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bjxtju

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files attached...

bjxtju 0
May 15 2012

I am really appriciated for your help. The attachments are the Maple files. I use Galerkin method to solve a function. File 1 is the original program. In File 2，the last two terms for a(s) are swaped and it gives me the numrical solutions.Thanks. @acer

1.mw2.mw

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restart;
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R:=1;B:=1;n:=3;y:=4;

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 (1)
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da:=diff(a(s),s)=1/R+c1*s*(s/R-Pi)+c2*(s*(s/R-Pi))^2+c3*(s*(s/R-Pi))^3;

diff(a(s), s) = 1+c1*s*(s-Pi)+c2*s^2*(s-Pi)^2+c3*s^3*(s-Pi)^3

 (2)
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dsolve({da,-a(0)=a(Pi*R)-Pi},a(s));

a(s) = (1/7)*c3*s^7-(1/2)*Pi*c3*s^6+(3/5)*s^5*c3*Pi^2+(1/5)*s^5*c2-(1/4)*s^4*Pi^3*c3-(1/2)*s^4*Pi*c2+(1/3)*s^3*Pi^2*c2+(1/3)*s^3*c1-(1/2)*Pi*c1*s^2+s+(1/280)*c3*Pi^7+(1/12)*Pi^3*c1-(1/60)*Pi^5*c2

 (3)
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a(s):=1/7*c3*s^7-1/2*Pi*c3*s^6+3/5*s^5*c3*Pi^2+1/5*s^5*c2-1/4*s^4*Pi^3*c3-1/2*s^4*Pi*c2+1/3*s^3*Pi^2*c2+1/3*s^3*c1-1/2*Pi*c1*s^2+s+1/280*c3*Pi^7-1/60*Pi^5*c2+1/12*Pi^3*c1;

(1/7)*c3*s^7-(1/2)*Pi*c3*s^6+(3/5)*s^5*c3*Pi^2+(1/5)*s^5*c2-(1/4)*s^4*Pi^3*c3-(1/2)*s^4*Pi*c2+(1/3)*s^3*Pi^2*c2+(1/3)*s^3*c1-(1/2)*Pi*c1*s^2+s+(1/280)*c3*Pi^7+(1/12)*Pi^3*c1-(1/60)*Pi^5*c2

 (4)
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eqn:=subs({m=(1-n)/y^2},R^2*diff(a(s),s,s)+B/2*sin(a(s))-B^2/8*m*sin(2*a(s)));

2*c1*s-Pi*c1+4*c2*s^3-6*c2*s^2*Pi+2*c2*s*Pi^2+6*c3*s^5-15*c3*s^4*Pi+12*c3*s^3*Pi^2-3*c3*s^2*Pi^3+(1/2)*sin((1/7)*c3*s^7-(1/2)*Pi*c3*s^6+(3/5)*s^5*c3*Pi^2+(1/5)*s^5*c2-(1/4)*s^4*Pi^3*c3-(1/2)*s^4*Pi*c2+(1/3)*s^3*Pi^2*c2+(1/3)*s^3*c1-(1/2)*Pi*c1*s^2+s+(1/280)*c3*Pi^7+(1/12)*Pi^3*c1-(1/60)*Pi^5*c2)+(1/64)*sin((2/7)*c3*s^7-Pi*c3*s^6+(6/5)*s^5*c3*Pi^2+(2/5)*s^5*c2-(1/2)*s^4*Pi^3*c3-s^4*Pi*c2+(2/3)*s^3*Pi^2*c2+(2/3)*s^3*c1-Pi*c1*s^2+2*s+(1/140)*c3*Pi^7+(1/6)*Pi^3*c1-(1/30)*Pi^5*c2)

 (5)
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eqn1:=Int(eqn*s*(s/R-Pi),s=0..Pi*R)=0;

