Lösungen Laplace-Transformationen

Lösungen Laplace-Transformationen

?*Laplace*

System`

InverseLaplaceTransform LaplaceTransform

?Laplace*

LaplaceTransform @ expr, t, s D gives the Laplace

transform of expr. LaplaceTransform @ expr, 8 t1, t2, ... < , 8 s1,

s2, ... <D gives the multidimensional Laplace transform of expr. Mehr…

?InverseLaplace*

InverseLaplaceTransform @ expr, s, t D gives the inverse Laplace transform of expr. InverseLaplaceTransform @ expr, 8 s1, s2, ... < , 8 t1, t2, ... <D gives the multidimensional inverse Laplace transform of expr. Mehr…

1

Ÿ a

transf = LaplaceTransform[(y''[t]+ w

y[t]==Sin[t]),t,s]/.{LaplaceTransform[y[t],t,s]->Y[s]}

-s y @ 0 D + s ² Y @ s D + w Y @ s D - y

@ 0 D Š 1

€€€€€€€€€€€€€€€

1 + s ²

Ÿ b

solv = Solve[transf,{Y[s]}] // Flatten 9 Y @ s D ® €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ 1 + s y @ 0 D + H 1 s €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ ³ + y s @ ² 0 L H D + s ² y

+ wL @ 0 D €€€€€€€€€€€€€€€€€€€€€€€€€€€ + s ² y

@ 0 D =

Y1[s_]:=Y[s] /. solv Y1[s]

1 + s y @ 0 D + s ³ y @ 0 D + y

@ 0 D + s ² y

@ 0 D

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ H 1 €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ + s ² L H s ² + wL €€€€€€€€€€€€€€€€€€€€€€€€€€€

Y2[s_]:=Y1[s] /. {y[0]->1,y'[0]->1}

Y2[s]

2 + s + s ² + s ³

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ H 1 + s ² L H s ² + w L

LMMathAna1_05.nb 1

Ÿ c

InverseLaplaceTransform[Y2[s],s,t]

H-1 + wL •!!!! w Cos A t •!!!! w E + •!!!! w Sin @ t D + H-2 + wL Sin A t •!!!! w E

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ €€€€€€€€€€€€€€€

H-1 + wL •!!!! w

Ÿ d

(InverseLaplaceTransform[Y2[s],s,t] /. w-> 1/(2 p))//Simplify

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ •!!!! 2 H-1 1 + 2 pL i

k jjj•!!!! 2 H -1 + 2 p L Cos A €€€€€€€€€€€€€€ •!!!!!!!! 2 t p E + 2 •!!!! p i

k jjj - •!!!!!!!! 2 p Sin @ t D + H -1 + 4 p L Sin A €€€€€€€€€€€€€€ •!!!!!!!! 2 t p Ey { zzzy

{ zzz Plot[%,{t,0,12 p}];

5 10 15 20 25 30 35

-6 -4 -2 2 4 6

Plot[%%,{t,0,26 p}];

20 40 60 80

-6 -4 -2 2 4 6

Ÿ d

Ueberlagerung von periodischen Funktionen.

LMMathAna1_05.nb 2

2

Ÿ a

E^(-s Pi/2) LaplaceTransform[Cos[t],t,s]

ã

s

€€€€€€€€€€€€€€€€€

1 + s ²

Ÿ b

LaplaceTransform[E^(t Pi/2) Cos[t],t,s]

- €€€€

₂ + s

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€

1 + H- €€€€

₂ + s L ²

Ÿ c

LaplaceTransform[t^3 Cos[t],t,s]

6 H 1 - 6 s ² + s ⁴ L

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€

H 1 + s ² L ⁴

Ÿ d

LaplaceTransform[Sin[2t]/t,t,s]

ArcTan A 2

€€€€ s E

Ÿ e

LaplaceTransform[(Cos[t-1])/t,t,s]

- 1

€€€€ 2 Cos @ 1 D H 2 EulerGamma + Log @ 1 + s ² DL + ArcCot @ s D Sin @ 1 D So weit gehen die bei uns behandelten Regeln nicht... ==> Handarbeit!

Ÿ e1

LaplaceTransform[(Cos[t]-1)/t,t,s]

-¥

LMMathAna1_05.nb 3

Ÿ f

LaplaceTransform[Evaluate[Integrate[Sin[l] Cos[t-l],{l,0,t}]],t,s] // Simplify

€€€€€€€€€€€€€€€€€€€€€€€ s H 1 + s ² L ²

LaplaceTransform[Sin[t],t,s]*LaplaceTransform[Cos[t],t,s] // Simplify

€€€€€€€€€€€€€€€€€€€€€€€ s H 1 + s ² L ²

Ÿ g

f[t_]:=t-Floor[t]; Plot[f[t],{t,-3,3},AspectRatio->Automatic];

-3 -2 -1 1 2 3

0.2 0.4 0.6 0.8 1

LaplaceTransform[f[t],t,s]

€€€€€€€ 1

s ² - LaplaceTransform @ Floor @ t D , t, s D r@ s_ D := E ^ H-s * 1 L ;

1 • H 1 - r@ s DL Integrate @ E ^ H-s tL * t, 8t, 0, 1 <D 1 - ã

H 1 + s L

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ H 1 - ã

L s ²

LMMathAna1_05.nb 4