2268ab9b-c3d0-440d-8919-a25afdecba1a 32 6 9 decimal

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t q t sqrt :

Dirac Integral

s L t q t s laplace maple :

One of the solution contains Dirac Integral. Smath/Maple are not companions for that one.

s L π sqrt 2 s 32 /^* /

s F Y0 s - m - (* Y1 - s L - s 2^s m * + n + (/ (- :

Y0 Y1 m n t fnct s F t 0 > assume s t invlaplace maple 11 line :

t f Y0 Y1 m n t fnct :

Y0 0 :

Y1 0 :

m 0.5 - :

n 1 :

m n C 14 n * - m 2^+ (/ :

Y0 m n t sol_1 m n C Y0 0.5 m * t * - exp * m 2^* 0.5 4 n * m 2^- (sqrt * t * cos * (* :

Y0 m n t sol_2 m n C Y0 m * 0.5 m * t * - exp * 4 n * m 2^- (sqrt * 0.5 4 n * m 2^- (sqrt * t * sin * (* :

m n t sol_3 m n C 24 n * m 2^- (sqrt * u sqrt 0.5 m * t u - (* - exp * 0.5 4 n * m 2^- (sqrt * t u - (* sin * u 0 t int * (* :

Y1 m n t sol_4 m n C 2 Y1 * 0.5 m * t * - exp * 4 n * m 2^- (sqrt * 0.5 4 n * m 2^- (sqrt * t * sin * (* :

Y0 m n t sol_5 m n C 4 Y0 * 0.5 m * t * - exp * n * 0.5 4 n * m 2^- (sqrt * t * cos * (* :

Y0 Y1 m n t Sol Y0 m n t sol_1 Y0 m n t sol_2 - 1 m n t sol_3 * - Y1 m n t sol_4 - Y0 m n t sol_5 - :

Y0 0 : Y1 0 : m 0.5 - : n 1 : 14 mat

t f Y0 Y1 m n t Sol :

m n C 2 * 4 n * m 2^- (sqrt * 1.032796 -

u sqrt 0.5 m * t u - (* - exp * 0.5 4 n * m 2^- (sqrt * t u - (* sin * u 0 t int

Laplace integrand for Dirac x u integrand u sqrt 0.5 m * x u - (* - exp * 0.5 4 n * m 2^- (sqrt * x u - (* sin * :

u π 2 / :

0 x ≤ ( x 15 ≤ ( & ( x u integrand N/A if

x f x u integrand (u diff :

x φ 0 x ≤ (x 15 ≤ (& (1 x f * N/A if :

Dirac integrand

x φ

Apply the 'Transit procedure'

Transit the scalar f(x) uu F x φ :

Observe: eval(,) z Σ 1 uu F eval uu 0 x int * :

re-transit
plot canvas x Σ z Σ :

in-situ interpolation 0 Σ 0

0 x ≤ ( x 15 ≤ ( & x Σ if 0 2 1 sys

Phantom scale. Adjust the scale 
at some point of your preference. scale 0.05 :

D time -> 15 sec x 1012 -^15 0.05 range : j 1 x rows range y j el x j el Σ eval : for x scale y * augment 41 line :

Solution of the inverselaplace component in the forcing sqrt(t). The Dirac integral is the cumulative. Independently of the apparent amplitude of the "cumulative", it depends upon 'u' in integrand(x,u). In this demo, 'u' is set π/2 [some kind of conventional]. Phantom scale is 'user' . The solution of this component is invariant wrt Y0, Y1. For a complete solution over initials [Y0, Y1], sum all other respective solutions. sol_3(m,n,t) does not have a generalised symbolic expansion.

D 0 2 1 sys

n x Φ 1 π / n x * sin x / * :

1001012 -^Φ 31.830989

Dirac Δ(x) completeness

100 x Φ

Mathcad 11: The Dirac Delta is returned from Maple Δ(x) where Δ(0)= ∞ ... Δ(x) = 0 everywhere else

TheIntegralof x Δ x ∞ - ∞ int : 1 ≡