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  1. import numpy as np
  2. import math
  3. import matplotlib.pyplot as plt
  4.  
  5. left = -1
  6. right = 1
  7. k = 1
  8. n = 12
  9. plt.grid()
  10.  
  11.  
  12. def phi(x, i):
  13.     return x**i * (1 - x ** 2)
  14.  
  15.  
  16. def a(x):
  17.     return math.sin(k)
  18.  
  19.  
  20. def b(x):
  21.     return math.cos(k)
  22.  
  23.  
  24. def f(x):
  25.     return -1
  26.  
  27.  
  28. def p(x):
  29.     return 1 + x * x * b(x)
  30.  
  31.  
  32. def GetBasisCoefficient(j, x):
  33.     if j == 0:
  34.         return -2 + (1 - x ** 2) * p(x)
  35.     elif j == 1:
  36.         return -6 * x + (x - x ** 3) * p(x)
  37.     else:
  38.         return j * (j - 1) * x ** (j - 2) - (j + 1) * (j + 2) * x ** j \
  39.                      + (x ** j - x ** (j + 2)) * p(x)
  40.  
  41.  
  42. def AddSolutionToPlot(X, A, y):
  43.     res = np.linalg.solve(A, y)
  44.     func = lambda x: sum(res[i] * (x ** i * (1 - x ** 2)) for i in range(n))
  45.     Y = [func(x) for x in X]
  46.     plt.plot(X, Y)
  47.  
  48.  
  49. def SolveCollocation():
  50.     X = np.linspace(left, right, n)
  51.     y = [f(x) for x in X]
  52.     A = [[0] * n for _ in range(n)]
  53.     for i in range(n):
  54.         for j in range(n):
  55.             A[i][j] = GetBasisCoefficient(j, X[i])
  56.     AddSolutionToPlot(X, A, y)
  57.  
  58.  
  59. def SolveLSMIntegral():
  60.     X = np.linspace(left, right, n)
  61.     h = (right - left) / n
  62.     A = [[0] * n for _ in range(n)]
  63.     y = [0 for _ in range(n)]
  64.     for i in range(n):
  65.         y[i] = np.sum(f(X) * GetBasisCoefficient(i, X)) * h
  66.         for j in range(n):
  67.             A[i][j] = np.sum(GetBasisCoefficient(j, X) * GetBasisCoefficient(i, X)) * h
  68.     AddSolutionToPlot(X, A, y)
  69.  
  70.  
  71. def SolveLSMDiscrete():
  72.     X = np.linspace(left, right, n)
  73.     A = [[0] * n for _ in range(n)]
  74.     y = [0 for _ in range(n)]
  75.     for i in range(n):
  76.         y[i] = np.sum([f(X) * GetBasisCoefficient(i, X)])
  77.         for j in range(n):
  78.             A[i][j] = np.sum(GetBasisCoefficient(j, X) * GetBasisCoefficient(i, X))
  79.     AddSolutionToPlot(X, A, y)
  80.  
  81.  
  82. def SolveGalerkin():
  83.     X = np.linspace(left, right, n)
  84.     A = [[0] * n for _ in range(n)]
  85.     h = (right - left) / n
  86.     y = [0 for _ in range(n)]
  87.     for i in range(n):
  88.         y[i] = np.sum(f(X) * phi(X, i)) * h
  89.         for j in range(n):
  90.             A[i][j] = np.sum(phi(X, i) * GetBasisCoefficient(j, X)) * h
  91.     AddSolutionToPlot(X, A, y)
  92.  
  93.  
  94. SolveCollocation()
  95. SolveLSMIntegral()
  96. SolveLSMDiscrete()
  97. SolveGalerkin()
  98. plt.show()
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