq  t < \infty $
 +
and absolutely integrable in any finite interval. The integral
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833010.png" /></td> </tr></table>
+
$$
 +
= f \star g  = \int\limits _ { 0 } ^ { t }  f( t - \tau ) g( \tau )  d \tau
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833011.png" /></td> </tr></table>
+
is called the convolution of the functions  $  f, g \in K $.
 +
With the usual addition operation and the operation of convolution,  $  K $
 +
becomes a ring without zero divisors (Titchmarsh's theorem, 1924). Elements of the quotient field  $  P $
 +
of this ring are called operators and are written as  $  a/b $;  
 +
the fact that division in  $  K $
 +
is not always possible is precisely the source of a new concept, operators, which generalizes the concept of a function. To indicate the necessary difference in operational calculus between the concepts of a function and of its value at a point, the following notation is used:
 +
 
 +
$$
 +
\{ f( t) \}  \textrm{ for  the  function  }  f  \textrm{ of  a
 +
variable  }  t ;
 +
$$
 +
 
 +
$$
 +
f( t)  \textrm{ for  the  value  of  }  \{
 +
f( t) \}  \textrm{ at  a  point  }  t.
 +
$$
  
 
===Examples of operators.===
 
===Examples of operators.===
  
 +
1)  $  e = \{ 1 \} $
 +
is the integration operator:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833012.png" /> is the integration operator:
+
$$
 
+
\{ 1 \} \{ f \}  = \left \{ \int\limits _ { 0 } ^ { t }  f( \tau )  d \tau
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833013.png" /></td> </tr></table>
+
\right \} .
 +
$$
  
 
Moreover,
 
Moreover,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833014.png" /></td> </tr></table>
+
$$
 +
e  ^ {p}  = \left \{
 +
\frac{t  ^ {p-} 1 }{\Gamma ( p) }
 +
\right
 +
\}
 +
$$
  
 
and, in particular,
 
and, in particular,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833015.png" /></td> </tr></table>
+
$$
 +
e  ^ {n} \{ f \}  = \
 +
\int\limits _ { 0 } ^ { t }  dt \dots \int\limits _ { 0 } ^ { t }  f( t)  dt  ( n
 +
\textrm{ - fold  } ) =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833016.png" /></td> </tr></table>
+
$$
 +
= \
 +
\int\limits _ { 0 } ^ { t } 
 +
\frac{( t- \tau )  ^ {n-} 1 }{( n- 1) ! }
 +
f( \tau )  d \tau
 +
$$
  
 
This is the Cauchy formula, a generalization of which to the case of an arbitrary (non-integer) index serves to define fractional integration.
 
This is the Cauchy formula, a generalization of which to the case of an arbitrary (non-integer) index serves to define fractional integration.
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833017.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833018.png" /> is a constant function) is a numerical operator; insofar as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833021.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833022.png" />, numerical operators behave as ordinary numbers. Thus the operator is a generalization not only of a function, but also of a number; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833023.png" /> is the unit of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833024.png" />.
+
2) $  [ \alpha ] = \{ \alpha \} / \{ 1 \} $(
 +
where $  \alpha $
 +
is a constant function) is a numerical operator; insofar as $  [ \alpha ] [ \beta ] = [ \alpha \beta ] $,  
 +
$  [ \alpha + \beta ] = [ \alpha ] + [ \beta ] $,  
 +
$  [ \alpha ] \{ f \} = \{ \alpha f \} $,  
 +
while $  \{ \alpha \} \{ \beta \} = \{ \alpha \beta t \} $,  
 +
numerical operators behave as ordinary numbers. Thus the operator is a generalization not only of a function, but also of a number; $  [ 1] $
 +
is the unit of the ring $  K $.
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833025.png" /> is the differentiation operator, the inverse of the integration operator. So, if a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833026.png" /> has a derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833027.png" />, then
+
3) $  s = [ 1] / e $
 +
is the differentiation operator, the inverse of the integration operator. So, if a function $  a( t) = \{ a( t) \} $
 +
has a derivative $  a  ^  \prime  ( t) $,  
 +
then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833028.png" /></td> </tr></table>
+
$$
 +
s \{ a \}  = \{ a  ^  \prime  \} + [ a( 0)]
 +
$$
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833029.png" /></td> </tr></table>
+
$$
 +
\{ a  ^ {(} n) \}  = \
 +
s  ^ {n} \{ a \} - s  ^ {n-} 1 [ a( 0)] - \dots - [ a  ^ {n-} 1 ( 0)].
 +
$$
  
