%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
import seaborn as sns
from scipy.integrate import odeint
from IPython.html.widgets import interact, fixed

"""
Explanation: Ordinary Differential Equations Exercise 1
Imports
End of explanation
"""


np.zeros?

def solve_euler(derivs, y0, x):
    """Solve a 1d ODE using Euler's method.
    
    Parameters
    ----------
    derivs : function
        The derivative of the diff-eq with the signature deriv(y,x) where
        y and x are floats.
    y0 : float
        The initial condition y[0] = y(x[0]).
    x : np.ndarray, list, tuple
        The array of times at which of solve the diff-eq.
    
    Returns
    -------
    y : np.ndarray
        Array of solutions y[i] = y(x[i])
    """
    y = np.empty(len(x))
    y[0] = y0
    n = 0
    while n < len(x)-1:
        h = x[n+1] - x[n]
        y[n+1] = y[n] + h*derivs(y[n],x[n])
        n += 1
    return y

assert np.allclose(solve_euler(lambda y, x: 1, 0, [0,1,2]), [0,1,2])

"""
Explanation: Euler's method
Euler's method is the simplest numerical approach for solving a first order ODE numerically. Given the differential equation
$$ \frac{dy}{dx} = f(y(x), x) $$
with the initial condition:
$$ y(x_0)=y_0 $$
Euler's method performs updates using the equations:
$$ y_{n+1} = y_n + h f(y_n,x_n) $$
$$ h = x_{n+1} - x_n $$
Write a function solve_euler that implements the Euler method for a 1d ODE and follows the specification described in the docstring:
End of explanation
"""


def solve_midpoint(derivs, y0, x):
    """Solve a 1d ODE using the Midpoint method.
    
    Parameters
    ----------
    derivs : function
        The derivative of the diff-eq with the signature deriv(y,x) where y
        and x are floats.
    y0 : float
        The initial condition y[0] = y(x[0]).
    x : np.ndarray, list, tuple
        The array of times at which of solve the diff-eq.
    
    Returns
    -------
    y : np.ndarray
        Array of solutions y[i] = y(x[i])
    """
    y = np.empty(len(x))
    y[0] = y0
    n = 0
    while n < len(x)-1:
        h = x[n+1] - x[n]
        y[n+1] = y[n] + h*derivs(y[n] + h*derivs(y[n],x[n])/2,x[n] + h/2)
        n += 1
    return y

assert np.allclose(solve_midpoint(lambda y, x: 1, 0, [0,1,2]), [0,1,2])

"""
Explanation: The midpoint method is another numerical method for solving the above differential equation. In general it is more accurate than the Euler method. It uses the update equation:
$$ y_{n+1} = y_n + h f\left(y_n+\frac{h}{2}f(y_n,x_n),x_n+\frac{h}{2}\right) $$
Write a function solve_midpoint that implements the midpoint method for a 1d ODE and follows the specification described in the docstring:
End of explanation
"""


def solve_exact(x):
    """compute the exact solution to dy/dx = x + 2y.
    
    Parameters
    ----------
    x : np.ndarray
        Array of x values to compute the solution at.
    
    Returns
    -------
    y : np.ndarray
        Array of solutions at y[i] = y(x[i]).
    """
    y = np.empty(len(x))
    for i in range(len(x)):
        y[i] = 0.25*np.exp(2*x[i]) - 0.5*x[i] - 0.25
    return y

assert np.allclose(solve_exact(np.array([0,1,2])),np.array([0., 1.09726402, 12.39953751]))

"""
Explanation: You are now going to solve the following differential equation:
$$
\frac{dy}{dx} = x + 2y
$$
which has the analytical solution:
$$
y(x) = 0.25 e^{2x} - 0.5 x - 0.25
$$
First, write a solve_exact function that compute the exact solution and follows the specification described in the docstring:
End of explanation
"""


N = 10
x = np.linspace(0,1.0,N)
y0 = 0.0
derivs = lambda y, x: x + 2*y

y_euler = solve_euler(derivs,y0,x)
y_midpoint = solve_midpoint(derivs,y0,x)
y_odeint = odeint(derivs,y0,x)
y_exact = solve_exact(x)

plt.plot(x, y_euler, label='Euler')
plt.plot(x, y_midpoint, label='Midpoint')
plt.plot(x, y_odeint, label='odeint')
plt.plot(x, y_exact, label='Exact')
plt.legend(loc=2);

assert True # leave this for grading the plots

"""
Explanation: In the following cell you are going to solve the above ODE using four different algorithms:

Euler's method
Midpoint method
odeint
Exact

Here are the details:

Generate an array of x values with $N=11$ points over the interval $[0,1]$ ($h=0.1$).
Define the derivs function for the above differential equation.
Using the solve_euler, solve_midpoint, odeint and solve_exact functions to compute
  the solutions using the 4 approaches.

Visualize the solutions on a sigle figure with two subplots:

Plot the $y(x)$ versus $x$ for each of the 4 approaches.
Plot $\left|y(x)-y_{exact}(x)\right|$ versus $x$ for each of the 3 numerical approaches.

Your visualization should have legends, labeled axes, titles and be customized for beauty and effectiveness.
While your final plot will use $N=10$ points, first try making $N$ larger and smaller to see how that affects the errors of the different approaches.
End of explanation
"""


