Add Euler method (from master) (#2148)

* readded EulerMethod.java after sync

* add package

Maths/EulerMethod.java

@@ -0,0 +1,105 @@
+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;
+  }
+}