 
Hence, for example,
 
Hence, for example,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833030.png" /></td> </tr></table>
+
$$
 +
\{ e ^ {\alpha t } \}  =
 +
\frac{1}{s- a }
 +
.
 +
$$
  
Of course, a non-differentiable function can be multiplied by the differentiation operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833031.png" />; however, the result will be, in general, an operator.
+
Of course, a non-differentiable function can be multiplied by the differentiation operator $  s $;  
 +
however, the result will be, in general, an operator.
  
4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833032.png" /> is the algebraic derivative. It extends to arbitrary operators in the usual way. It appears that the action of this operator on a function of the differentiation operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833033.png" /> coincides with differentiation with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833034.png" />.
+
4) $  D \{ f \} = \{ - tf( t) \} $
 +
is the algebraic derivative. It extends to arbitrary operators in the usual way. It appears that the action of this operator on a function of the differentiation operator $  s $
 +
coincides with differentiation with respect to $  s $.
  
 
Operational calculus provides suitable methods for the solution of linear differential equations, both ordinary and partial. For example, the solution of the equation
 
Operational calculus provides suitable methods for the solution of linear differential equations, both ordinary and partial. For example, the solution of the equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833035.png" /></td> </tr></table>
+
$$
 +
\alpha _ {n} x  ^ {(} n) + \dots + \alpha _ {0} x  = f,\ \
 +
\alpha _ {i} = \textrm{ const } ,\ \
 +
i= 0 \dots n,
 +
$$
  
satisfying the initial conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833036.png" /> automatically reduces to an algebraic equation. It is expressed symbolically by the formula
+
satisfying the initial conditions $  x( 0) = \gamma _ {0} \dots x  ^ {(} n- 1) ( 0) = \gamma _ {n-} 1 $
 +
automatically reduces to an algebraic equation. It is expressed symbolically by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833037.png" /></td> </tr></table>
+
$$
 +
=
 +
\frac{\beta _ {n-} 1 s  ^ {n-} 1 + \dots + \beta _ {0} + f }{\alpha _ {n} s  ^ {n} + \dots + \alpha _ {0} }
 +
,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833038.png" /></td> </tr></table>
+
$$
 +
\beta _ {v}  = \alpha _ {v+} 1 \gamma _ {0} + \dots + \alpha _ {n} \gamma _ {n-} v- 1 .
 +
$$
  
The solution in its usual form is obtained by decomposition into elementary fractions with respect to the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833039.png" />, with subsequent inverse transformation by referring to appropriate function tables.
+
The solution in its usual form is obtained by decomposition into elementary fractions with respect to the variable $  s $,  
 +
with subsequent inverse transformation by referring to appropriate function tables.
  
 
In the use of operational calculus for partial differential equations (as well as for more general pseudo-differential equations), a differential and integral calculus of operator functions, i.e. functions with operator values, is employed. The concepts of continuity, derivative, convergence of series, integrals, etc., must be developed for these functions.
 
