📜流体力学电磁学运动学动力学化学和电路中欧拉法

📜流体力学电磁学运动学动力学化学和电路中欧拉法示例：Python重力弹弓流体晃动微分方程模型和交直流电阻电容电路

✒️多语言实现欧拉法和修正欧拉法

在数学和计算科学中，欧拉方法（也称为前向欧拉方法）是一种用于求解具有给定初值的常微分方程的一阶数值程序。考虑一个微分方程 $dy/dx=f(x,y)$，初始条件为 $y(x0)=y0$，则该方程的逐次逼近可由下式给出：
$$y(n+1)=y(n)+h\ast f(x(n),y(n))$$
其中 $h=(x(n)-x(0))/n$， $h $表示步长。选择较小的 $h$​ 值会导致更准确的结果和更多的计算时间。

例如，考虑微分方程 $$dy/dx=(x+y+xy)$$，初始条件为 $y(0)=1$，步长为 $h=0.025$。求$y(0.1)$​。

解：$f(x,y)=(x+y+xy)$

$x0=0,y0=1,h=0.025$

现在我们可以使用欧拉公式计算 $y_1$
$$\begin{array}{rl}& y_1=y0+h\ast f(x0,y0)\\ & y_1=1+0.025\ast (0+1+0\ast 1)\\ & y_1=1.025\\ & y(0.025)=\mathrm{1.025.}\end{array}$$
类似地我们可以计算 $y(0.050),y(0.075),\dots$ $y(0.1)$​

$y(0.1)=1.11167$

Python实现：

def func( x, y ):return (x + y + x * y)def euler( x0, y, h, x ):temp = -0while x0 < x:temp = yy = y + h * func(x0, y)x0 = x0 + hprint("Approximate solution at x = ", x, " is ", "%.6f"% y)x0 = 0
y0 = 1
h = 0.025
x = 0.1euler(x0, y0, h, x)

C++实现：

#include <iostream>
using namespace std;float func(float x, float y)
{return (x + y + x * y);
}void euler(float x0, float y, float h, float x)
{float temp = -0;while (x0 < x) {temp = y;y = y + h * func(x0, y);x0 = x0 + h;}cout << "Approximate solution at x = "<< x << " is " << y << endl;
}int main()
{float x0 = 0;float y0 = 1;float h = 0.025;float x = 0.1;euler(x0, y0, h, x);return 0;
}

C#实现：

using System;class GFG {static float func(float x, float y){return (x + y + x * y);}static void euler(float x0, float y, float h, float x){while (x0 < x) {y = y + h * func(x0, y);x0 = x0 + h;}Console.WriteLine("Approximate solution at x = "+ x + " is " + y);}public static void Main(){float x0 = 0;float y0 = 1;float h = 0.025f;float x = 0.1f;euler(x0, y0, h, x);}
}

Java实现：

import java.io.*;class Euler {float func(float x, float y){return (x + y + x * y);}void euler(float x0, float y, float h, float x){float temp = -0;while (x0 < x) {temp = y;y = y + h * func(x0, y);x0 = x0 + h;}System.out.println("Approximate solution at x = "+ x + " is " + y);}public static void main(String args[]) throws IOException{Euler obj = new Euler();float x0 = 0;float y0 = 1;float h = 0.025f;float x = 0.1f;obj.euler(x0, y0, h, x);}
}

JavaScript实现：

<script>function func(x, y){return (x + y + x * y);}function euler(x0, y, h, x){let temp = -0;while (x0 < x) {temp = y;y = y + h * func(x0, y);x0 = x0 + h;}document.write("Approximate solution at x = "+ x + " is " + y);}let x0 = 0;let y0 = 1;let h = 0.025;let x = 0.1;euler(x0, y0, h, x);</script>

预测校正器或修正欧拉法

Python实现

def f(x, y):v = y - 2 * x * x + 1;return v;def predict(x, y, h):y1p = y + h * f(x, y);return y1p;def correct(x, y, x1, y1, h):e = 0.00001;y1c = y1;while (abs(y1c - y1) > e + 1):y1 = y1c;y1c = y + 0.5 * h * (f(x, y) + f(x1, y1));return y1c;def printFinalValues(x, xn, y, h):while (x < xn):x1 = x + h;y1p = predict(x, y, h);y1c = correct(x, y, x1, y1p, h);x = x1;y = y1c;print("The final value of y at x =",int(x), "is :", y);if __name__ == '__main__':x = 0; y = 0.5;xn = 1;h = 0.2;printFinalValues(x, xn, y, h);

