LAPLACE: Laplace Transforms

REDUCE

PrevTail

Tail

20.30 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: [Kaz87].

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 above operators use 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(Euler_Gamma * il!&)/il!&,

    invlap(log(Euler_Gamma * ~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⟩ w.r.t. ⟨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.

PrevTail

Front