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July 21, 2021 07:12
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Milane Method Implementation in Python
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#!/usr/bin/python3.6 | |
# A finite-difference method for the solution of the Cauchy problem for systems of first-order ordinary differential equations: | |
# | |
# dy/dx = f(x, y) | |
# | |
# The method uses the finite-difference formula: | |
# yi - yi-1 = 2hf(xi-1, yi-1). | |
# xi = x0+ih, i = 0,1,2,3,..... | |
# | |
# The predictor-corrector Milne method uses a pair of finite-difference formulas: | |
# a predictor | |
# yi = yi-4 + 4h(2y'i-3 - y'i-2 + 2y'i-1)/3, i = 4,5,6,7,...., | |
# and a corrector | |
# yi = yi-2 + h(y'i-2+4y'i-1+y'i) | |
def dybydx(x, y): | |
fx = (x + y) | |
return fx | |
def milnep(x, y, delx, ydash, n): | |
for i in range(4): | |
ydash.append(dybydx(x[i], y[i])) | |
i = 4 | |
while i < n: | |
tmp = y[i-4] + (4*delx*(2*ydash[i-3]-ydash[i-2]+2*ydash[i-1])/3) | |
y.append(tmp) | |
x.append(x[i-1] + delx) | |
ydash.append(dybydx(x[i], y[i])) | |
print("x%i = %f and y%i = %f."% (i, x[i], i, y[i])) | |
i = i + 1 | |
def milnec(x, y, delx, ydash, n): | |
print("\nBy corrector's method: ") | |
i = 4 | |
while i < n: | |
ycheck = y[i-2] + delx*(ydash[i-2]+4*ydash[i-1]+ydash[i])/3 | |
print("y%i = %f" % (i, ycheck)) | |
i = i + 1 | |
x = list() | |
y = list() | |
ydash = list() | |
for i in range(4): | |
tmp = float(input("Enter the value of x%i: "% i)) | |
x.append(tmp) | |
tmp = float(input("Enter the value of y%i: "% i)) | |
y.append(tmp) | |
delx = float(input("Enter the value of Δx: ")) | |
n = int(input("Enter the no. of iterations: ")) + 4 | |
milnep(x, y, delx, ydash, n) | |
milnec(x, y, delx, ydash, n) |
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