How to numerically integrate differential equations with boundary conditions at infinity
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I am trying the numerically integrate the Nielsen-Olesen equation for vortices:
beginequation
a''(r)-frac1ra'(r)-e(n+ea(r))f^2=0
endequation
beginequation
f''(r)+frac1rf'(r)-frac1r^2(n+ea(r))^2f=2lambda f(f^2-phi_0^2)
endequation
where $f$ and $a$ depend on $r$ (in cylindrical coordinates) and the other terms are constants. These coupled differential equations have the following boundary conditions:
beginequation
f(0)=a(0)=0
endequation
beginequation
lim_rtoinftyf(r)=phi_0
endequation
beginequation
lim_rtoinftya(r)=-fracne
endequation
My question is more general though. I've seen some algorithms on the numerical recipes book for differential equations with boundary conditions. However they don't explain how to proceed when the boundary conditions are at infinity. Does anybody know any reference for studying these kind of problems? Or is a simple solutions such as setting a large value for $r$ (but not to large so that the program doesn't take too much computation time) enough?
differential-equations
migrated from physics.stackexchange.com Aug 10 at 19:45
This question came from our site for active researchers, academics and students of physics.
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I am trying the numerically integrate the Nielsen-Olesen equation for vortices:
beginequation
a''(r)-frac1ra'(r)-e(n+ea(r))f^2=0
endequation
beginequation
f''(r)+frac1rf'(r)-frac1r^2(n+ea(r))^2f=2lambda f(f^2-phi_0^2)
endequation
where $f$ and $a$ depend on $r$ (in cylindrical coordinates) and the other terms are constants. These coupled differential equations have the following boundary conditions:
beginequation
f(0)=a(0)=0
endequation
beginequation
lim_rtoinftyf(r)=phi_0
endequation
beginequation
lim_rtoinftya(r)=-fracne
endequation
My question is more general though. I've seen some algorithms on the numerical recipes book for differential equations with boundary conditions. However they don't explain how to proceed when the boundary conditions are at infinity. Does anybody know any reference for studying these kind of problems? Or is a simple solutions such as setting a large value for $r$ (but not to large so that the program doesn't take too much computation time) enough?
differential-equations
migrated from physics.stackexchange.com Aug 10 at 19:45
This question came from our site for active researchers, academics and students of physics.
1
$$r = fracz1-z$$ moves the point at infinity to $z=1$ and this becomes a regular singular point of the differential equations.
– Count Iblis
Aug 9 at 20:50
add a comment |Â
up vote
0
down vote
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up vote
0
down vote
favorite
I am trying the numerically integrate the Nielsen-Olesen equation for vortices:
beginequation
a''(r)-frac1ra'(r)-e(n+ea(r))f^2=0
endequation
beginequation
f''(r)+frac1rf'(r)-frac1r^2(n+ea(r))^2f=2lambda f(f^2-phi_0^2)
endequation
where $f$ and $a$ depend on $r$ (in cylindrical coordinates) and the other terms are constants. These coupled differential equations have the following boundary conditions:
beginequation
f(0)=a(0)=0
endequation
beginequation
lim_rtoinftyf(r)=phi_0
endequation
beginequation
lim_rtoinftya(r)=-fracne
endequation
My question is more general though. I've seen some algorithms on the numerical recipes book for differential equations with boundary conditions. However they don't explain how to proceed when the boundary conditions are at infinity. Does anybody know any reference for studying these kind of problems? Or is a simple solutions such as setting a large value for $r$ (but not to large so that the program doesn't take too much computation time) enough?
differential-equations
I am trying the numerically integrate the Nielsen-Olesen equation for vortices:
beginequation
a''(r)-frac1ra'(r)-e(n+ea(r))f^2=0
endequation
beginequation
f''(r)+frac1rf'(r)-frac1r^2(n+ea(r))^2f=2lambda f(f^2-phi_0^2)
endequation
where $f$ and $a$ depend on $r$ (in cylindrical coordinates) and the other terms are constants. These coupled differential equations have the following boundary conditions:
beginequation
f(0)=a(0)=0
endequation
beginequation
lim_rtoinftyf(r)=phi_0
endequation
beginequation
lim_rtoinftya(r)=-fracne
endequation
My question is more general though. I've seen some algorithms on the numerical recipes book for differential equations with boundary conditions. However they don't explain how to proceed when the boundary conditions are at infinity. Does anybody know any reference for studying these kind of problems? Or is a simple solutions such as setting a large value for $r$ (but not to large so that the program doesn't take too much computation time) enough?
differential-equations
asked Aug 9 at 19:05
user191374
migrated from physics.stackexchange.com Aug 10 at 19:45
This question came from our site for active researchers, academics and students of physics.
migrated from physics.stackexchange.com Aug 10 at 19:45
This question came from our site for active researchers, academics and students of physics.
1
$$r = fracz1-z$$ moves the point at infinity to $z=1$ and this becomes a regular singular point of the differential equations.
– Count Iblis
Aug 9 at 20:50
add a comment |Â
1
$$r = fracz1-z$$ moves the point at infinity to $z=1$ and this becomes a regular singular point of the differential equations.
– Count Iblis
Aug 9 at 20:50
1
1
$$r = fracz1-z$$ moves the point at infinity to $z=1$ and this becomes a regular singular point of the differential equations.
– Count Iblis
Aug 9 at 20:50
$$r = fracz1-z$$ moves the point at infinity to $z=1$ and this becomes a regular singular point of the differential equations.
– Count Iblis
Aug 9 at 20:50
add a comment |Â
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1
$$r = fracz1-z$$ moves the point at infinity to $z=1$ and this becomes a regular singular point of the differential equations.
– Count Iblis
Aug 9 at 20:50