|
| 1 | +packageMaths; |
| 2 | + |
| 3 | +importjava.util.ArrayList; |
| 4 | +importjava.util.function.BiFunction; |
| 5 | + |
| 6 | +/** |
| 7 | + * In mathematics and computational science, the Euler method (also called forward Euler method) is |
| 8 | + * a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given |
| 9 | + * initial value. It is the most basic explicit method for numerical integration of ordinary |
| 10 | + * differential equations. The method proceeds in a series of steps. At each step the y-value is |
| 11 | + * calculated by evaluating the differential equation at the previous step, multiplying the result |
| 12 | + * with the step-size and adding it to the last y-value: y_n+1 = y_n + stepSize * f(x_n, y_n). |
| 13 | + * (description adapted from https://en.wikipedia.org/wiki/Euler_method ) (see also: |
| 14 | + * https://www.geeksforgeeks.org/euler-method-solving-differential-equation/ ) |
| 15 | + */ |
| 16 | +publicclassEulerMethod { |
| 17 | + |
| 18 | +/** Illustrates how the algorithm is used in 3 examples and prints the results to the console. */ |
| 19 | +publicstaticvoidmain(String[]args) { |
| 20 | +System.out.println("example 1:"); |
| 21 | +BiFunction<Double,Double,Double>exampleEquation1 = (x,y) ->x; |
| 22 | +ArrayList<double[]>points1 =eulerFull(0,4,0.1,0,exampleEquation1); |
| 23 | +assertpoints1.get(points1.size() -1)[1] ==7.800000000000003; |
| 24 | +points1.forEach( |
| 25 | +point ->System.out.println(String.format("x: %1$f; y: %2$f",point[0],point[1]))); |
| 26 | + |
| 27 | +// example from https://en.wikipedia.org/wiki/Euler_method |
| 28 | +System.out.println("\n\nexample 2:"); |
| 29 | +BiFunction<Double,Double,Double>exampleEquation2 = (x,y) ->y; |
| 30 | +ArrayList<double[]>points2 =eulerFull(0,4,0.1,1,exampleEquation2); |
| 31 | +assertpoints2.get(points2.size() -1)[1] ==45.25925556817596; |
| 32 | +points2.forEach( |
| 33 | +point ->System.out.println(String.format("x: %1$f; y: %2$f",point[0],point[1]))); |
| 34 | + |
| 35 | +// example from https://www.geeksforgeeks.org/euler-method-solving-differential-equation/ |
| 36 | +System.out.println("\n\nexample 3:"); |
| 37 | +BiFunction<Double,Double,Double>exampleEquation3 = (x,y) ->x +y +x *y; |
| 38 | +ArrayList<double[]>points3 =eulerFull(0,0.1,0.025,1,exampleEquation3); |
| 39 | +assertpoints3.get(points3.size() -1)[1] ==1.1116729841674804; |
| 40 | +points3.forEach( |
| 41 | +point ->System.out.println(String.format("x: %1$f; y: %2$f",point[0],point[1]))); |
| 42 | + } |
| 43 | + |
| 44 | +/** |
| 45 | + * calculates the next y-value based on the current value of x, y and the stepSize the console. |
| 46 | + * |
| 47 | + * @param xCurrent Current x-value. |
| 48 | + * @param stepSize Step-size on the x-axis. |
| 49 | + * @param yCurrent Current y-value. |
| 50 | + * @param differentialEquation The differential equation to be solved. |
| 51 | + * @return The next y-value. |
| 52 | + */ |
| 53 | +publicstaticdoubleeulerStep( |
| 54 | +doublexCurrent, |
| 55 | +doublestepSize, |
| 56 | +doubleyCurrent, |
| 57 | +BiFunction<Double,Double,Double>differentialEquation) { |
| 58 | +if (stepSize <=0) { |
| 59 | +thrownewIllegalArgumentException("stepSize should be greater than zero"); |
| 60 | + } |
| 61 | +doubleyNext =yCurrent +stepSize *differentialEquation.apply(xCurrent,yCurrent); |
| 62 | +returnyNext; |
| 63 | + } |
| 64 | + |
| 65 | +/** |
| 66 | + * Loops through all the steps until xEnd is reached, adds a point for each step and then returns |
| 67 | + * all the points |
| 68 | + * |
| 69 | + * @param xStart First x-value. |
| 70 | + * @param xEnd Last x-value. |
| 71 | + * @param stepSize Step-size on the x-axis. |
| 72 | + * @param yStart First y-value. |
| 73 | + * @param differentialEquation The differential equation to be solved. |
| 74 | + * @return The points constituting the solution of the differential equation. |
| 75 | + */ |
| 76 | +publicstaticArrayList<double[]>eulerFull( |
| 77 | +doublexStart, |
| 78 | +doublexEnd, |
| 79 | +doublestepSize, |
| 80 | +doubleyStart, |
| 81 | +BiFunction<Double,Double,Double>differentialEquation) { |
| 82 | +if (xStart >=xEnd) { |
| 83 | +thrownewIllegalArgumentException("xEnd should be greater than xStart"); |
| 84 | + } |
| 85 | +if (stepSize <=0) { |
| 86 | +thrownewIllegalArgumentException("stepSize should be greater than zero"); |
| 87 | + } |
| 88 | + |
| 89 | +ArrayList<double[]>points =newArrayList<double[]>(); |
| 90 | +double[]firstPoint = {xStart,yStart}; |
| 91 | +points.add(firstPoint); |
| 92 | +doubleyCurrent =yStart; |
| 93 | +doublexCurrent =xStart; |
| 94 | + |
| 95 | +while (xCurrent <xEnd) { |
| 96 | +// Euler method for next step |
| 97 | +yCurrent =eulerStep(xCurrent,stepSize,yCurrent,differentialEquation); |
| 98 | +xCurrent +=stepSize; |
| 99 | +double[]point = {xCurrent,yCurrent}; |
| 100 | +points.add(point); |
| 101 | + } |
| 102 | + |
| 103 | +returnpoints; |
| 104 | + } |
| 105 | +} |