Numerical and Symbolic Inverse Laplace Transform

526ae638-319d-46f8-b844-c00208042099 9 4 5 false false 0 decimal Numerical and Symbolic Inverse Laplace Transform

XY# x# cinterp XY# 1 col XY# 2 col x# cinterp 11 line :

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

1 F t.i t.e N P ILap a ns nd 13 mat P : m ns nd + 1 + : 12 mat t N 1 matrix : ft t : t.o 105 -^t.i 21 mat max eval : Δt t.e t.o - N 1 - / : 14 mat α m 1 matrix : β α : γ nd 1 matrix : 13 mat n 1 m range : α n el a n 1 - (π * i * + eval : β n el 1 - (n 1 +^: 13 mat n h nd 1 + Γ 2 nd^nd 2 + n - Γ * n Γ * / eval : n 2 nd range : γ 1 el 1 h : γ n el n h γ n 1 - el + : 13 mat n ns 3 + (m range : β 1 el 0.5 β 1 el * : β n el β n el γ m n - 1 + el * : 13 mat β e a^β * eval : k 1 N range t k el t.o k 1 - (Δt * + : αt α t k el / : βt β t k el / : 13 mat ft k el βt αt F * Re vectorize eval sum : 21 line for t ft augment eval 101 line :

1 F t.i t.e N ILap s# F t.i t.e N 6201913 mat ILap 11 line :

ζ Γ Γ 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 : z ζ 1 - : zh 1 - 2 ζ * + 2 / : zg zh 607128 / + : 13 mat 2 π * sqrt zg 0.5 zh *^(2^* e zg^/ Γ 1 el Γ k 1 + el z k + / k k Clear 14 sum + (* eval 31 line :

Valid for Re(ζ)>0

Heaviside x Φ 1 x sign + 2 / :

ode ic fnc t s LapOde r 1 ic rows range : c 1 ic cols range : 12 mat ODE L f F A# Clear f r el fnc r 1 el num2str (t num2str) concat str2num : F r el fnc r 2 el num2str (s num2str) concat str2num : A# r c el f r el t c 1 - diff t 0 ≡ ic r c el atvalue Maxima : ODE r el ode r el t s laplace Maxima : L r el f r el t s laplace : 12 mat S ODE L Solve : I r el S r el s t ilt Maxima : 12 mat x# Su s# x# : k# 1 ic rows range s# F k# el L k# el s# subst Maxima : for s# 31 line : ODE Su S Su I f F L 16 mat 91 line :

Maxima Laplace

Initial value problem

ode t f t 2 diff t g t diff + 3 t f * + 15 t - exp * ≡ t g t 2 diff 4 t f t diff * - 3 t g * + 152 t * sin * ≡ 21 mat :

ic 3548 - 2755 - 22 mat :

0 f 0 f' 0 g 0 g' 22 mat

fnc f F g G 22 mat :

d\002D\domain s\002D\domain 12 mat

Symbolic Solver call

ODE ode ic fnc t s LapOde :

Laplace transform of the ode

ODE 1 el s 35 - s s F * + s G + (* 37 s F + (* + 151 s + / ≡ 553 s 2^+ (s G * + 435 - s s F * + (* - 27 s * - 304 s 2^+ / ≡ 21 mat

Freq-domain solution, used in numerical solution

ODE 2 el s F 626 - 85 s 2^* + (* 153 s * + (* s 3^185 s 407 s 13 - 35 s * + (* + (* + (* + 21 s + (* 187 s 4^* + (* s 2^49 s 49 s 3^1 s + (* + (* + (* + / ≡ s G 33250 - 3 s 6^* + (* 674 s * - (* s 2^787 - 2 s * 305 - 2514 s * - (s * + (* + (* + 21 s + (* 187 s 4^* + (* s 2^49 s 49 s 3^1 s + (* + (* + (* + / ≡ 21 sys

Time-domain solution

ODE 3 el t f 3152 t cos * 3 t * sin - (* e t^* + (* 22 t * cos * e t^* + e t^/ ≡ t g 31 - 102 t sin * - 3 t * cos + (* e t^* + (* 2 t * sin e t^* + e t^/ ≡ 21 mat

Plots: numerical and symbolic solutions

ODE 2 el Assign ODE 3 el Assign ti te N 13 mat 0162513 mat : f s F ti te N ILap : t fn f t cinterp : g s G ti te N ILap : t gn g t cinterp : 71 line

x f x ti te lele ( * x g x ti te lele ( * f o 5 blue augment g o 5 red augment x fn x ti te lele ( * x gn x ti te lele ( * 8 1 sys

Increase N for better numerical adjust.

Alvaro