7.2.3. Improved Euler’s Method (Heun’s Method)

Try these first

1. 1.

   What is the main advantage of Heun's Method over Euler's Method?

   Hint

   Consider the aspect of slope calculations.

2. 2.

   What does ODE stand for?

   Hint

   Think about the types of equations involving derivatives.

3. 3.

   What is the main purpose of Heun's Method?

   • To provide an accurate solution to linear equations
   • To improve the accuracy of Euler's Method
   • To simplify ordinary differential equations

   Hint

   Think about why we need numerical methods.

4. 4.

   True or False: Heun’s Method requires twice the function evaluations compared to Euler's Method.

   • True
   • False

   Hint

   Consider how many times we calculate the function in each method.

5. 5.

   Using Heun's Method, calculate an approximation of the solution for the ODE dy/dx = y - x^2 + 1 with initial value y(0) = 0.5 at x = 0.2; use a step size of 0.1.

   Hint

   Break the problem down into steps: evaluate the function, predict, and then correct.

6. 6.

   A population is modeled by the equation dP/dt = rP where P(0) = 1000, r=0.03. Predict the population at t=1 using Heun's Method with h=0.1.

   Hint

   Apply the Heun's method formula and keep track of your computations carefully.