JavaAlgorithms/Maths/EulerMethod.java

package Maths;

import java.util.ArrayList;
import java.util.function.BiFunction;

/**
 * In mathematics and computational science, the Euler method (also called forward Euler method) is
 * a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given
 * initial value. It is the most basic explicit method for numerical integration of ordinary
 * differential equations. The method proceeds in a series of steps. At each step the y-value is
 * calculated by evaluating the differential equation at the previous step, multiplying the result
 * with the step-size and adding it to the last y-value: y_n+1 = y_n + stepSize * f(x_n, y_n).
 * (description adapted from https://en.wikipedia.org/wiki/Euler_method ) (see also:
 * https://www.geeksforgeeks.org/euler-method-solving-differential-equation/ )
 */
public class EulerMethod {

  /** Illustrates how the algorithm is used in 3 examples and prints the results to the console. */
  public static void main(String[] args) {
    System.out.println("example 1:");
    BiFunction<Double, Double, Double> exampleEquation1 = (x, y) -> x;
    ArrayList<double[]> points1 = eulerFull(0, 4, 0.1, 0, exampleEquation1);
    assert points1.get(points1.size() - 1)[1] == 7.800000000000003;
    points1.forEach(
        point -> System.out.println(String.format("x: %1$f; y: %2$f", point[0], point[1])));

    // example from https://en.wikipedia.org/wiki/Euler_method
    System.out.println("\n\nexample 2:");
    BiFunction<Double, Double, Double> exampleEquation2 = (x, y) -> y;
    ArrayList<double[]> points2 = eulerFull(0, 4, 0.1, 1, exampleEquation2);
    assert points2.get(points2.size() - 1)[1] == 45.25925556817596;
    points2.forEach(
        point -> System.out.println(String.format("x: %1$f; y: %2$f", point[0], point[1])));

    // example from https://www.geeksforgeeks.org/euler-method-solving-differential-equation/
    System.out.println("\n\nexample 3:");
    BiFunction<Double, Double, Double> exampleEquation3 = (x, y) -> x + y + x * y;
    ArrayList<double[]> points3 = eulerFull(0, 0.1, 0.025, 1, exampleEquation3);
    assert points3.get(points3.size() - 1)[1] == 1.1116729841674804;
    points3.forEach(
        point -> System.out.println(String.format("x: %1$f; y: %2$f", point[0], point[1])));
  }

  /**
   * calculates the next y-value based on the current value of x, y and the stepSize the console.
   *
   * @param xCurrent Current x-value.
   * @param stepSize Step-size on the x-axis.
   * @param yCurrent Current y-value.
   * @param differentialEquation The differential equation to be solved.
   * @return The next y-value.
   */
  public static double eulerStep(
      double xCurrent,
      double stepSize,
      double yCurrent,
      BiFunction<Double, Double, Double> differentialEquation) {
    if (stepSize <= 0) {
      throw new IllegalArgumentException("stepSize should be greater than zero");
    }
    double yNext = yCurrent + stepSize * differentialEquation.apply(xCurrent, yCurrent);
    return yNext;
  }

  /**
   * Loops through all the steps until xEnd is reached, adds a point for each step and then returns
   * all the points
   *
   * @param xStart First x-value.
   * @param xEnd Last x-value.
   * @param stepSize Step-size on the x-axis.
   * @param yStart First y-value.
   * @param differentialEquation The differential equation to be solved.
   * @return The points constituting the solution of the differential equation.
   */
  public static ArrayList<double[]> eulerFull(
      double xStart,
      double xEnd,
      double stepSize,
      double yStart,
      BiFunction<Double, Double, Double> differentialEquation) {
    if (xStart >= xEnd) {
      throw new IllegalArgumentException("xEnd should be greater than xStart");
    }
    if (stepSize <= 0) {
      throw new IllegalArgumentException("stepSize should be greater than zero");
    }

    ArrayList<double[]> points = new ArrayList<double[]>();
    double[] firstPoint = {xStart, yStart};
    points.add(firstPoint);
    double yCurrent = yStart;
    double xCurrent = xStart;

    while (xCurrent < xEnd) {
      // Euler method for next step
      yCurrent = eulerStep(xCurrent, stepSize, yCurrent, differentialEquation);
      xCurrent += stepSize;
      double[] point = {xCurrent, yCurrent};
      points.add(point);
    }

    return points;
  }
}