REDUCE

16.34 LAPLACE: Laplace transforms

This package can calculate ordinary and inverse Laplace transforms of expressions. Documentation is in plain text.

Authors: C. Kazasov, M. Spiridonova, V. Tomov.

Reference:

Christomir Kazasov, Laplace Transformations in REDUCE 3, Proc. Eurocal ’87, Lecture Notes in Comp. Sci., Springer-Verlag (1987) 132-133.

 
 
Some hints on how to use to use this package:
 
Syntax:
 
LAPLACE(< exp >,< var - s >,< var - t > )
 
INVLAP(< exp >,< var - s >,< var - t >)
 
where < exp > is the expression to be transformed, < var -s > is the source variable (in most cases < exp > depends explicitly of this variable) and < var -t > is the target variable. If < var - t > is omitted, the package uses an internal variable lp!& or il!&, respectively.
 
The following switches can be used to control the transformations:

lmon:

If on, sin, cos, sinh and cosh are converted by LAPLACE into exponentials,

lhyp:

If on, expressions e˜x are converted by INVLAP into hyperbolic functions sinh and cosh,

ltrig:

If on, expressions e˜x are converted by INVLAP into trigonometric functions sin and cos.

 
The system can be extended by adding Laplace transformation rules for single functions by rules or rule sets.  In such a rule the source variable MUST be free, the target variable MUST be il!& for LAPLACE and lp!& for INVLAP and the third parameter should be omitted.  Also rules for transforming derivatives are entered in such a form.

Examples:

 
    let {laplace(log(~x),x) => -log(gam * il!&)/il!&,  
 
    invlap(log(gam * ~x)/x,x) => -log(lp!&)};  
 
 
    operator f;  
 
    let{  
 
    laplace(df(f(~x),x),x) => il!&*laplace(f(x),x) - sub(x=0,f(x)),  
 
    laplace(df(f(~x),x,~n),x) => il!&**n*laplace(f(x),x) -  
 
    for i:=n-1 step -1 until 0 sum  
 
    sub(x=0, df(f(x),x,n-1-i)) * il!&**i  
 
    when fixp n,  
 
    laplace(f(~x),x) = f(il!&)  
 
    };  

Remarks about some functions:
 
The DELTA and GAMMA functions are known.
ONE is the name of the unit step function.
INTL is a parametrized integral function

intl(< expr >,< var >,0,< obj.var >)

which means "Integral of < expr > wrt.  < var > taken from 0 to < obj.var >", e.g. intl(2*y2,y,0,x) which is formally a function in x.  
 
We recommend reading the file LAPLACE.TST for a further introduction.