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Hi,

my skills in numerics are very limited, thus maybe this is a silly question. Is it possible, in sage or python, to interpolate a numerical solution of a differential equation and use it as a symbolic expression? Let me explain my problem. I have the following differential equation and numerical solution

z = function('z')(x)
de = diff(z,x) == sqrt(-36*(2*x/sqrt(x^2 + 5) - (x^2 - 1)*x/(x^2 + 5)^(3/2))^2 + (x^2 + 5)/(x^2 - 1)^2)
var('nz')
de_sol = desolve_rk4(de.rhs(),dvar=nz,ics=[0,0],ivar=x,end_points=.93, step=.01)

I would like to use the above solution as z(x) and (at first) plot a parametric surface. For illustrating, I will use z = -log(1-x), that has a similar shape.

th = var('theta')
z = -log(1-x)
r = -6*(x^2 - 1)/sqrt(x^2 + 5)
parametric_plot3d([r*cos(th),r*sin(th),z],(th,0,2*pi), (x,0,.93))

I tried:

• spline: it fits greatly, but the output cannot be used as symbolic. Parametric plot is not allowed.
• lagrange_polynomial: it gives me a polynomial fit that can be used for parametric plot, but the interpolation fails "close" to the divergence at x=1.
• I've tried to follow this post, but 1) I didn't find how to pass my de_sol to maxima (maxima('p:de_sol') doesn't work) and 2) I don't know how to turn the maxima charfun into sage piecewise. Maxima documentation doesn't help that much.

Any idea?

Hi,

my skills in numerics are very limited, thus maybe this is a silly question. Is it possible, in sage or python, to interpolate a numerical solution of a differential equation and use it as a symbolic expression? Let me explain my problem. I have the following differential equation and numerical solution

z = function('z')(x)
de = diff(z,x) == sqrt(-36*(2*x/sqrt(x^2 + 5) - (x^2 - 1)*x/(x^2 + 5)^(3/2))^2 + (x^2 + 5)/(x^2 - 1)^2)
var('nz')
de_sol = desolve_rk4(de.rhs(),dvar=nz,ics=[0,0],ivar=x,end_points=.93, step=.01)

I would like to use the above solution as z(x) and (at first) plot a parametric surface. For illustrating, I will use z = -log(1-x), that has a similar shape.

th = var('theta')
z = -log(1-x)
r = -6*(x^2 - 1)/sqrt(x^2 + 5)
parametric_plot3d([r*cos(th),r*sin(th),z],(th,0,2*pi), (x,0,.93))

I tried:

• spline: it fits greatly, but the output cannot be used as symbolic. Parametric plot is not allowed.
• lagrange_polynomial: it gives me a polynomial fit that can be used for parametric plot, but the interpolation fails "close" to the divergence at x=1.
• I've tried to follow this post, but 1) I didn't find out how to pass my de_sol de_sol to maxima (maxima('p:de_sol') doesn't work) and 2) I don't know how to turn the maxima charfun charfun into sage piecewise. piecewise. Maxima documentation doesn't help that much.

Any idea?

Hi,

my skills in numerics are very limited, thus maybe this is a silly question. Is it possible, in sage or python, to interpolate a numerical solution of a differential equation and use it as a symbolic expression? Let me explain my problem. I have the following differential equation and numerical solution

z = function('z')(x)
de = diff(z,x) == sqrt(-36*(2*x/sqrt(x^2 + 5) - (x^2 - 1)*x/(x^2 + 5)^(3/2))^2 + (x^2 + 5)/(x^2 - 1)^2)
var('nz')
de_sol = desolve_rk4(de.rhs(),dvar=nz,ics=[0,0],ivar=x,end_points=.93, step=.01)

I would like to use the above solution as z(x) and (at first) plot a parametric surface. For illustrating, I will use z = -log(1-x), that has a similar shape.

th = var('theta')
z = -log(1-x)
r = -6*(x^2 - 1)/sqrt(x^2 + 5)
parametric_plot3d([r*cos(th),r*sin(th),z],(th,0,2*pi), (x,0,.93))

I tried:

• spline: it fits greatly, but the output cannot be used as symbolic. Parametric plot is not allowed.
• lagrange_polynomial: it gives me a polynomial fit that can be used for parametric plot, but the interpolation fails "close" to the divergence at x=1.
• I've tried to follow this post, but 1) I didn't find out how to pass my de_sol to maxima (maxima('p:de_sol') (maxima('p:de_sol') doesn't work) and 2) I don't know how to turn the maxima charfun into sage piecewise. Maxima documentation doesn't help that much.

Any idea?