python - 使用 scipy 的 solve_bvp 求解具有两个边界条件的一阶 BVP

Question

我正在使用 scipy 的 BVP 求解器:

http://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.solve_bvp.html

我遇到的问题是，边界条件的数量只能与方程的数量一样多。我只有一个方程，但我有两个边界条件。这怎么能解决？

MWE

>>> import numpy as np
>>> from scipy.integrate import solve_bvp
>>> 
>>> x = np.linspace(0, 1, 100)
>>> dydx = lambda x,y: y*np.sin(x)
>>> 
>>> result = solve_bvp(dydx, 
...     lambda ya,yb: np.array([ (ya[0]-1)**2 + (yb[0]-1)**2 ]),
...     x, [np.ones(len(x))], max_nodes=100000, tol=1e-9)
>>> 
>>> result
       message: 'The algorithm converged to the desired accuracy.'
         niter: 2
             p: None
 rms_residuals: array([  3.48054730e-10,   3.47134800e-10,   3.46220750e-10,
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           sol: <scipy.interpolate.interpolate.PPoly object at 0x2ad860930d58>
        status: 0
       success: True
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如您所见，y 与 y(x=0) = y(x=1) = 1 的边界条件相去甚远。

Answer 1

如果您为一阶 ODE 指定两个边界条件 y(0)=1 和 y(1)=1，则通常问题是超定的，没有解。如果您仅指定初始条件 y(0)=y0，则会遇到一阶初始值问题。事实上，在这种情况下，您可以推导出精确解:y(x) = y0*exp(-cos(x))。