In the use of operational calculus for partial differential equations (as well as for more general pseudo-differential equations), a differential and integral calculus of operator functions, i.e. functions with operator values, is employed. The concepts of continuity, derivative, convergence of series, integrals, etc., must be developed for these functions.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833040.png" /> be a function defined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833042.png" />. A parametric operator function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833043.png" /> is defined by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833044.png" />; it places operators of a certain type — functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833045.png" /> — in correspondence with the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833046.png" /> being considered. An operator function is said to be continuous for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833047.png" /> if it can be represented as the product of an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833048.png" /> and a parametric function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833049.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833050.png" /> is continuous in the ordinary sense.
+
Let $  f( \lambda , t) $
 +
be a function defined for $  t \geq  0 $
 +
and $  \lambda \in [ a, b] $.  
 +
A parametric operator function $  f( \lambda ) $
 +
is defined by the formula $  f( \lambda ) = \{ f( \lambda , t) \} $;  
 +
it places operators of a certain type — functions in $  t $—  
 +
in correspondence with the values of $  \lambda $
 +
being considered. An operator function is said to be continuous for $  \lambda \in [ a, b] $
 +
if it can be represented as the product of an operator $  q $
 +
and a parametric function $  f _ {1} ( \lambda ) = \{ f _ {1} ( \lambda , t) \} $
 +
such that $  f _ {1} ( \lambda , t) $
 +
is continuous in the ordinary sense.
  
 
===Examples.===
 
===Examples.===
  
 +
1) Using the parametric function  $  h( \lambda ) = \{ h( \lambda , t) \} $:
 +
 +
$$
 +
h( \lambda , t)  =  \left \{
  
1) Using the parametric function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833051.png" />:
+
\begin{array}{cll}
 +
0  &\textrm{ for }  &0 \leq  t < \lambda ,  \\
 +
t - \lambda  &\textrm{ for }  &0 \leq  \lambda \leq  t,  \\
 +
\end{array}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833052.png" /></td> </tr></table>
+
\right .$$
  
 
the Heaviside function is defined:
 
the Heaviside function is defined:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833053.png" /></td> </tr></table>
+
$$
 +
H( \lambda )  = s \{ h( \lambda , t) \} .
 +
$$
  
 
The values of the hyperbolic exponential function
 
The values of the hyperbolic exponential function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833054.png" /></td> </tr></table>
+
$$
 +
e ^ {- \lambda s }  \equiv  sH( \lambda )  = s  ^ {2} \{ h(
 +
\lambda , t) \}
 +
$$
  
are called shift operators, since multiplication of a given function by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833055.png" /> requires the displacement of its graph over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833056.png" /> in the positive direction of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833057.png" />-axis.
+
are called shift operators, since multiplication of a given function by $  e ^ {- \lambda s } $
 +
requires the displacement of its graph over $  \lambda $
 +
in the positive direction of the $  t $-
 +
axis.
  
 
2) The solution of the heat equation
 
2) The solution of the heat equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833058.png" /></td> </tr></table>
+
$$
 +
 
 +
\frac{\partial  x }{\partial  t }
 +
  =
 +
\frac{\partial  ^ {2} x }{\partial
 +
\lambda  ^ {2} }
 +
 
 +
$$
  
 
can be expressed using the parabolic exponential function (which is also a parametric operator function):
 
can be expressed using the parabolic exponential function (which is also a parametric operator function):
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833059.png" /></td> </tr></table>
+
$$
 +
e ^ {- \lambda \sqrt s }  = \left \{
  
3) A periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833060.png" /> with period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833061.png" /> has the representation:
+
\frac \lambda {2 \sqrt \pi t  ^ {3} }
 +
  \mathop{\rm exp} \left (
 +
-
 +
\frac{\lambda  ^ {2} }{4t}
 +
\right ) \right \} .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833062.png" /></td> </tr></table>
+
3) A periodic function  $  f( t) $
 +
with period  $  2 \lambda _ {0} $
 +
has the representation:
  
4) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833063.png" /> has numerical values in the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833064.png" />, then
+
$$
 +
\{ f \}  =
 +
\frac{\int\limits _ { 0 } ^ { {2 }  \lambda _ {0} } e ^ {-
 +
\lambda s } f( \lambda )
 +
d \lambda }{1 - e ^ {- 2 \lambda _ {0} s } }
 +
.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833065.png" /></td> </tr></table>
+
4) If  $  f( \lambda ) $
 +
has numerical values in the interval  $  [ \lambda _ {1} , \lambda _ {2} ] $,
 +
then
  
i.e. multiplication of the given function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833066.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833067.png" /> with subsequent integration entails a truncation of its graph. In particular,
+
$$
 +
\int\limits _ { \lambda _ {1} } ^ { {\lambda _ 2 } } e ^ {-
 +
\lambda s } f( \lambda )  d \lambda  = \
 +
\left \{
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833068.png" /></td> </tr></table>
+
\begin{array}{cl}
 +
f( \lambda ),  &\lambda _ {1} < t < \lambda _ {2} ,  \\
 +
0,  &0 \leq  t <
 +
\lambda _ {1} , t > \lambda _ {2} ,  \\
 +
\end{array}
  
Thus, with each function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833069.png" /> for which the integral being considered is convergent there is a corresponding analytic function:
+
\right .$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833070.png" /></td> </tr></table>
+
i.e. multiplication of the given function  $  \{ f \} $
 +
by  $  e ^ {- \lambda s } $
 +
with subsequent integration entails a truncation of its graph. In particular,
  
its Laplace transform. As a result, a fairly broad class of operators is described by functions of one parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833071.png" />; moreover, this formal similarity is defined more exactly in mathematical terms by establishing a definite isomorphism.
+
$$
 +
\int\limits _ { 0 } ^  \infty  e ^ {- \lambda s } f( \lambda )  d \lambda
 +
= \{ f( t) \} .
 +
$$
  
There are various generalizations of operational calculus; for example, operational calculus of differential operators other than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833072.png" />, for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833073.png" />, which is based on other function rings with a properly defined product.
+
Thus, with each function  $  f( t) $
 +
for which the integral being considered is convergent there is a corresponding analytic function:
 +
 
 +
$$
 +
F( s)  =  \int\limits _ { 0 } ^  \infty  e  ^ {-} st f( t)  dt,
 +
$$
 +
 
 +
its Laplace transform. As a result, a fairly broad class of operators is described by functions of one parameter  $  s $;
 +
moreover, this formal similarity is defined more exactly in mathematical terms by establishing a definite isomorphism.
 +
 
 +
There are various generalizations of operational calculus; for example, operational calculus of differential operators other than $  s= d/dt $,  
 +
for example, $  b= d/dt( t( d/dt)) $,  
 +
which is based on other function rings with a properly defined product.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.A. Ditkin,  A.P. Prudnikov,  "Handbook of operational calculus" , Moscow  (1965)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Mikusiński,  "Operational calculus" , Pergamon  (1959)  (Translated from Polish)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.A. Ditkin,  A.P. Prudnikov,  "Handbook of operational calculus" , Moscow  (1965)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Mikusiński,  "Operational calculus" , Pergamon  (1959)  (Translated from Polish)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
A second edition of [[#References|[2]]] has recently appeared, [[#References|[a1]]], [[#References|[a2]]]. In the examples of parametric operator functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833074.png" /> above, use is made of a differential and integral calculus for operators. For more details on the truncation of an operator function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833075.png" /> see [[#References|[a2]]], Part V, Chapt. 1, § 5. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833076.png" /> an operator of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068330/o06833077.png" /> is identified with a Schwartz distribution with support bounded from below.
+
A second edition of [[#References|[2]]] has recently appeared, [[#References|[a1]]], [[#References|[a2]]]. In the examples of parametric operator functions $  f ( \lambda ) $
 +
above, use is made of a differential and integral calculus for operators. For more details on the truncation of an operator function $  f ( \lambda ) $
 +
see [[#References|[a2]]], Part V, Chapt. 1, § 5. For $  f \in K $
 +
an operator of the form $  e ^ {- \lambda s } f $
 +
is identified with a Schwartz distribution with support bounded from below.
  
