Math5620

Author: David Merkley

Language: Python

Description/Purpose: This code solves an IVP using the Predictor Corrector Method. Then it solves for the exact solution and plots the error.

Implementation/Code:

from matplotlib import pyplot as plt
import numpy as np

# Function to get Exact solution
def ExactSol(t):
    return np.e ** ((1/2) * (t**2) + t)

# The initial value problem looked at
def IVP(u, t):
    return (t + 1) * u

# All of these k functions are to evaluate the Ufinal function
def k1(U, t):
    return IVP(U, t)

def k2(U, t):
    return IVP(U + (0.05 * k1(U, t)), t + 0.05)

#  Returns a solution to the IVP with the Runge Kutta Method
def Ufinal(U, t):
    return U + (0.05/2 * (k1(U, t) + k2(U, t)))

def main():
    # Initiates lists required to plot the error of the solution
    listU = [1]
    listExact = [1]
    listError = []

    #  For loop to get the runge kutta solution and exact solution along with it
    for n in range(1, 21):
        listU.append(Ufinal(listU[n-1], (n-1) * 0.05))
        listExact.append(ExactSol(n * 0.05))

    # For loop to get the error of the solution
    for listU_i, listExact1_i in zip(listU, listExact):
        listError.append(listExact1_i - listU_i)

    # Plotting the Errors and Graphs for Comparison
    x1 = np.linspace(0, 1, 21)
    plt.title("Predictor Corrector Method vs Exact Solution")
    plt.xticks(x1)
    plt.plot(x1, listU, marker='o', label='Exact Solution')
    plt.plot(x1, listExact, marker='o', label='Predictor Corrector Method')
    plt.legend()
    plt.show()
    plt.title("Error of Predictor Corrector Method")
    plt.xticks(x1)
    plt.plot(x1, listError, marker='o', label='Error')
    plt.legend()
    plt.show()

main()

Output: