<p>Numerical Inverse Laplace Transform</p>

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Gamma function. Valid for Re(ζ)>0 ζ Γ G 0.99999999999999709182 57.156235665862923517 59.597960355475491248 - 14.136097974741747174 0.49191381609762019978 - 0.33994649984811888699 104 -^* 0.46523628927048575665 104 -^* 0.98374475304879564677 104 -^* - 0.15808870322491248884 103 -^* 0.21026444172410488319 103 -^* - 0.21743961811521264320 103 -^* 0.16431810653676389022 103 -^* - 0.84418223983852743293 104 -^* 0.26190838401581408670 104 -^* - 0.36899182659531622704 105 -^* 151 mat : zg ζ 543128 / + : κ Clear 12 mat 2 π * sqrt zg 0.5 ζ 0.5 - (*^(2^* e zg^/ G 1 el G κ el ζ κ + 2 - / κ 215 sum + (* eval 31 line :

Based on algorithm of Oscar: https://en.smath.info/forum/yaf_postst1341_Signals-and-Systems.aspx

a ns nd ILap.αβ m ns nd + 1 + : α m 1 matrix : β α : γ nd 1 matrix : 14 mat μ 1 m range : nδ 2 nd range : nσ ns 3 + (m range : 13 mat α μ el a μ 1 - (π * i * + eval : β μ el 1 - (μ 1 +^e a^* eval : 12 mat γ 1 el 2 nd -^: γ nδ el nd 1 + Γ 2 nd^nd 2 + nδ - Γ * nδ Γ * / γ nδ 1 - el + eval : 12 mat β 1 el 0.5 β 1 el * : β nσ el β nσ el γ m nσ - 1 + el * eval : 12 mat α β μ 13 mat 61 line :

Inverse Laplace transform of F(s) 1 F t.i t.e N ILap t N 1 matrix : ft t : Fs αL rows 1 matrix : 13 mat t.o 105 -^t.e t.i - UnitsOf * t.i 12 mat max eval : Δt t.e t.o - N 1 - / : 12 mat n 1 N range t n el t.o n 1 - (Δt * + : Fs μL el βL μL el t n el / αL μL el t n el / F * eval : 12 mat ft n el Fs sum Re eval : 21 line for t ft augment eval 41 line :

XY x cinterp X Y 12 mat XY 1 col XY 2 col 12 mat : U X 1 el UnitsOf Y 1 el UnitsOf 12 mat : U 2 el X U 1 el / Y U 2 el / x U 1 el / cinterp * X min eval x ≤ (* x X max eval ≤ (* 31 line :

α and β values αL βL μL 13 mat 62019 ILap.αβ :

plot x f x f.s ft . 12 green augment 31 sys :

N 25 :

s F 10 s 2^2 s * + 10 + / 1 s / * : t f.s 1 t - exp 3 / 33 t * cos * 3 t * sin + (* - : ft s F 06 N ILap : t f ft t cinterp : 41 sys

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From Wolfram demonstations: http://demonstrations.wolfram.com/ComparingFourMethodsOfNumericalInversionOfLaplaceTransformsN/

s F 2 s 3^/ 1 s 2^1 + (/ + : t f.s t 2^t sin + : ft s F 05 N ILap : t f ft t cinterp : 41 sys

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s F s 1 s 2^+ (2^/ : t f.s 12 / t * t sin * : ft s F 010 N ILap : t f ft t cinterp : 41 sys

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s F e 254 s * / -^π sqrt * 2 s * 25 - (* 4 s 52 /^* / : t f.s t sqrt 5 t sqrt * cos * : ft s F 010 N ILap : t f ft t cinterp : 41 sys

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Alvaro