how to solve this ODE
qiu zhanlong
I am kind of new to the BVP solver， I found it is very confusing for me to understand what is and how to format the singular points, the boundary_conditions_derivative, and function_derivative basing on the few examples online.
My current ODE problem may require me to implement the singular points, boundary_conditions_derivative, and function_derivative into my code. Thus I need to seek help from you guys, and any suggestion is appreciated.
Below is the ODE and the boundary conditions I have
This is the my BVP code, which can only find the trivial solution, namely y(x)=0, other than that, it always generates error code, stating that the Newton iteration failed.
import numpy
import scikits.bvp_solver as bvp
epsilon=10**(-6)
b=0.3
k=10
def Fode(x,y):
return numpy.array([y[1],y[2],y[3],(((y[1]**2-y[0]*y[2])/16.0+(y[1]**2)/4.0)*y[2]-y[0]*(k**2)*((k**2)*(y[2]**2)/4.0+(x**2-b**2)/2.0))/epsilon-(-2*(k**2)*y[2]+(k**4)*y[0])])
def Fbc(ya,yb):
bca=numpy.array([ya[2],epsilon*(-ya[3]+2*(k**2)*ya[1])+ya[1]*((ya[1]**2)/4.0+(ya[1]**2-ya[0]*ya[2])/16.0)])
bcb=numpy.array([yb[2],epsilon*(-yb[3]+2*(k**2)*yb[1])+yb[1]*((yb[1]**2)/4.0+(yb[1]**2-yb[0]*yb[2])/16.0)])
return bca, bcb
def guess_y(x):
if x<0.3:
return numpy.array([(2*(b**2-x**2))**(0.5)/k,
-2**(0.5)*x/(k**(b**2-x**2)**(0.5)),
-(2)**(0.5)*b**2/(k*(b**2-x**2)**(1.5)),
-3*(2)**(0.5)*b**2*x/(k*(b**2-x**2)**(2.5))])
else:
return numpy.array([0.0,0.0,0.0,0.0])
problem=bvp.ProblemDefinition (num_ODE=4, num_parameters=0,
num_left_boundary_conditions=2,boundary_points=(0,0.5),
function=Fode,boundary_conditions=Fbc)
solution=bvp.solve(bvp_problem=problem,solution_guess=guess_y,trace=1)
Robert Schroll
> This is the my BVP code, which can only find the trivial solution,
> namely y(x)=0, other than that, it always generates error code,
> stating that the Newton iteration failed.
work is to start with a guess for the solution and then refine it to
reduce the error. As with any minimizer, there are regions of the
domain that will not converge to the correct solution. Choosing a good
initial guess is crucial.
> epsilon=10**(-6)
>
I haven't played with you system at all, but I suspect that this is the
problem. Having such a small parameter probably means that you'll have
a small features in your solution which will likely be missing from any
large-scale initial guess. I suspect you need a better initial guess
that respects that.
Have you tried solving the system with a larger epsilon, 0.1 or 0.01?
In my experience, this will increase the likelihood of a successful
solution. Once you have a solution for a large epsilon, you might try
using that as an initial guess for a smaller epsilon, and with a little
luck you may be able to ratchet yourself down to 10^-6.
You may also be able to take advantage of knowledge of the expected
form of a solution. If you know that some feature should scale like a
given power of epsilon, you can rescale your solution at large epsilon
before using it at smaller epsilon.
Good luck,
Robert