<gh_stars>0
import numpy as np
from scipy.integrate import solve_ivp
import matplotlib.pyplot as plt




#  Define our ODEs
def equations(s, theta):
    theta_prime = np.zeros(2)
    
    c = 8.360795454545455e-07

    theta_prime[0] = theta[1]
    theta_prime[1] = -c * np.cos(theta[0])

    return theta_prime




#  Constants
c = 8.360795454545455e-07
K = 0.0006693734347137342
L = 600

#  Define range of integration
x = np.linspace(0, L, 100)
xspan = (x[0],x[-1])

#  We use the small angle approximation for our initial guess
initial_guess = c * L + K

#  Then we adjust our guess until the derivative boundary condition is matched
#theta_0 = [0, initial_guess]
theta_0 = [0, 0.001145]

#  Solve the system
sol = solve_ivp(equations, xspan, theta_0, t_eval=x)

#  Plot the derivative and desired boundary value
plt.plot(x, sol.y[1,:])
plt.plot(L, K, 'o')
plt.title('Confirm Derivative Boundary Condition')
plt.grid(True)
plt.xlabel('s')
plt.ylabel(r'$d\theta/ds$')
plt.savefig('Derivative_Boundary_condition.png')
#  Check to see of our approximate solution is close.  I multiplied both bu 1000 to make the numbers easier to compare by eye.
print('Desired boundary = ', 10000 * K)
print('fFrom solver: ', 10000 * sol.y[1, -1])




plt.plot(x, sol.y[0,:])
plt.grid(True)
plt.xlabel('s')
plt.ylabel(r'$\theta$')
plt.title('Solution')
plt.savefig('Solution.png')


#  Desired tolerance
tol = 1e-6

#  Correct derivative is between these two numbers
low =  0.001
high = 0.002

#  Counter and maximum numer of iterations to avoid infinite loops
max_iters = 100
count = 0

while count <= max_iters:
    count = count + 1
    
    #  Derivative guess is average of bounds
    theta_prime0 = np.mean([low, high])    
    theta_0 = [0, theta_prime0 ]

    #  Solve the system
    sol = solve_ivp(equations, xspan, theta_0, t_eval = x)

    #  For ease of use, extract the function derivative values from the solution object.
    theta_prime = sol.y[1, :]

    #  Calculate percent error and break if within tolerance
    err = np.abs(theta_prime[-1] - K) / K
    if np.abs(err) <= tol:
        break
    
    #  Adjust our bounds if we are not within tolerance
    if theta_prime[-1] > K:
        high = theta_prime0
    else:
        low = theta_prime0
            

print('Correct initial derivative = ', theta_prime0)



#  solve and plot the final results

theta_0 = [0, theta_prime0]
sol = solve_ivp(equations, xspan, theta_0, t_eval=x)

fig, (ax1, ax2) = plt.subplots(1,2)

ax1.plot(x, sol.y[0,:], 'k')
ax1.grid(True)
ax1.set_xlabel('s')
ax1.set_ylabel(r'$\theta$')
ax1.set_title('Solution')

ax2.plot(x, sol.y[1,:], 'k')
ax2.plot(x[-1], K, 'o')
ax2.grid(True)
ax2.set_xlabel('s')
ax2.set_ylabel(r'$d\theta/ds$')
ax2.set_title('Check Derivative Boundary')

plt.tight_layout()
plt.savefig('final.png')




import numpy as np
from scipy.integrate import solve_ivp
import matplotlib.pyplot as plt




#  Define our ODEs
def equations(s, theta):
    theta_prime = np.zeros(2)
    
    c = 8.360795454545455e-07

    theta_prime[0] = theta[1]
    theta_prime[1] = -c * np.cos(theta[0])

    return theta_prime




#  Constants
c = 8.360795454545455e-07
K = 0.0006693734347137342
L = 600

#  Define range of integration
x = np.linspace(0, L, 100)
xspan = (x[0],x[-1])

#  We use the small angle approximation for our initial guess
initial_guess = c * L + K

#  Then we adjust our guess until the derivative boundary condition is matched
#theta_0 = [0, initial_guess]
theta_0 = [0, 0.001145]

#  Solve the system
sol = solve_ivp(equations, xspan, theta_0, t_eval=x)

#  Plot the derivative and desired boundary value
plt.plot(x, sol.y[1,:])
plt.plot(L, K, 'o')
plt.title('Confirm Derivative Boundary Condition')
plt.grid(True)
plt.xlabel('s')
plt.ylabel(r'$d\theta/ds$')
plt.savefig('Derivative_Boundary_condition.png')
#  Check to see of our approximate solution is close.  I multiplied both bu 1000 to make the numbers easier to compare by eye.
print('Desired boundary = ', 10000 * K)
print('fFrom solver: ', 10000 * sol.y[1, -1])




plt.plot(x, sol.y[0,:])
plt.grid(True)
plt.xlabel('s')
plt.ylabel(r'$\theta$')
plt.title('Solution')
plt.savefig('Solution.png')


#  Desired tolerance
tol = 1e-6

#  Correct derivative is between these two numbers
low =  0.001
high = 0.002

#  Counter and maximum numer of iterations to avoid infinite loops
max_iters = 100
count = 0

while count <= max_iters:
    count = count + 1
    
    #  Derivative guess is average of bounds
    theta_prime0 = np.mean([low, high])    
    theta_0 = [0, theta_prime0 ]

    #  Solve the system
    sol = solve_ivp(equations, xspan, theta_0, t_eval = x)

    #  For ease of use, extract the function derivative values from the solution object.
    theta_prime = sol.y[1, :]

    #  Calculate percent error and break if within tolerance
    err = np.abs(theta_prime[-1] - K) / K
    if np.abs(err) <= tol:
        break
    
    #  Adjust our bounds if we are not within tolerance
    if theta_prime[-1] > K:
        high = theta_prime0
    else:
        low = theta_prime0
            

print('Correct initial derivative = ', theta_prime0)



#  solve and plot the final results

theta_0 = [0, theta_prime0]
sol = solve_ivp(equations, xspan, theta_0, t_eval=x)

fig, (ax1, ax2) = plt.subplots(1,2)

ax1.plot(x, sol.y[0,:], 'k')
ax1.grid(True)
ax1.set_xlabel('s')
ax1.set_ylabel(r'$\theta$')
ax1.set_title('Solution')

ax2.plot(x, sol.y[1,:], 'k')
ax2.plot(x[-1], K, 'o')
ax2.grid(True)
ax2.set_xlabel('s')
ax2.set_ylabel(r'$d\theta/ds$')
ax2.set_title('Check Derivative Boundary')

plt.tight_layout()
plt.savefig('final.png')
#  Define our ODEs #  Constants #  Define range of integration #  We use the small angle approximation for our initial guess #  Then we adjust our guess until the derivative boundary condition is matched #theta_0 = [0, initial_guess] #  Solve the system #  Plot the derivative and desired boundary value #  Check to see of our approximate solution is close.  I multiplied both bu 1000 to make the numbers easier to compare by eye. #  Desired tolerance #  Correct derivative is between these two numbers #  Counter and maximum numer of iterations to avoid infinite loops #  Derivative guess is average of bounds #  Solve the system #  For ease of use, extract the function derivative values from the solution object. #  Calculate percent error and break if within tolerance #  Adjust our bounds if we are not within tolerance #  solve and plot the final results