V-Lab @ ANDC

Modified Euler's method for Ordinary Differential Equation

Aim

To Solve Ordinary Differential Equation using Modified Euler's method

Theory

What is Modified Euler's rule ?

The Euler forward scheme may be very easy to implement but it can't give accurate solutions. A very small step size is required for any meaningful result. In this scheme, since, the starting point of each sub-interval is used to find the slope of the solution curve, the solution would be correct only if the function is linear. So an improvement over this is to take the arithmetic average of the slopes at xi and xi+1(that is, at the end points of each sub-interval). The scheme so obtained is called modified Euler's method. It works first by approximating a value to yi+1 and then improving it by making use of average slope.
yi+1 = yi+ h/2 (y'i + y'i+1)
= yi + h/2(f(xi, yi) + f(xi+1, yi+1))

If Euler's method is used to find the first approximation of yi+1 then
yi+1 = yi + 0.5h(fi + f(xi+1, yi + hfi))

Truncation error:

yi+1 = yi + h y'i + h2yi'' /2 + h3yi''' /3! + h4yiiv /4! + ...
fi+1 = y'i+1 = y'i + h y''i + h2yi'''' /2 + h3yiiv /3! + h4yiv /4! + ...
By substituting these expansions in the Modified Euler formula gives

Procedure

  1. Define function f(x)
  2. Enter initial value of x0 and y0
  3. Enter the value of x for which y is to be calculated
  4. Enter number of iterations
  5. calculate step size (h)= (x-x0)/n
  6. Using for loop:
    i=1:n
    x(i+1) = x(i) + h
    yp(i+1) = y(i) + h * f(x(i),y(i))
  7. y(i+1) = y(i) + h * 0.5 * f[x(i),y(i) + f(x(i+1),yp(i+1))]
  8. Display the answer

Python Code

                        
                            # Modified Euler method in Python
                            import numpy as np
                            import matplotlib.pyplot as plt
                            
                            plt.style.use('seaborn-poster')
                            # Define parameters
                            f = lambda t, s: np.exp(-t) # ODE
                            h = 0.1 # Step size
                            t = np.arange(0, 1 + h, h) # Numerical grid
                            s0 = -1 # Initial Condition
                            
                            s = np.zeros(len(t))
                            sp= np.zeros(len(t))
                            s[0] = s0
                            sp[0] = s0
                            
                            for i in range(0, len(t) - 1):
                                s[i + 1] = s[i] + h*f(t[i], s[i])
                                sp[i+1]= s[i] + h*0.5*(f(t[i],s[i]) + f(t[i+1], s[i+1]))
                            
                            print(sp)
                            plt.figure(figsize = (12, 8))
                            plt.plot(t, s, 'bo--', label='Approximate')
                            plt.plot(t, -np.exp(-t), 'g', label='Exact')
                            plt.title('Approximate and Exact Solution \
                            for Simple ODE')
                            plt.xlabel('t')
                            plt.ylabel('f(t)')
                            plt.grid()
                            plt.legend(loc='lower right')
                            plt.show()

Observations

Take Observations from the method and tabulate it for the given intervals.

Plot a graph also. (Value vs Function).

Result

Hence we have calculated the value of Ordinary Differential Equation using Modified Euler's method.

References

Modified Euler's method Goseeko

Modified Euler's method Wikipedia