Int((2*c1*s-Pi*c1+4*c2*s^3-6*c2*s^2*Pi+2*c2*s*Pi^2+6*c3*s^5-15*c3*s^4*Pi+12*c3*s^3*Pi^2-3*c3*s^2*Pi^3+(1/2)*sin((1/7)*c3*s^7-(1/2)*Pi*c3*s^6+(3/5)*s^5*c3*Pi^2+(1/5)*s^5*c2-(1/4)*s^4*Pi^3*c3-(1/2)*s^4*Pi*c2+(1/3)*s^3*Pi^2*c2+(1/3)*s^3*c1-(1/2)*Pi*c1*s^2+s+(1/280)*c3*Pi^7+(1/12)*Pi^3*c1-(1/60)*Pi^5*c2)+(1/64)*sin((2/7)*c3*s^7-Pi*c3*s^6+(6/5)*s^5*c3*Pi^2+(2/5)*s^5*c2-(1/2)*s^4*Pi^3*c3-s^4*Pi*c2+(2/3)*s^3*Pi^2*c2+(2/3)*s^3*c1-Pi*c1*s^2+2*s+(1/140)*c3*Pi^7+(1/6)*Pi^3*c1-(1/30)*Pi^5*c2))*s*(s-Pi), s = 0 .. Pi) = 0

 (6)
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eqn2:=Int(eqn*(s*(s/R-Pi))^2,s=0..Pi*R)=0;

Int((2*c1*s-Pi*c1+4*c2*s^3-6*c2*s^2*Pi+2*c2*s*Pi^2+6*c3*s^5-15*c3*s^4*Pi+12*c3*s^3*Pi^2-3*c3*s^2*Pi^3+(1/2)*sin((1/7)*c3*s^7-(1/2)*Pi*c3*s^6+(3/5)*s^5*c3*Pi^2+(1/5)*s^5*c2-(1/4)*s^4*Pi^3*c3-(1/2)*s^4*Pi*c2+(1/3)*s^3*Pi^2*c2+(1/3)*s^3*c1-(1/2)*Pi*c1*s^2+s+(1/280)*c3*Pi^7+(1/12)*Pi^3*c1-(1/60)*Pi^5*c2)+(1/64)*sin((2/7)*c3*s^7-Pi*c3*s^6+(6/5)*s^5*c3*Pi^2+(2/5)*s^5*c2-(1/2)*s^4*Pi^3*c3-s^4*Pi*c2+(2/3)*s^3*Pi^2*c2+(2/3)*s^3*c1-Pi*c1*s^2+2*s+(1/140)*c3*Pi^7+(1/6)*Pi^3*c1-(1/30)*Pi^5*c2))*s^2*(s-Pi)^2, s = 0 .. Pi) = 0

 (7)
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eqn3:=Int(eqn*(s*(s/R-Pi))^3,s=0..Pi*R)=0;

Int((2*c1*s-Pi*c1+4*c2*s^3-6*c2*s^2*Pi+2*c2*s*Pi^2+6*c3*s^5-15*c3*s^4*Pi+12*c3*s^3*Pi^2-3*c3*s^2*Pi^3+(1/2)*sin((1/7)*c3*s^7-(1/2)*Pi*c3*s^6+(3/5)*s^5*c3*Pi^2+(1/5)*s^5*c2-(1/4)*s^4*Pi^3*c3-(1/2)*s^4*Pi*c2+(1/3)*s^3*Pi^2*c2+(1/3)*s^3*c1-(1/2)*Pi*c1*s^2+s+(1/280)*c3*Pi^7+(1/12)*Pi^3*c1-(1/60)*Pi^5*c2)+(1/64)*sin((2/7)*c3*s^7-Pi*c3*s^6+(6/5)*s^5*c3*Pi^2+(2/5)*s^5*c2-(1/2)*s^4*Pi^3*c3-s^4*Pi*c2+(2/3)*s^3*Pi^2*c2+(2/3)*s^3*c1-Pi*c1*s^2+2*s+(1/140)*c3*Pi^7+(1/6)*Pi^3*c1-(1/30)*Pi^5*c2))*s^3*(s-Pi)^3, s = 0 .. Pi) = 0