C++实现

#include <bits/stdc++.h>
using namespace std;double f(double x, double y)
{double v = y - 2 * x * x + 1;return v;
}double predict(double x, double y, double h)
{double y1p = y + h * f(x, y);return y1p;
}double correct(double x, double y,double x1, double y1,double h)
{double e = 0.00001;double y1c = y1;do {y1 = y1c;y1c = y + 0.5 * h * (f(x, y) + f(x1, y1));} while (fabs(y1c - y1) > e);return y1c;
}void printFinalValues(double x, double xn,double y, double h)
{while (x < xn) {double x1 = x + h;double y1p = predict(x, y, h);double y1c = correct(x, y, x1, y1p, h);x = x1;y = y1c;}cout << "The final value of y at x = "<< x << " is : " << y << endl;
}int main()
{double x = 0, y = 0.5;double xn = 1;double h = 0.2;printFinalValues(x, xn, y, h);return 0;
}

C#实现

using System;class GFG
{static double f(double x, double y)
{double v = y - 2 * x * x + 1;return v;
}static double predict(double x, double y, double h)
{double y1p = y + h * f(x, y);return y1p;
}static double correct(double x, double y,double x1, double y1,double h)
{double e = 0.00001;double y1c = y1;do{y1 = y1c;y1c = y + 0.5 * h * (f(x, y) + f(x1, y1));}while (Math.Abs(y1c - y1) > e);return y1c;
}static void printFinalValues(double x, double xn,double y, double h)
{while (x < xn) {double x1 = x + h;double y1p = predict(x, y, h);double y1c = correct(x, y, x1, y1p, h);x = x1;y = y1c;}Console.WriteLine("The final value of y at x = "+x + " is : " + Math.Round(y, 5));
}static void Main()
{double x = 0, y = 0.5;double xn = 1;double h = 0.2;printFinalValues(x, xn, y, h);
}
}

Java实现

import java.text.*;class GFG
{static double f(double x, double y)
{double v = y - 2 * x * x + 1;return v;
}static double predict(double x, double y, double h)
{double y1p = y + h * f(x, y);return y1p;
}static double correct(double x, double y,double x1, double y1,double h)
{double e = 0.00001;double y1c = y1;do{y1 = y1c;y1c = y + 0.5 * h * (f(x, y) + f(x1, y1));}while (Math.abs(y1c - y1) > e);return y1c;
}static void printFinalValues(double x, double xn,double y, double h)
{while (x < xn) {double x1 = x + h;double y1p = predict(x, y, h);double y1c = correct(x, y, x1, y1p, h);x = x1;y = y1c;}DecimalFormat df = new DecimalFormat("#.#####");System.out.println("The final value of y at x = "+x + " is : "+df.format(y));
}public static void main (String[] args) 
{double x = 0, y = 0.5;double xn = 1;double h = 0.2;printFinalValues(x, xn, y, h);
}
}

JavaScript实现

<script>function f(x , y) {var v = y - 2 * x * x + 1;return v;}function predict(x , y , h) {var y1p = y + h * f(x, y);return y1p;}function correct(x , y , x1 , y1 , h) {var e = 0.00001;var y1c = y1;do {y1 = y1c;y1c = y + 0.5 * h * (f(x, y) + f(x1, y1));} while (Math.abs(y1c - y1) > e);return y1c;}function printFinalValues(x , xn , y , h) {while (x < xn) {var x1 = x + h;var y1p = predict(x, y, h);var y1c = correct(x, y, x1, y1p, h);x = x1;y = y1c;}document.write("The final value of y at x = " + x + " is : " + y.toFixed(5));}var x = 0, y = 0.5;var xn = 1;var h = 0.2;printFinalValues(x, xn, y, h);
</script>

👉参阅：计算思维 | 亚图跨际