 
The notion of a Schwartz distribution and Mikusiński operator do not include each other, but both generalize the idea of a function and its derivatives.
 
The notion of a Schwartz distribution and Mikusiński operator do not include each other, but both generalize the idea of a function and its derivatives.

Latest revision as of 14:54, 7 June 2020


One of the methods of mathematical analysis which in many cases makes it possible to reduce the study of differential operators, pseudo-differential operators and certain types of integral operators (cf. Differential operator; Integral operator; Pseudo-differential operator) and the solution of equations containing them, to an examination of simpler algebraic problems. The development and systematic use of operational calculus began with the work of O. Heaviside (1892), who proposed formal rules for dealing with the differentiation operator $ d/dt $ and solved a number of applied problems. However, he did not give operational calculus a mathematical basis; this was done with the aid of the Laplace transform; J. Mikusiński (1953) put operational calculus into algebraic form, using the concept of a function ring. The most general concept of an operational calculus is obtained using generalized functions (cf. Generalized function).

The simplest variant of operational calculus is as follows. Let $ K $ be the set of functions (with real or complex values) given in the domain $ 0 \leq t < \infty $ and absolutely integrable in any finite interval. The integral

$$ h = f \star g = \int\limits _ { 0 } ^ { t } f( t - \tau ) g( \tau ) d \tau $$

is called the convolution of the functions $ f, g \in K $. With the usual addition operation and the operation of convolution, $ K $ becomes a ring without zero divisors (Titchmarsh's theorem, 1924). Elements of the quotient field $ P $ of this ring are called operators and are written as $ a/b $; the fact that division in $ K $ is not always possible is precisely the source of a new concept, operators, which generalizes the concept of a function. To indicate the necessary difference in operational calculus between the concepts of a function and of its value at a point, the following notation is used:

$$ \{ f( t) \} \textrm{ for the function } f \textrm{ of a variable } t ; $$

$$ f( t) \textrm{ for the value of } \{ f( t) \} \textrm{ at a point } t. $$

Examples of operators.

1) $ e = \{ 1 \} $ is the integration operator:

$$ \{ 1 \} \{ f \} = \left \{ \int\limits _ { 0 } ^ { t } f( \tau ) d \tau \right \} . $$

Moreover,

$$ e ^ {p} = \left \{ \frac{t ^ {p-} 1 }{\Gamma ( p) } \right \} $$

and, in particular,

$$ e ^ {n} \{ f \} = \ \int\limits _ { 0 } ^ { t } dt \dots \int\limits _ { 0 } ^ { t } f( t) dt ( n \textrm{ - fold } ) = $$

$$ = \ \int\limits _ { 0 } ^ { t } \frac{( t- \tau ) ^ {n-} 1 }{( n- 1) ! } f( \tau ) d \tau $$

This is the Cauchy formula, a generalization of which to the case of an arbitrary (non-integer) index serves to define fractional integration.

2) $ [ \alpha ] = \{ \alpha \} / \{ 1 \} $( where $ \alpha $ is a constant function) is a numerical operator; insofar as $ [ \alpha ] [ \beta ] = [ \alpha \beta ] $, $ [ \alpha + \beta ] = [ \alpha ] + [ \beta ] $, $ [ \alpha ] \{ f \} = \{ \alpha f \} $, while $ \{ \alpha \} \{ \beta \} = \{ \alpha \beta t \} $, numerical operators behave as ordinary numbers. Thus the operator is a generalization not only of a function, but also of a number; $ [ 1] $ is the unit of the ring $ K $.

