Numerical and Symbolic Inverse Laplace Transform

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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 / :

restart MaximaControl Restart complete.

ode ic fnc t s LapOde r 1 ic rows range : c 1 ic cols range : 12 mat ODE L f F A# κ Clear t 0 > assume Maxima s 0 > assume Maxima 12 mat 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 k 19 range ODE r el 1 - (k^t k^* f r el * L r el s k diff ODE r el subst Maxima : for a b Sol r 1 ≡ a b Solve 11 mat a b Solve if 11 line : x# Su s# x# : k# 1 ic rows range s# F k# el L k# el s# subst Maxima : for s# 31 line : S ODE L Sol : I r el S r el s t ilt Maxima : 12 mat ODE Su S Su I f Sol f F L 16 mat 121 line :

Maxima Laplace

Initial value problem: linear ODE but with polynomial coeffs, not only constants.

ode t x t 2 diff 6 t * t 4^- 16 / t x * t 4 + 2 / t x t diff * - t t cos * - ≡ 11 mat :

ic 1 - 112 mat :

0 x 0 x' 12 mat

fnc x X 12 mat :

t\002D\domain s\002D\domain 12 mat

Symbolic Solver call

ODE ode ic fnc t s LapOde :

Laplace transform of the ode (mixing time and freq domain)

ODE 1 el 1 - s 1 s s X * + (* + 21 s 2^+ (2^3 t * t x * 4 s t * t x * 41 s s X * + (* + s X - (* - (* 81 s 2^- (* + (* 1 s 2^+ (2^t 4^* t x * - 161 s 2^+ (2^* / ≡ 11 mat

ODE1 ode 1 el t s laplace Maxima :

ODE1 1 - s 1 s t x t s laplace * + (* + 24 s t x t s laplace s diff * 41 s t x t s laplace * + (* - t x t s laplace + (* 3 t x t s laplace s diff * - (1 s 2^+ (2^* 81 s 2^- (* + (* t x t s laplace s diff s diff s diff s diff 1 s 2^+ (2^* - 161 s 2^+ (2^* / ≡

ODE1 1 - (t * t x * t x t s laplace s diff ODE1 subst Maxima :

ODE1 1 - (4^t 4^* t x * t x t s laplace s 4 diff ODE1 subst Maxima :

ODE11 s X t x t s laplace ODE1 subst Maxima :

ODE11 1 - s 1 s s X * + (* + 21 s 2^+ (2^3 t * t x * 4 s t * t x * 41 s s X * + (* + s X - (* - (* 81 s 2^- (* + (* 1 s 2^+ (2^t 4^* t x * - 161 s 2^+ (2^* / ≡

Freq-domain solution

ODE 2 el s X 1 s 2^2 s 2^+ (* + (t * t 3^23 - 4 s * + (* + (* t x * 16 s * 1 s 3 s 2 s 1 s + (* + (* + (* + (* + 81 - s 4 s 2^8 s 32 s * 2 s + (* + (* + (* + (* + (* / - ≡ 11 mat

S ODE1 t x t s laplace Solve :

S t x t s laplace 1 s 2^2 s 2^+ (* + (t 3^23 - 4 s * + (* + (* t * t x * 16 s * 1 s 3 s 2 s 1 s + (* + (* + (* + (* + 81 - s 3^8 s 32 s * 2 s + (* + (* + (* + 4 s * + (* / - ≡

Time-domain symbolic solution

ODE 3 el t x 81336 t 6 sqrt * 2 / sinh * 296 sqrt * t 6 sqrt * 2 / cosh * - (* 86 sqrt * 1725 t * + (* e t^* t sin * - 26 sqrt * 124 - 75 t * + (* e t^* t cos * + (* 625 t 14 - t 3^+ (t 6 sqrt * 2 / sinh * 46 sqrt * t 6 sqrt * 2 / cosh * + (* 86 sqrt * e t^* + (* / ≡ 11 mat

ODE3 S s t ilt Maxima :

ODE3 Maxima t x 8468625 t * t x * + (* - 625 t * 6 - t 3^+ (* t x * + (t 6 sqrt * 2 / sinh * 46 sqrt * 754625 t * t x * + (t 6 sqrt * 2 / cosh * 16 t sin * e t^* 1725 t * + (* + 300 t * t cos * e t^* - (* + 1984 t cos * e t^* 6 sqrt * + 5000 e t^* 6 sqrt * / - ≡

Numerical RK solution

ode x'' 6 t 2^* t 4^- 16 / x * t 4 + 2 / x' * - t t cos * - ≡ (≡

U x x' 21 mat ≡

dU dt / x' x'' 21 mat ≡

t u D u 2 el 6 t * t 4^- 16 / u 1 el * t 4 + 2 / u 2 el * - t t cos * - 21 mat :

Plots: numerical and symbolic solutions

ti te 12 mat 01212 mat :

ODE 3 el Assign xy ic transpose ti te 100 t U D Rkadapt : 21 line

x x x ti te lele ( * xy 1 col xy 2 col augment 2 1 sys xy 2 col xy 3 col augment ic o augment 2 1 sys

Alvaro

s X 2 s 2^* s 2^1 + (2^/ 1 s 2^1 + / - :

s X s t ilt Maxima t t cos *