Re: laplace transform

• To: mathgroup at smc.vnet.net
• Subject: [mg8571] Re: [mg8520] laplace transform
• From: "w.meeussen" <meeussen.vdmcc at vandemoortele.be>
• Date: Sat, 6 Sep 1997 23:16:32 -0400
• Sender: owner-wri-mathgroup at wolfram.com

```At 02:20 4-09-97 -0400, Jaime Y. Hernandez Jr. wrote:
>Dear Mathgroup Members,
>
>I'd like to clarify my question regarding the Inverse Laplace transform
>question I raised. I'm actually solving something like this,
>
>        In[2]:=InverseLaplaceTransform[1/(s^k+a),s,t]
>
>        Out[2]=\!\(InverseLaplaceTransform[1\/\(a + s\^k\), s, t]\)
>
>This is actual mathematica output. Mathematica always returns what I input
>no matter what value (fractional value between 0 and 1) of k I place,
>except if you change s^k with Sqrt[s], i.e. k=1/2. If one places k=0.5 it
>does not work. Is the result for using Sqrt[s] only a formula within
>mathematica? Anyway, if you have any suggestions on how to solve such a
>problem, again I'd be very grateful.
>And sorry about the lack of details earlier.
>
>Jaime
>
>
>

for fractional s I found nothing new,

but for integer s there are solutions to be found after tinkering:
(* along the way, I hit on what seems a bug in FullSimplify that needs
verification *)

Also, how impossible would it be to write a "Induction" package that
automates the procedure a human does : incrementing a parameter (like n
below) and "guess" the progression?

check it out:

In[1]:=
<<Calculus`LaplaceTransform`
In[2]:=
InverseLaplaceTransform[1/(s^k+a),s,t]
Out[2]=
1
InverseLaplaceTransform[------, s, t]
k
a + s
In[50]:=
Clear[testeven,testodd]
In[51]:=
testeven[n_] := (-1)^(1/n)*Sum[((-1)^((k - 1)/n)/E^((-1)^(k/n)*a^(1/n)*t) -
(-1)^((k - 1)/n)*E^((-1)^(k/n)*a^(1/n)*t))/
(n*a^(1 - 1/n)), {k, 1, n, 2}]
In[52]:=
testodd[n_] := (a^(-1 + 1/n)*(E^(-(a^(1/n)*t)) + Sum[(-1)^(k/n + k)*Exp[(-1)^(
k/n + k + 1)*a^(1/n)*t], {k, 1, n - 1}]))/n

In[53]:=
checkit=FullSimplify[testeven[n]]
(*   this simplification seems to go wrong !!!  *)
Out[53]=
1/n  1/n
1/n  -1 + 1/n        2 (-1)    a    t             -1 + n
(-1)    a         (-1 + E                ) (1 + Floor[------])
2
-(--------------------------------------------------------------)
1/n  1/n
(-1)    a    t
E               n
In[54]:=
InverseLaplaceTransform[1/(s^k+a),s,t]/.k->2;
Simplify[testeven[2]-%]
Out[55]=
0
In[56]:=
InverseLaplaceTransform[1/(s^k+a),s,t]/.k->3;
Simplify[testodd[3]-%]
Out[57]=
0
In[58]:=
InverseLaplaceTransform[1/(s^k+a),s,t]/.k->4;
Simplify[testeven[4]-%]
Out[59]=
0
In[60]:=
InverseLaplaceTransform[1/(s^k+a),s,t]/.k->5;
Simplify[testodd[5]-%]
Out[61]=
0
In[62]:=
Table[{testeven[n],check},{a,1,2},{t,1,2},{n,6}]//N//Chop//TableForm
Out[62]//TableForm=
-2.3504                -2.3504                  -7.25372
-7.25372
0.841471               0.841471                 0.909297
0.909297
-0.400067 - 0.481238 I 0.766801 - 0.962476 I    -1.47567 - 0.398749 I  1.88447
- 0.797498 I
0.166468               0.166468 - 0.991669 I    1.30799                1.30799
- 1.73474 I
-0.235043 - 0.24274 I  -0.201004 - 1.25272 I    -0.726783 - 0.943507 I
0.800543 - 3.00598 I
0.00833331             -0.408236 - 1.01019 I    0.266615
-0.0547257 - 2.86354 I

-7.25372               -7.25372                 -54.5798
-54.5798
0.698456               0.698456                 0.21784                0.21784

-0.357 - 0.337363 I    0.647332 - 0.674726 I    -1.83001 + 0.0109456 I 1.52504
+ 0.0218911 I
0.16627                0.16627 - 0.695329 I     1.28274                1.28274
- 1.04107 I
-0.16295 - 0.179241 I  -0.0946172 - 0.8482 I    -0.568425 - 0.712088 I
0.917239 - 2.06479 I
0.00833328             -0.240382 - 0.659902 I   0.266564
0.180615 - 1.97796 I
In[65]:=
LaplaceTransform[testeven[4],t,s]//Simplify
Out[65]=
1
------
4
a + s
In[67]:=
LaplaceTransform[testodd[5],t,s]//FullSimplify
Out[67]=
1
------
5
a + s

wouter.

Dr. Wouter L. J. MEEUSSEN
eu000949 at pophost.eunet.be
w.meeussen.vdmcc at vandemoortele.be

```