3) $ s = [ 1] / e $ is the differentiation operator, the inverse of the integration operator. So, if a function $ a( t) = \{ a( t) \} $ has a derivative $ a ^ \prime ( t) $, then

$$ s \{ a \} = \{ a ^ \prime \} + [ a( 0)] $$

and

$$ \{ a ^ {(} n) \} = \ s ^ {n} \{ a \} - s ^ {n-} 1 [ a( 0)] - \dots - [ a ^ {n-} 1 ( 0)]. $$

Hence, for example,

$$ \{ e ^ {\alpha t } \} = \frac{1}{s- a } . $$

Of course, a non-differentiable function can be multiplied by the differentiation operator $ s $; however, the result will be, in general, an operator.

4) $ D \{ f \} = \{ - tf( t) \} $ is the algebraic derivative. It extends to arbitrary operators in the usual way. It appears that the action of this operator on a function of the differentiation operator $ s $ coincides with differentiation with respect to $ s $.

Operational calculus provides suitable methods for the solution of linear differential equations, both ordinary and partial. For example, the solution of the equation

$$ \alpha _ {n} x ^ {(} n) + \dots + \alpha _ {0} x = f,\ \ \alpha _ {i} = \textrm{ const } ,\ \ i= 0 \dots n, $$

satisfying the initial conditions $ x( 0) = \gamma _ {0} \dots x ^ {(} n- 1) ( 0) = \gamma _ {n-} 1 $ automatically reduces to an algebraic equation. It is expressed symbolically by the formula

$$ x = \frac{\beta _ {n-} 1 s ^ {n-} 1 + \dots + \beta _ {0} + f }{\alpha _ {n} s ^ {n} + \dots + \alpha _ {0} } , $$

$$ \beta _ {v} = \alpha _ {v+} 1 \gamma _ {0} + \dots + \alpha _ {n} \gamma _ {n-} v- 1 . $$

The solution in its usual form is obtained by decomposition into elementary fractions with respect to the variable $ s $, with subsequent inverse transformation by referring to appropriate function tables.

In the use of operational calculus for partial differential equations (as well as for more general pseudo-differential equations), a differential and integral calculus of operator functions, i.e. functions with operator values, is employed. The concepts of continuity, derivative, convergence of series, integrals, etc., must be developed for these functions.

Let $ f( \lambda , t) $ be a function defined for $ t \geq 0 $ and $ \lambda \in [ a, b] $. A parametric operator function $ f( \lambda ) $ is defined by the formula $ f( \lambda ) = \{ f( \lambda , t) \} $; it places operators of a certain type — functions in $ t $— in correspondence with the values of $ \lambda $ being considered. An operator function is said to be continuous for $ \lambda \in [ a, b] $ if it can be represented as the product of an operator $ q $ and a parametric function $ f _ {1} ( \lambda ) = \{ f _ {1} ( \lambda , t) \} $ such that $ f _ {1} ( \lambda , t) $ is continuous in the ordinary sense.

Examples.

1) Using the parametric function $ h( \lambda ) = \{ h( \lambda , t) \} $:

$$ h( \lambda , t) = \left \{ \begin{array}{cll} 0 &\textrm{ for } &0 \leq t < \lambda , \\ t - \lambda &\textrm{ for } &0 \leq \lambda \leq t, \\ \end{array} \right .$$

the Heaviside function is defined:

$$ H( \lambda ) = s \{ h( \lambda , t) \} . $$

The values of the hyperbolic exponential function

$$ e ^ {- \lambda s } \equiv sH( \lambda ) = s ^ {2} \{ h( \lambda , t) \} $$

are called shift operators, since multiplication of a given function by $ e ^ {- \lambda s } $ requires the displacement of its graph over $ \lambda $ in the positive direction of the $ t $- axis.

