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4
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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0.21902122, 0.22189114, 0.22476599, 0.22764585, 0.23053076,
0.23342079, 0.236316 , 0.23921645, 0.2421222 , 0.24503332,
0.24794987, 0.2508719 , 0.25379948, 0.25673268, 0.25967155,
0.26261615, 0.26556655, 0.2685228 , 0.27148497, 0.27445313,
0.27742732, 0.28040762, 0.28339409, 0.28638678, 0.28938576,
0.29239109, 0.29540283, 0.29842105, 0.3014458 , 0.30447715,
0.30751515, 0.31055988, 0.31361139, 0.31666974, 0.31973499,
0.32280722, 0.32588647, 0.32897281, 0.3320663 , 0.33516701,
0.33827498, 0.3413903 , 0.34451301, 0.34764319, 0.35078088,
0.35392616, 0.35707908, 0.3602397 , 0.3634081 , 0.36658432,
0.36976843, 0.37296049, 0.37616057, 0.37936872, 0.382585 ,
0.38580948, 0.38904223, 0.39228329, 0.39553273, 0.39879061,
0.402057 , 0.40533195, 0.40861553, 0.4119078 , 0.41520881,
0.41851863, 0.42183733, 0.42516495, 0.42850157, 0.43184723,
0.43520202, 0.43856597, 0.44193917, 0.44532166, 0.4487135 ,
0.45211476, 0.45552551, 0.45894578, 0.46237566, 0.4658152 ,
0.46926446, 0.47272349, 0.47619237, 0.47967114, 0.48315988,
0.48665863, 0.49016747, 0.49368644, 0.49721562, 0.50075505,
0.5043048 , 0.50786493, 0.5114355 , 0.51501656, 0.51860818,
0.52221041, 0.52582331, 0.52944695, 0.53308138, 0.53672666,
0.54038285, 0.54405001, 0.54772819, 0.55141745, 0.55511786,
0.55882946, 0.56255232, 0.5662865 , 0.57003205, 0.57378903,
0.5775575 , 0.58133751, 0.58512912, 0.58893239, 0.59274738,
0.59657414, 0.60041272, 0.60426319, 0.6081256 , 0.61200001,
0.61588646, 0.61978503, 0.62369576, 0.6276187 , 0.63155392,
0.63550147, 0.6394614 , 0.64343376, 0.64741862, 0.65141602,
0.65542602, 0.65944867, 0.66348403, 0.66753215, 0.67159308,
0.67566687, 0.67975358, 0.68385327, 0.68796597, 0.69209174,
0.69623064, 0.70038272, 0.70454802, 0.7087266 , 0.7129185 ,
0.71712379, 0.7213425 , 0.72557469, 0.72982041, 0.7340797 ,
0.73835262, 0.74263921, 0.74693953, 0.75125361, 0.75558151,
0.75992327, 0.76427895, 0.76864858, 0.77303222, 0.7774299 ,
0.78184168, 0.7862676 , 0.79070771, 0.79516204, 0.79963065,
0.80411358, 0.80861086, 0.81312256, 0.81764869, 0.82218932,
0.82674447, 0.8313142 , 0.83589854, 0.84049753, 0.84511122,
0.84973964, 0.85438283, 0.85904083, 0.86371368, 0.86840142,
0.87310408, 0.8778217 , 0.88255432, 0.88730198, 0.89206471,
0.89684254, 0.90163551, 0.90644365, 0.911267 , 0.91610559,
0.92095945, 0.92582862, 0.93071312, 0.93561298, 0.94052825,
0.94545894, 0.95040508, 0.95536671, 0.96034386, 0.96533654,
0.97034479, 0.97536863, 0.98040809, 0.98546319, 0.99053396,
0.99562042, 1.0007226 , 1.00584051, 1.01097418, 1.01612363,
1.02128888, 1.02646995, 1.03166686, 1.03687962, 1.04210827,
1.0473528 , 1.05261324, 1.0578896 ]])
如您所见，y 与 y(x=0) = y(x=1) = 1 的边界条件相去甚远。
最佳答案
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))。
关于python - 使用 scipy 的 solve_bvp 求解具有两个边界条件的一阶 BVP，我们在Stack Overflow上找到一个类似的问题： https://stackoverflow.com/questions/39626681/
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