Step by Step solution about How to resolve the algorithm Euler method step by step in the Java programming language

The provided Java code demonstrates the Euler method for numerically solving ordinary differential equations (ODEs). The Euler method is a first-order numerical integration method that approximates the solution of an ODE by using the derivative of the solution at the current time step to estimate the solution at the next time step.

The euler method in the code takes a callable f, an initial value y0, a starting time a, an end time b, and a step size h as parameters. It then iteratively applies the Euler method to solve the ODE, printing the solution at each time step. The DONE message is printed once the solution has been computed for all time steps.

In the main method, the euler method is used to solve the Newton cooling equation, which describes the rate of change of the temperature of an object as it cools. The cooling equation is given by:

dT/dt = -k(T - T_a)

where:

• T is the temperature of the object
• T_a is the ambient temperature
• k is a positive constant

In the code, the Cooling class implements the callable interface and provides an implementation of the compute method that returns the derivative of the Newton cooling equation. The step size is varied to demonstrate the effect of step size on the accuracy of the solution.

Here is a step-by-step explanation of the code:

1. The euler method takes a callable f, an initial value y0, a starting time a, an end time b, and a step size h as parameters.
2. It initializes the time t to a and the solution y to y0.
3. It enters a loop that continues until t is greater than or equal to b.
4. Inside the loop, it prints the current time t and the current solution y.
5. It updates the time t by adding the step size h.
6. It updates the solution y by adding the step size h multiplied by the derivative of the solution at the current time t, which is computed by calling the compute method of the callable f.
7. Once the loop has completed, it prints the message "DONE" to indicate that the solution has been computed for all time steps.

In the main method:

1. A callable cooling is created that implements the Newton cooling equation.
2. An array steps is created that contains the step sizes to be used.
3. A loop is entered that iterates over the step sizes in the steps array.
4. For each step size, the euler method is called to solve the Newton cooling equation with the given step size. The initial temperature is set to 100.0, the starting time is set to 0, the end time is set to 100, and the step size is set to the current value in the steps array.
5. The solution is printed to the console.

public class Euler {
  private static void euler (Callable f, double y0, int a, int b, int h) {
    int t = a;
    double y = y0;
    while (t < b) {
      System.out.println ("" + t + " " + y);
      t += h;
      y += h * f.compute (t, y);
    }
    System.out.println ("DONE");
  }

  public static void main (String[] args) {
    Callable cooling = new Cooling ();
    int[] steps = {2, 5, 10};
    for (int stepSize : steps) {
      System.out.println ("Step size: " + stepSize);
      euler (cooling, 100.0, 0, 100, stepSize);
    }
  }
}

// interface used so we can plug in alternative functions to Euler
interface Callable {
  public double compute (int time, double t);
}

// class to implement the newton cooling equation
class Cooling implements Callable {
  public double compute (int time, double t) {
    return -0.07 * (t - 20);
  }
}