2) The solution of the heat equation

$$ \frac{\partial x }{\partial t } = \frac{\partial ^ {2} x }{\partial \lambda ^ {2} } $$

can be expressed using the parabolic exponential function (which is also a parametric operator function):

$$ e ^ {- \lambda \sqrt s } = \left \{ \frac \lambda {2 \sqrt \pi t ^ {3} } \mathop{\rm exp} \left ( - \frac{\lambda ^ {2} }{4t} \right ) \right \} . $$

3) A periodic function $ f( t) $ with period $ 2 \lambda _ {0} $ has the representation:

$$ \{ f \} = \frac{\int\limits _ { 0 } ^ { {2 } \lambda _ {0} } e ^ {- \lambda s } f( \lambda ) d \lambda }{1 - e ^ {- 2 \lambda _ {0} s } } . $$

4) If $ f( \lambda ) $ has numerical values in the interval $ [ \lambda _ {1} , \lambda _ {2} ] $, then

$$ \int\limits _ { \lambda _ {1} } ^ { {\lambda _ 2 } } e ^ {- \lambda s } f( \lambda ) d \lambda = \ \left \{ \begin{array}{cl} f( \lambda ), &\lambda _ {1} < t < \lambda _ {2} , \\ 0, &0 \leq t < \lambda _ {1} , t > \lambda _ {2} , \\ \end{array} \right .$$

i.e. multiplication of the given function $ \{ f \} $ by $ e ^ {- \lambda s } $ with subsequent integration entails a truncation of its graph. In particular,

$$ \int\limits _ { 0 } ^ \infty e ^ {- \lambda s } f( \lambda ) d \lambda = \{ f( t) \} . $$

Thus, with each function $ f( t) $ for which the integral being considered is convergent there is a corresponding analytic function:

$$ F( s) = \int\limits _ { 0 } ^ \infty e ^ {-} st f( t) dt, $$

its Laplace transform. As a result, a fairly broad class of operators is described by functions of one parameter $ s $; moreover, this formal similarity is defined more exactly in mathematical terms by establishing a definite isomorphism.

There are various generalizations of operational calculus; for example, operational calculus of differential operators other than $ s= d/dt $, for example, $ b= d/dt( t( d/dt)) $, which is based on other function rings with a properly defined product.

References

[1] V.A. Ditkin, A.P. Prudnikov, "Handbook of operational calculus" , Moscow (1965) (In Russian)
[2] J. Mikusiński, "Operational calculus" , Pergamon (1959) (Translated from Polish)

Comments

A second edition of [2] has recently appeared, [a1], [a2]. In the examples of parametric operator functions $ f ( \lambda ) $ above, use is made of a differential and integral calculus for operators. For more details on the truncation of an operator function $ f ( \lambda ) $ see [a2], Part V, Chapt. 1, § 5. For $ f \in K $ an operator of the form $ e ^ {- \lambda s } f $ is identified with a Schwartz distribution with support bounded from below.

The notion of a Schwartz distribution and Mikusiński operator do not include each other, but both generalize the idea of a function and its derivatives.

The term "operational calculus" is also used in the sense of functional calculus; i.e. a homomorphism of a certain algebra of functions into an algebra of operators. Finally, the phrase "operational calculus" or "operator calculusoperator calculus" occurs in the context of the time-ordered operator calculus (Feynman–Dyson time-ordered operator calculus) developed in the 1950's for the study of quantum electrodynamics [a4], [a5], and relating to product integrals (cf. Product integral), [a6].

References

[a1] J. Mikusiński, "Operational calculus" , 1 , PWN & Pergamon (1987) (Translated from Polish)
[a2] J. Mikusiński, Th.K. Boehme, "Operational calculus" , II , PWN & Pergamon (1987)
[a3] B. van der Pol, H. Bremmer, "Operational calculus based on the two-sided Laplace integral" , Cambridge Univ. Press (1959)
[a4] R.P. Feynman, "An operator calculus having applications in quantum electrodynamics" Phys. Rev. , 84 (1951) pp. 108–128
[a5] T.L. Gill, W.W. Zachary, "Time-ordered operators and Feynman–Dyson algebras" J. Math. Phys. , 28 (1987) pp. 1459–1470
[a6] J.D. Dollard, Ch.N. Friedman, "Product integration" , Addison-Wesley (1979)
How to Cite This Entry:
Operational calculus. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Operational_calculus&oldid=49341
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article