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package com.thealgorithms.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;
}
}