Numerical Inverse Laplace Transform

a807e1ea-e61f-46a4-9080-70fd2ae62934 103 4 5 false false 0 decimal Numerical Inverse Laplace Transform

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 :

equrep

Euler Mascheroni constant γ 0.577215664901532860606 :

f# a# b# AUXFS f# num2str (a# num2str, b# num2str) concat str2num 11 line :

f# a# b# c# AUXFS f# num2str (a# num2str, b# num2str, c# num2str) concat str2num 11 line :

expr# Rules# ApplyERRules ans# expr# : n# 1 Rules# rows range ans# equrep ans# Rules# n# 1 el Rules# n# 2 el AUXFS : for ans# 31 line :

expr# Rules# ApplyNERRules ans# expr# : N# 120 range old# ans# : ans# ans# Rules# ApplyERRules : old# num2str ans# num2str ≡ break continue if 31 line for ans# 31 line :

expr# equal2EQ ans# expr# num2str : ans# \2261\ findstr 1 - ≡ ans# ans# \2261\0 concat : n# ans# \2261\ findstr 1 el : 12 mat n# ans# \2261\ findstr 1 el : if ans# 1 n# 1 - substr str2num ans# n# 1 + ans# strlen n# - substr str2num 12 mat 31 line :

a# x# depends diff a# x# AUXFS num2str 0 ≠ 11 line :

a# x# isfree a# x# depends ¬ 11 line :

laplace

LT LT(a) LT.0(a) 013 mat :

LT LT sqrt(a) a^(1/2) sqrt 13 mat stack :

LT LT nthroot(a,b) a^b nthroot 13 mat stack :

LT LT LT(sin(a)) LT.φω(sin(a),a,(s*sin(lφ)+lω*cos(lφ))/(s^2+lω^2) sin 13 mat stack :

LT LT LT(cos(a)) LT.φω(cos(a),a,(s*cos(lφ)-lω*sin(lφ))/(s^2+lω^2) cos 13 mat stack :

LT LT LT(sinh(a)) LT.φω(sinh(a),a,(s*sinh(lφ)+lω*cosh(lφ))/(s^2-lω^2) sinh 13 mat stack :

LT LT LT(cosh(a)) LT.φω(cosh(a),a,(s*cosh(lφ)+lω*sinh(lφ))/(s^2-lω^2) cosh 13 mat stack :

LT LT LT(exp(a)) LT.φω(exp(a),a,e^lφ/(s-lω) exp 13 mat stack :

LT LT LT(ln(a)) LT.φω(ln(a),a,(ln(lω)-ln(s)-'γ)/s ln 13 mat stack :

LT.α sin cos sinh cosh 14 mat :

lk 14 range LT LT LT(exp(a)* LT.α lk el (b)) concat LT.α(exp(a), LT.α lk el (b),a,b) concat exp* LT.α lk el concat 13 mat stack : for

LT LT LT(-a) 0-LT(a) minus 13 mat stack :

LT LT LT(a^n*b) LT.n(a,n,b) nshift 13 mat stack :

LT LT LT(a^b) LT.pot(a,b) pot 13 mat stack :

LT LT LT(a*b) LT.prod(a,b) product 13 mat stack :

LT LT LT(a*(b+c)) LT(a*b)+LT(a*c) expand 13 mat stack :

LT LT LT(a+b) LT(a)+LT(b) plus 13 mat stack :

LT.O t s^(-2) 12 mat :

LT.O LT.O x(t) X(s) 12 mat stack :

LT.O LT.O u(t) 1/s 12 mat stack :

LT.O LT.O δ(t) 112 mat stack :

LT.O LT.O diff(x(t),t) s*X(s) 12 mat stack :

LT.O LT.O diff(diff(x(t),t),t) s^2*X(s)-x'.o-s*x.o 12 mat stack :

LT.O LT.O diff(diff(diff(x(t),t),t),t) s^3*X(s)-x''.o-s*x'.o-s^2*x.o 12 mat stack :

expr# lt t s t x s X lω lφ Clear a# LT.0 a# t depends row# LT.O a# num2str 1 findrows : (0 ≡ a# LT row# 2 el str2num if a# s 1 -^* if 11 line : A# B# a# b# LT.α lω diff b# t AUXFS : lα diff a# t AUXFS : lα t depends lω t depends | A# B# * LT s tr# B# lt : s lα - tr# 21 line if 31 line : f# a# tr# LT.φω lω diff a# t AUXFS : lφ a# lω t * - : 12 mat a# t depends lω t depends f# LT tr# if a# s 1 -^* if 21 line : a# n# b# LT.n n# t isfree a# num2str t ≡ (& 1 - (n#^b# lt s n# diff * a# n#^b# * LT if 11 line : a# n# LT.pot n# IsDefined a# num2str t ≡ (& n#! (s n# 1 + (-^* a# num2str e ≡ n# exp LT a# n#^LT if if 11 line : a# b# LT.prod a# t isfree a# b# LT * b# t isfree b# a# LT * a# num2str t ≡ b# LT s diff - b# num2str t ≡ a# LT s diff - a# b# * LT if if if if 11 line : expr# LT LT ApplyNERRules 81 line :

Laplace examples

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 :

t f.s 1 t - exp 2 / 3 t * cos 5 t * sin + (* - :

s F t f.s lt : 2251 s + (2^+ (* 91 s + (2^+ (* 591 s + (2^+ (* 1 s + (251 s + (2^+ (* + (s * - 291 s + (2^+ (* 251 s + (2^+ (* s * /

ft s F 06 N ILap : t f ft t cinterp : 21 sys

plot

t f.s t 2^t sin + :

s F t f.s lt : s 3^21 s 2^+ (* + s 3^1 s 2^+ (* /

ft s F 05 N ILap : t f ft t cinterp : 21 sys

plot

t f.s 0.5 t * - exp 3 t * sin * :

s F t f.s lt : 123612 s * + (2^+ /

ft s F 06 N ILap : t f ft t cinterp : 21 sys

plot

t f.s t 5 t * cos * :

s F t f.s lt : 25 s 2^- 25 s 2^+ (2^/ -

ft s F 05 N ILap : t f ft t cinterp : 21 sys

plot

t f.s t 2^t 5 / sinh * :

s F t f.s lt : 2505 s * 0 sinh * 0 cosh + (1 - 25 s 2^* + (* 101 - 25 s 2^* + (0 sinh * 105 s * 0 sinh * 0 cosh + (* s * - (* s * + (* 1 - 25 s 2^* + (3^/ -

ft s F 05 N ILap : t f ft t cinterp : 21 sys

plot

Alvaro