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Program 1

Euler Algorithm

Euler's Method for Solving ODE

To solve the first-order ODE given by:

\(\frac{dy}{dx} = f(x, y)\)

with an initial value:

\(y(x_0) = y_0\)

Euler's method uses the following formula to approximate the value of \(y\) at each step:

\(y_{i+1} = y_i + h \cdot f(x_i, y_i)\)

Where:

\(h\) is the step size (how far you move along the x-axis at each step).
\(y_i\) is the current value of \(y\) at \(x_i\).
\(f(x_i, y_i)\) is the slope at the point \((x_i, y_i)\).

Steps to Implement:

Write down the ODE and the initial value.
Choose a step size \(h\). Smaller steps usually give better results.
Start with the initial value \(y_0\).
Use the formula to calculate \(y\) at each step \(x_i\).

Algorithm: Euler's Method for Solving ODE

Input:

Function \(f(x, y)\) representing the ODE \(\frac{dy}{dx} = f(x, y)\).
Initial condition \(x_0\), \(y_0\) where \(y(x_0) = y_0\).
Step size \(h\).
End value \(x_{\text{end}}\) where the solution is to be approximated.

Output:

Approximate values of \(y\) at discrete points \(x_0, x_1, x_2, \dots, x_n\) where \(x_n \leq x_{\text{end}}\).

Steps:

Initialization:
- Set \(x = x_0\).
- Set \(y = y_0\).
- Initialize a list \(x_values\) to store the values of \(x\).
- Initialize a list \(y_values\) to store the values of \(y\).
- Add the initial values \(x_0\) and \(y_0\) to \(x_values\) and \(y_values\).
Iterate:
- While \(x < x_{\text{end}}\):
Termination:
- Continue the iteration until \(x\) reaches or exceeds \(x_{\text{end}}\).
Output the Results:
- Return the lists \(x_(values)\) and \(y_(values)\), which contain the discrete points \(x\) and the corresponding approximated values \(y\).

Example:

Given the ODE:

\(\frac{dy}{dx} = x + y\)

with the initial condition:

\(y(0) = 1\)

Let’s find the value of \(y\) at \(x = 0.2\) with \(h = 0.1\).

import numpy as np
import matplotlib.pyplot as plt

# Define the ODE as a function
def f(x, y):
    return x + y

# Euler's method implementation using a for loop
def euler_method(f, x0, y0, h, x_end):
    # Initialize the lists to store x and y values
    x_values = [x0]
    y_values = [y0]
    
    # Determine the number of steps
    n_steps = int((x_end - x0) / h)
    
    # Iteratively apply Euler's method using a for loop
    for _ in range(n_steps):
        # Calculate the next y value
        y_new = y_values[-1] + h * f(x_values[-1], y_values[-1])
        
        # Calculate the next x value
        x_new = x_values[-1] + h
        
        # Append the new values to the lists
        x_values.append(x_new)
        y_values.append(y_new)
    
    return x_values, y_values

# Parameters
x0 = 0          # Initial x value
y0 = 1          # Initial y value
h = 0.1         # Step size
x_end = 2       # End value of x

# Solve the ODE using Euler's method
x_values, y_values = euler_method(f, x0, y0, h, x_end)

# Print the results
print(f"x values: {x_values}, y values: {y_values}")

# Plot the results
plt.plot(x_values, y_values, 'b-', label="Euler's Method Approximation")
plt.xlabel('x')
plt.ylabel('y')
plt.title("Solving ODE using Euler's Method")
plt.legend()
plt.grid(True)
plt.savefig('plot.png')
plt.show()

Output 1

Ordinary Differential Equation-1