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Simulate some PhysicsPhenonmenon Apply ODE Method.
Phạm Bá Duy Phạm Xuân Huy
Bùi !n "#nh $% Xuân &'n(
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)i*st de(*ee ODE+ dy/dx=f(x,y)
Eule* method is a nume*ical method tosol,e o*dina*y di--e*ential
euation /ODE0.
1nlitial ,alue p*o2lem
dy/dt=v(t) (3.1)t 1=to, y 1=yo.
t 2=t 1+∆t
y 2= ??? y 2 = y 1 + ∆t*v(t)
3epeat this p*ocedu*e4 in
(ene*al y n+1=y n+v(t n ). ∆t .
5uantity 6t is calledstepsi7e and can 2e chosen.
"hat is Eule* method.
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"*uncation E**o*
$ocal t*uncation e**o* in each step isp*opo*tional to h.
E**o* p*opa(ates th*ou(h steps o-
computin(4 it is much
2i((e*.8ei89:M/8eo8;Bp/A=/>i?>o00
8@-89:A 8y8 9:
B:/
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Eule*?*ome* method
/t; ∆t 0:,/t0 ; ∆t *a(t)
X(t+ ∆t ) = x(t) + ∆t *v(t+ ∆t )
Approximate etter for o!"i##atin$ !y!tem.
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Eule*?3icha*son method /0
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Eule*?3icha*son Method /C0
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ODE odin(
Basic steps:Step 1. de-ine f(x,y).Step 2. input
initial ,alue >o and yo.
Step 3. input step si7e 6t and num2e* o- step n.
Step 4. calculate >4 y and y+-o* i:+n
>:>;hG
y:y;h=-/>4y0G
y:y;h=yGend
Step 5. output > and y.
Step 6. end
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Eule* Method Matla2 ode
IModi-ied Eule* Methods ? P*o2lem J. -unction -:Eul/>4,4t0 :
G m : G >K:KG ,K:KG n : KKG dt : K.G
> : 7e*os/n40G >/0 : >KG , : 7e*os/n40G ,/0 : KG t :
7e*os/n40G t/0 : K.KG
-o* step:+n? ,/step;0 : ,/step0 ? dt=/>/step00G >/step;0 :
>/step0 ; dt=,/step0G t/step;0 : t/step0 ; dtG end
cl-G plot/t4>4L*0G le(end/>?/cm04,?/cmla2el /"ime /s00G
yla2el /Amplitude0G title/Displacement ,s time4)ontSi7e4N0G (*idG
hold onG
