Question

Design and construct a computer program in one of the approved languages (C++) that will illustrate the use of a fourth-order explicit Runge-Kutta method of your own design. In other words, you will first have to solve the Runge-Kutta equations of condition for the coefficients of a fourth-order Runge-Kutta method. See the Mathematica notebook on solving the equations for 4th order RK method. That notebook can be found at rk4Solution.nb . PLEASE DO NOT USE a[1] = 1/2 or a[2] = 1/2. In general, you should pick a[1] and a[2] to be distinct values greater than zero and less than one. Then, you will use these coefficients in a computer program to solve the ordinary differential equation below. Be sure to follow the documentation and programming style policies of the Computer Science Department.

The initial value problem to be solved is the following: x'(t) = 3 x2 cos(5 t) subject to the initial condition: x(0) = 1.0 Obtain a numerical solution to this problem over the range from t=0.0 to t=2.0 for seven different values of the stepsize, h=0.1, 0.05 , 0.025 , 0.0125 , 0.00625 , 0.003125 , and 0.0015625 . In other words, make seven runs with 20, 40, 80, 160, 320, 640, and 1280 steps, respectively. For each run, print out the value of h, then a table of t and x, and then the error at t=2. You may use the following very precise value for your "true answer" in order to compute the error at t=2: 0.753913186469598763502963347. The true solution of this differential equation resembles the following plot of x(t) as a function of t.

Answer 1

PLEASE GIVE IT A THUMBS UP, I SERIOUSLY NEED ONE, IF YOU NEED ANY MODIFICATION THEN LET ME KNOW, I WILL DO IT FOR YOU

import math


def dxdt(t, x):
    return (3*(x**2)*math.cos(5*t))

# Finds value of x for a given t using step size h
# and initial value x0 at t0.


def rungeKutta(t0, x0, t, h):
    # Count number of iterations using step size or
    # step height h
    n = (int)((t - t0)/h)
    # Iterate for number of iterations
    x = x0
    for i in range(1, n + 1):
        "Applx Runge Kutta Formulas to find nett value of x"
        k1 = h * dxdt(t0, x)
        k2 = h * dxdt(t0 + 0.5 * h, x + 0.5 * k1)
        k3 = h * dxdt(t0 + 0.5 * h, x + 0.5 * k2)
        k4 = h * dxdt(t0 + h, x + k3)

        # Update nett value of x
        x = x + (1.0 / 6.0)*(k1 + 2 * k2 + 2 * k3 + k4)

        # Update nett value of t
        t0 = t0 + h
    return x


# Driver method
t0 = 0.0
x = 1.0
t = 2.0
a = [0.1, 0.05, 0.025, 0.0125, 0.00625, 0.003125, 0.0015625]
i = 1
for i in range(len(a)):
    print("The value of x at with step-size %.7f is: %.27f" %
          (a[i], rungeKutta(t0, x, t, a[i])))

print("Error at t=2 is: ", 0.753913186469598763502963347 -
      rungeKutta(t0, x, 2, 0.0015625))