 (8)
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C:=fsolve({eqn1,eqn2,eqn3},{c1,c2,c3});

fsolve({Int((2*c1*s-Pi*c1+4*c2*s^3-6*c2*s^2*Pi+2*c2*s*Pi^2+6*c3*s^5-15*c3*s^4*Pi+12*c3*s^3*Pi^2-3*c3*s^2*Pi^3+(1/2)*sin((1/7)*c3*s^7-(1/2)*Pi*c3*s^6+(3/5)*s^5*c3*Pi^2+(1/5)*s^5*c2-(1/4)*s^4*Pi^3*c3-(1/2)*s^4*Pi*c2+(1/3)*s^3*Pi^2*c2+(1/3)*s^3*c1-(1/2)*Pi*c1*s^2+s+(1/280)*c3*Pi^7+(1/12)*Pi^3*c1-(1/60)*Pi^5*c2)+(1/64)*sin((2/7)*c3*s^7-Pi*c3*s^6+(6/5)*s^5*c3*Pi^2+(2/5)*s^5*c2-(1/2)*s^4*Pi^3*c3-s^4*Pi*c2+(2/3)*s^3*Pi^2*c2+(2/3)*s^3*c1-Pi*c1*s^2+2*s+(1/140)*c3*Pi^7+(1/6)*Pi^3*c1-(1/30)*Pi^5*c2))*s*(s-Pi), s = 0 .. Pi) = 0, Int((2*c1*s-Pi*c1+4*c2*s^3-6*c2*s^2*Pi+2*c2*s*Pi^2+6*c3*s^5-15*c3*s^4*Pi+12*c3*s^3*Pi^2-3*c3*s^2*Pi^3+(1/2)*sin((1/7)*c3*s^7-(1/2)*Pi*c3*s^6+(3/5)*s^5*c3*Pi^2+(1/5)*s^5*c2-(1/4)*s^4*Pi^3*c3-(1/2)*s^4*Pi*c2+(1/3)*s^3*Pi^2*c2+(1/3)*s^3*c1-(1/2)*Pi*c1*s^2+s+(1/280)*c3*Pi^7+(1/12)*Pi^3*c1-(1/60)*Pi^5*c2)+(1/64)*sin((2/7)*c3*s^7-Pi*c3*s^6+(6/5)*s^5*c3*Pi^2+(2/5)*s^5*c2-(1/2)*s^4*Pi^3*c3-s^4*Pi*c2+(2/3)*s^3*Pi^2*c2+(2/3)*s^3*c1-Pi*c1*s^2+2*s+(1/140)*c3*Pi^7+(1/6)*Pi^3*c1-(1/30)*Pi^5*c2))*s^2*(s-Pi)^2, s = 0 .. Pi) = 0, Int((2*c1*s-Pi*c1+4*c2*s^3-6*c2*s^2*Pi+2*c2*s*Pi^2+6*c3*s^5-15*c3*s^4*Pi+12*c3*s^3*Pi^2-3*c3*s^2*Pi^3+(1/2)*sin((1/7)*c3*s^7-(1/2)*Pi*c3*s^6+(3/5)*s^5*c3*Pi^2+(1/5)*s^5*c2-(1/4)*s^4*Pi^3*c3-(1/2)*s^4*Pi*c2+(1/3)*s^3*Pi^2*c2+(1/3)*s^3*c1-(1/2)*Pi*c1*s^2+s+(1/280)*c3*Pi^7+(1/12)*Pi^3*c1-(1/60)*Pi^5*c2)+(1/64)*sin((2/7)*c3*s^7-Pi*c3*s^6+(6/5)*s^5*c3*Pi^2+(2/5)*s^5*c2-(1/2)*s^4*Pi^3*c3-s^4*Pi*c2+(2/3)*s^3*Pi^2*c2+(2/3)*s^3*c1-Pi*c1*s^2+2*s+(1/140)*c3*Pi^7+(1/6)*Pi^3*c1-(1/30)*Pi^5*c2))*s^3*(s-Pi)^3, s = 0 .. Pi) = 0}, {c1, c2, c3})

 (9)
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Download 1.mw

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> 

restart;
> 

R:=1;B:=1;n:=3;y:=4;

1

  

1

  

3

  

4

 (1)
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da:=diff(a(s),s)=1/R+c1*s*(s/R-Pi)+c2*(s*(s/R-Pi))^2+c3*(s*(s/R-Pi))^3;

diff(a(s), s) = 1+c1*s*(s-Pi)+c2*s^2*(s-Pi)^2+c3*s^3*(s-Pi)^3

 (2)
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#dsolve({da,-a(0)=a(Pi*R)-Pi},a(s));
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a(s):=1/7*c3*s^7-1/2*Pi*c3*s^6+3/5*s^5*c3*Pi^2+1/5*s^5*c2-1/4*s^4*Pi^3*c3-1/2*s^4*Pi*c2+1/3*s^3*Pi^2*c2+1/3*s^3*c1-1/2*Pi*c1*s^2+s+1/280*c3*Pi^7-1/60*Pi^5*c2+1/12*Pi^3*c1;

(1/7)*c3*s^7-(1/2)*Pi*c3*s^6+(3/5)*s^5*c3*Pi^2+(1/5)*s^5*c2-(1/4)*s^4*Pi^3*c3-(1/2)*s^4*Pi*c2+(1/3)*s^3*Pi^2*c2+(1/3)*s^3*c1-(1/2)*Pi*c1*s^2+s+(1/280)*c3*Pi^7-(1/60)*Pi^5*c2+(1/12)*Pi^3*c1

 (3)
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eqn:=subs({m=(1-n)/y^2},R^2*diff(a(s),s,s)+B/2*sin(a(s))-B^2/8*m*sin(2*a(s)));

6*c3*s^5-15*Pi*c3*s^4+12*s^3*c3*Pi^2+4*s^3*c2-3*s^2*Pi^3*c3-6*s^2*Pi*c2+2*s*Pi^2*c2+2*s*c1-Pi*c1+(1/2)*sin((1/7)*c3*s^7-(1/2)*Pi*c3*s^6+(3/5)*s^5*c3*Pi^2+(1/5)*s^5*c2-(1/4)*s^4*Pi^3*c3-(1/2)*s^4*Pi*c2+(1/3)*s^3*Pi^2*c2+(1/3)*s^3*c1-(1/2)*Pi*c1*s^2+s+(1/280)*c3*Pi^7-(1/60)*Pi^5*c2+(1/12)*Pi^3*c1)+(1/64)*sin((2/7)*c3*s^7-Pi*c3*s^6+(6/5)*s^5*c3*Pi^2+(2/5)*s^5*c2-(1/2)*s^4*Pi^3*c3-s^4*Pi*c2+(2/3)*s^3*Pi^2*c2+(2/3)*s^3*c1-Pi*c1*s^2+2*s+(1/140)*c3*Pi^7-(1/30)*Pi^5*c2+(1/6)*Pi^3*c1)

 (4)
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eqn1:=Int(eqn*s*(s/R-Pi),s=0..Pi*R)=0;

