Laplace transform involving time-shifts

[mforets]

Sage allows to compute the inverse Laplace transform through Maxima's ilt function,

    sage: var('s t')
    sage: inverse_laplace(1/s, s, t)
    1

An unevaluated expression is returned when no explicit inverse Laplace transform is computed, as in

    sage: inverse_laplace(exp(-s)/s, s, t)
    ilt(e^(-s)/s, s, t)

The result in this case is h(t-1), where h is the Heaviside step function. In Sage it is available as heaviside.

The problem in this ticket is to extend the current behavior of inverse_laplace to provide explicit expressions for proper real-rational functions with any number of real exponentials linear in the transform variable s (time-shifts) in the numerator. For consistency, the direct Laplace transform with a heaviside should also work as well.

These are some approaches:

(1) Implement an in-house solution, possibly in the lines of this answer.

(2) Add an algorithm flag that allows to choose sympy (similar to integration).

(3) Interface with Giac/XCAS. With this package installed, it is possible to do:

sage: giac('invlaplace(exp(-s)/s, s, t)')
Heaviside(t-1)

IMHO, a combination of (2)-(3) is the more robust approach. A small set of experiments show that (3) is, at the time of writing, more convenient than inverse_laplace_transform of SymPy in terms of quality of solution and execution time. Unfortunately, the giac interface does not currently support automatic translation back to the symbolic ring, as it does with SymPy objects via SR(..).

Any recommendations?

See also:

• How to find inverse Laplace transform
• laplace(heaviside(t),t,s) doesn't return 1/s
• Inverse Laplace transforms of shifts

CC: @kcrisman @paulmasson @frederichan-IMJPRG @rwst

Component: calculus

Keywords: laplace, transform, symbolics, giac, heaviside

Author: Marcelo Forets

Branch/Commit: f845269

Reviewer: Paul Masson, Ralf Stephan

Issue created by migration from https://trac.sagemath.org/ticket/22422

[mforets]

Branch: u/mforets/laplace_transform_involving_time_shifts

[mforets]

Commit: 2096388

[mforets]

comment:3

Needs info (please see commit message) to transform heaviside via _giac_(). Example:

    sage: f = heaviside(x); f, type(f)
    (heaviside(x), <type 'sage.symbolic.expression.Expression'>)
    sage: fg = f._giac_(); fg,  type(fg)
    (heaviside(x), <class 'sage.interfaces.giac.GiacElement'>)

But heaviside(x) doesn't seem to be understood by giac.

---

New commits:

2096388

Added algorithm='sympy' (needs work) and algorithm='giac' (needs info).

[mforets]

Description changed:

--- 
+++ 
@@ -14,7 +14,7 @@
 ```
 The result in this case is h(t-1), where h is the Heaviside step function. In Sage it is available as `heaviside`.
 
-This problem in this ticket is to extend the current behavior of `inverse_laplace` to provide explicit expressions for proper real-rational functions with any number of real exponentials linear in the transform variable s (time-shifts) in the numerator. For consistency, the direct Laplace transform with a heaviside should also work as well.
+The problem in this ticket is to extend the current behavior of `inverse_laplace` to provide explicit expressions for proper real-rational functions with any number of real exponentials linear in the transform variable s (time-shifts) in the numerator. For consistency, the direct Laplace transform with a heaviside should also work as well.
 
 These are some approaches:
 

[mforets]

Author: Marcelo Forets

[rwst]

comment:4

Code looking good so far but needs doctests for all code branches. Minor: you seem to change spacing, there are empty hunks in the patch, check your editor configuration.

[frederichan-IMJPRG]

comment:5

Replying to @mforets:

Needs info (please see commit message) to transform heaviside via _giac_(). Example:

    sage: f = heaviside(x); f, type(f)
    (heaviside(x), <type 'sage.symbolic.expression.Expression'>)
    sage: fg = f._giac_(); fg,  type(fg)
    (heaviside(x), <class 'sage.interfaces.giac.GiacElement'>)

But heaviside(x) doesn't seem to be understood by giac.

in giac it looks to be Heaviside, so you need to either translate the string or define it in giac:

giac("'heaviside:=Heaviside'")
sage: giac(f).integrate(x,-1,2)
2
sage: f
heaviside(x)
sage: type(f)
<type 'sage.symbolic.expression.Expression'>

similar with giacpy_sage which have a raw conversion to sage:

sage: from giacpy_sage import *
// Giac share root-directory:/home/fred-dev/sage/develop/sage.develop/local/share/giac/
// Giac share root-directory:/home/fred-dev/sage/develop/sage.develop/local/share/giac/
Help file /home/fred-dev/sage/develop/sage.develop/local/share/giac/doc/fr/aide_cas not found
Added 0 synonyms
sage: libgiac('heaviside:=Heaviside')
'Heaviside'
sage: f=heaviside(x)
sage: fg=libgiac(f)
sage: fg.integrate(x,-1,2)
2
sage: type(fg.integrate(x,-1,2))
<type 'giacpy_sage.Pygen'>
sage: SR(fg.integrate(x,-1,2))
2
sage: type(SR(fg.integrate(x,-1,2)))
<type 'sage.symbolic.expression.Expression'>
sage: (fg.integrate(x,-1,2)).sage()
2
sage: type((fg.integrate(x,-1,2)).sage())
<type 'sage.rings.integer.Integer'>

but with Heaviside will be sent to SR as a raw string so you will need some translation back to heaviside also.

[mforets]

comment:6

@rwst: ok, I am learning such things as doctesting in days84. for the spacing issue good that you pointed this out, I'll be more careful. BTW I use Atom editor, and thanks for the prompt feedback.

@frederichan-IMJPRG: thanks for the insight! I see that I could use the conversion at the script level. but then each new Sage function which for some reason needs it, would have to handle the conversion. does it make sense to define this at the level of the giac.py interface?

[sagetrac-git]

Changed commit from 2096388 to a3a43d2

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

a3a43d2

A solution for heaviside <-> Heaviside.