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ssllys

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5 years, 201 days
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Thank you so much. It really does work. ...

ssllys 35
August 24 2018
0 0

@Kitonum Thank you so much. It really does work. By the way, could you explain it, because I am confused about it a lot? I don't know why do like that? 

Thank you!! Could you give me a example ...

ssllys 35
July 26 2018
0 0

Thank you!! Could you give me an example or just give me a solution to my problem? Many Thx!

Many thanks to you....

ssllys 35
July 10 2018
0 0

@Mariusz Iwaniuk  Thank you very much. I need to be more careful. 

thanks a lot...

ssllys 35
May 31 2018
0 0

@Mariusz Iwaniuk Thank you very much for your prompt reply.

thank you for you reply....

ssllys 35
May 30 2018
0 0

@dharr you a kind man.

Thx....

ssllys 35
May 30 2018
0 0

@Rouben Rostamian   Thank you very much. Actually, I am a beginner of Maple. Thanks for your detailed answer.

Thank you for your code of numerical way...

ssllys 35
May 13 2018
0 0

@tomleslie  I had downloaded it and preparing to learn it. thank you very much.

Many thanks to you...

ssllys 35
May 13 2018
0 0

@Carl Love Actually, I was using the Multiple scale method to solve duffing equation. Now, I have conquered it already. The problem is that the edition of Maple, I updated the software and it does work. Thank you so much.

The original code....

ssllys 35
May 12 2018
0 0


 

NULL

> 

restart
> 

alias(epsilon = e, omega = w, omega[0] = w0, t[1] = t1, t[2] = t2):
> 

ode := (D@@2)(u)+2*mu*epsilon*D(u)+omega[0]^2*u+epsilon*omega[0]^2*u^3-epsilon*f*cos(omega*t) = 0;

(D@@2)(u)+2*mu*epsilon*D(u)+omega[0]^2*u+epsilon*omega[0]^2*u^3-epsilon*f*cos(omega*t) = 0

 (1)
> 

e_oredr := 1:
> 

ode := simplify(subs(D = sum('epsilon^(i-1)*D[i]', 'i' = 1 .. e_oredr+1), ode), {epsilon^(e_oredr+1) = 0});

(u^3*omega[0]^2+2*(epsilon*D[2]+D[1])(u)*mu-cos(omega*t)*f+2*D[1, 2](u))*epsilon+omega[0]^2*u+D[1, 1](u) = 0

 (2)
> 

simplify(collect(%, epsilon), {epsilon^(e_oredr+1) = 0});

(u^3*omega[0]^2+2*D[1](u)*mu-cos(omega*t)*f+2*D[1, 2](u))*epsilon+omega[0]^2*u+D[1, 1](u) = 0

 (3)
> 

NULL
> 

u := sum('v[i]*epsilon^i', 'i' = 0 .. e_oredr);

epsilon*v[1]+v[0]

 (4)
> 

ode := simplify(collect(ode, epsilon), {epsilon^2 = 0});

(v[0]^3*omega[0]^2+omega[0]^2*v[1]+2*D[1](v[0])*mu-cos(omega*t)*f+2*D[1, 2](v[0])+D[1, 1](v[1]))*epsilon+omega[0]^2*v[0]+D[1, 1](v[0]) = 0

 (5)
> 

for i from 0 to e_oredr do eq[i] := coeff(lhs(ode), epsilon, i) = 0 end do;

omega[0]^2*v[0]+D[1, 1](v[0]) = 0

  

v[0]^3*omega[0]^2+omega[0]^2*v[1]+2*D[1](v[0])*mu-cos(omega*t)*f+2*D[1, 2](v[0])+D[1, 1](v[1]) = 0

 (6)
> 

remove(has, lhs(eq[1]), cos);

v[0]^3*omega[0]^2+omega[0]^2*v[1]+2*D[1](v[0])*mu+2*D[1, 2](v[0])+D[1, 1](v[1])

  

v[0](t[1], t[2])^3*omega[0]^2+omega[0]^2*v[1](t[1], t[2])+2*(diff(v[0](t[1], t[2]), t[1]))*mu+2*(diff(diff(v[0](t[1], t[2]), t[1]), t[2]))+diff(diff(v[1](t[1], t[2]), t[1]), t[1])

 (7)
> 

eq[1] := %-convert(f*cos(sigma*t[2]+omega[0]*t[1]), 'exp');

v[0](t[1], t[2])^3*omega[0]^2+omega[0]^2*v[1](t[1], t[2])+2*(diff(v[0](t[1], t[2]), t[1]))*mu+2*(diff(diff(v[0](t[1], t[2]), t[1]), t[2]))+diff(diff(v[1](t[1], t[2]), t[1]), t[1])-f*((1/2)*exp(I*(sigma*t[2]+omega[0]*t[1]))+(1/2)*exp(-I*(sigma*t[2]+omega[0]*t[1])))

 (8)
> 

 

> 

v[0] := A(t[2])*cos(omega[0]*t[1]+B(t[2])):

proc (t1, t2) options operator, arrow; A(t2)*((1/2)*exp(I*(omega[0]*t1+B(t2)))+(1/2)*exp(-I*(omega[0]*t1+B(t2)))) end proc

 (9)
> 

NULL
> 

expand(eq[1])

Error, (in property/ConvertProperty) invalid input: PropRange uses a 2nd argument, b, which is missing

  
> 

collect(%, exp(I*w0*t1))

Error, (in collect) invalid 1st argument proc (t1, t2) options operator, arrow; A(t2)*((1/2)*exp(I*(w0*t1+B(t2)))+(1/2)*exp(-I*(w0*t1+B(t2)))) end proc

  
> 

coeff(%, exp(I*omega[0]*t[1])):
> 

map(proc (x) options operator, arrow; x*exp(-I*B(t[2])) end proc, %):
> 

combine(%, 'exp'):
> 

subs(I*B(t[2]) = I*sigma*t[2]-I*C(t[2]), B(t[2]) = sigma*t[2]-C(t[2]), %):
> 

conds := combine(%, 'exp');

0

 (10)
> 

 

NULL

NULL


 

Download Duffing_osilliator_primary_resonence1.mw

The original code.

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