Int((6*c3*s^5-15*Pi*c3*s^4+12*s^3*c3*Pi^2+4*s^3*c2-3*s^2*Pi^3*c3-6*s^2*Pi*c2+2*s*Pi^2*c2+2*s*c1-Pi*c1+(1/2)*sin((1/7)*c3*s^7-(1/2)*Pi*c3*s^6+(3/5)*s^5*c3*Pi^2+(1/5)*s^5*c2-(1/4)*s^4*Pi^3*c3-(1/2)*s^4*Pi*c2+(1/3)*s^3*Pi^2*c2+(1/3)*s^3*c1-(1/2)*Pi*c1*s^2+s+(1/280)*c3*Pi^7-(1/60)*Pi^5*c2+(1/12)*Pi^3*c1)+(1/64)*sin((2/7)*c3*s^7-Pi*c3*s^6+(6/5)*s^5*c3*Pi^2+(2/5)*s^5*c2-(1/2)*s^4*Pi^3*c3-s^4*Pi*c2+(2/3)*s^3*Pi^2*c2+(2/3)*s^3*c1-Pi*c1*s^2+2*s+(1/140)*c3*Pi^7-(1/30)*Pi^5*c2+(1/6)*Pi^3*c1))*s*(s-Pi), s = 0 .. Pi) = 0

 (5)
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eqn2:=Int(eqn*(s*(s/R-Pi))^2,s=0..Pi*R)=0;

Int((6*c3*s^5-15*Pi*c3*s^4+12*s^3*c3*Pi^2+4*s^3*c2-3*s^2*Pi^3*c3-6*s^2*Pi*c2+2*s*Pi^2*c2+2*s*c1-Pi*c1+(1/2)*sin((1/7)*c3*s^7-(1/2)*Pi*c3*s^6+(3/5)*s^5*c3*Pi^2+(1/5)*s^5*c2-(1/4)*s^4*Pi^3*c3-(1/2)*s^4*Pi*c2+(1/3)*s^3*Pi^2*c2+(1/3)*s^3*c1-(1/2)*Pi*c1*s^2+s+(1/280)*c3*Pi^7-(1/60)*Pi^5*c2+(1/12)*Pi^3*c1)+(1/64)*sin((2/7)*c3*s^7-Pi*c3*s^6+(6/5)*s^5*c3*Pi^2+(2/5)*s^5*c2-(1/2)*s^4*Pi^3*c3-s^4*Pi*c2+(2/3)*s^3*Pi^2*c2+(2/3)*s^3*c1-Pi*c1*s^2+2*s+(1/140)*c3*Pi^7-(1/30)*Pi^5*c2+(1/6)*Pi^3*c1))*s^2*(s-Pi)^2, s = 0 .. Pi) = 0

 (6)
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eqn3:=Int(eqn*(s*(s/R-Pi))^3,s=0..Pi*R)=0;

Int((6*c3*s^5-15*Pi*c3*s^4+12*s^3*c3*Pi^2+4*s^3*c2-3*s^2*Pi^3*c3-6*s^2*Pi*c2+2*s*Pi^2*c2+2*s*c1-Pi*c1+(1/2)*sin((1/7)*c3*s^7-(1/2)*Pi*c3*s^6+(3/5)*s^5*c3*Pi^2+(1/5)*s^5*c2-(1/4)*s^4*Pi^3*c3-(1/2)*s^4*Pi*c2+(1/3)*s^3*Pi^2*c2+(1/3)*s^3*c1-(1/2)*Pi*c1*s^2+s+(1/280)*c3*Pi^7-(1/60)*Pi^5*c2+(1/12)*Pi^3*c1)+(1/64)*sin((2/7)*c3*s^7-Pi*c3*s^6+(6/5)*s^5*c3*Pi^2+(2/5)*s^5*c2-(1/2)*s^4*Pi^3*c3-s^4*Pi*c2+(2/3)*s^3*Pi^2*c2+(2/3)*s^3*c1-Pi*c1*s^2+2*s+(1/140)*c3*Pi^7-(1/30)*Pi^5*c2+(1/6)*Pi^3*c1))*s^3*(s-Pi)^3, s = 0 .. Pi) = 0

 (7)
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C:=fsolve({eqn1,eqn2,eqn3},{c1,c2,c3});

{c1 = -198.6041770, c2 = -144.8614487, c3 = -26.89546880}

 (8)
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