Euler’s Method and Improved Euler’s Method

Euler’s method is a numerical technique for solving first-order initial value problems, such as ordinary differential equations (ODEs). The method approximates the solution of the ODE at a series of discrete points by using the initial condition and the slope of the function at each point.

Improved Euler’s method, also known as the Heun’s method, is an extension of Euler’s method that provides a more accurate approximation by using two slopes at each point: the slope at the beginning of the interval and the slope at the end. This approach results in a better estimation of the function’s behavior over the interval.

Both Euler’s method and Improved Euler’s method are considered explicit methods, meaning the approximated solution can be calculated directly from the known values. These methods are relatively simple and easy to implement, but they may not provide the best accuracy for some problems. More advanced numerical methods, such as Runge-Kutta methods, are often used when higher accuracy or stability is required.