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Cubic ODE: method Euler
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‎Cauchy problem for ODE/methodEulera/methodEulera.cpp‎

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#include <iostream>
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#define SIZE 10
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using namespace std;
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double function(double x, double y); // y'=3sin(2y)+x Начальное условие y(0)=2 на отрезке [0,1] h=0.1;
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double Runge_K(double x, double y, double h);
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int main()
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{
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setlocale(LC_ALL, "RU");
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double* nodeX = new double[SIZE+1]; //точки
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double* y = new double[SIZE+1]; // приближенное значений функций в этих точках
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double a = 0.0, b = 0.0;//отрезок
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cout << " Введите отрезок: начало : ";
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cin >> a;
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cout << " Введите отрезок: конец : ";
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cin >> b;
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cout << " Введите начальную точку x0 на отрезке: ";
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cin >> nodeX[0];
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cout << " Введите начальное условие : ";
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cin >> y[0];//nodeX[0];
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double h = 0.0; // если известны узлы то x[i-1]-x[i]
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int n = 0;
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cout << " Введите количество разбиений отрезка(целое): ";
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cin >> n;
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h = (b - a) / n;
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int select = 0;
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cout << " Выбирете метод для вычисления ОДУ : \n";
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cout << " 1.Метод Эйлера.\n 2. Метод Рунге-Кутты 2 порядка(частный случай Эйлера)\n 3.Метод Рунге-Кутты 4 порядка\n";
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cin >> select;
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switch (select)
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{
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case 1:
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cout << " Вычисление ОДУ методом Эйлера: 3sin(2y)+x" << endl; // Неявная формула
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for (int i = 1; i <= n; i++)
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{
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nodeX[i] = nodeX[i - 1] + h; //a + i * h;
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y[i] = y[i - 1] + h * function(nodeX[i - 1], y[i - 1]);
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cout << " Итерация: " << i << " --> X[" << i << "] = " << nodeX[i] << "--> Y[" << i << "] = " << y[i] << endl;
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}
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break;
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case 2:
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cout << " Вычисление ОДУ методов Рунге-Кутты второго порядка\n(частный случай Эйлера) :3sin(2y)+x " << endl;
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for (int i = 1; i <= n; i++)
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{
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cout << "Прогноз : " << i << " : ";
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nodeX[i] = nodeX[i - 1] + h;
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y[i] = y[i - 1] + h * function(nodeX[i - 1], y[i - 1]);
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cout << " Y[" << i << "] = " << y[i] << endl;
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cout << "Коррекция : " << i << " ";
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y[i] = y[i - 1] + (h / 2) * (function(nodeX[i - 1], y[i - 1]) + function(nodeX[i], y[i]));
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cout << " Y[" << i << "] = " << y[i] << " --> X[" << i << "]" << nodeX[i] << endl;
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}
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break;
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case 3:
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cout << " Вычисление ОДУ методов Рунге-Кутты 4 порядка :3sin(2y)+x " << endl;
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for (int i = 1; i <= n; i++)
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{
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nodeX[i] = nodeX[i - 1] + h;
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y[i] = y[i - 1] + (h / 6.0) * (Runge_K(nodeX[i - 1], y[i - 1], h));
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cout << " Итерация: " << i << " --> X[" << i << "] = " << nodeX[i] << "--> Y[" << i << "] = " << y[i] << endl;
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}
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break;
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case 4:
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cout << " Вычисление ОДУ методом Адамса-Баншорта(экстраполяционный) 4 порядка : 3sin(2y)+x " << endl; //неявный
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break;
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case 5:
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cout << " Вычисление ОДУ методом Адамса-Мультона 2 порядка : 3sin(2y)+x " << endl; //явный
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break;
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default:
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break;
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}
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/*абс погр = max[i](abs(y[i] при h, -y[i] при h/2))/2^p-1*/
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delete[] nodeX;
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delete[] y;
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return 0;
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}
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double Runge_K(double x, double y,double h)
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{
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double k1 = function(x, y);
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double k2 = 2.0 * function(x + h / 2.0, y + (h / 2.0) * k1);
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double k3 = 2.0 * function(x + h / 2.0, y + (h / 2.0) *k2);
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double k4 = function(x + h, y + h * k3);
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return k1+ k2 + k3 + k4;
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}
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double function(double x, double y)
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{
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return 3 * sin(2 * y) + x;
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}
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//function y''+3xy'-2y=1.5; [0.5,1] y(0.5)=1, y'(0.5)=0.5 h=0.1
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//y'=z; y(0.5)=1, z(0.5)=0.5
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// z'=-3xz+2y+1.5
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/*
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double function(x,y,z)
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{
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return -3*x*z+2y+1.5;
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}
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в main()
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{
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//ввод отрезка а,b
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//ввод х0 = a
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//ввод у0 =1
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//ввод z0 = 0.5
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//h = (b-a)/n
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for(int i=1;i<=n;i++)
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{
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x[i]=x[i-1]+h;
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y[i]=y[i-1]+h*function(x[i-1],y[i-1],z[i-1]);
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z[i]=z[i-1]+h*G(x[i-1],y[i-1],z[i-1]);
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}
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}
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*/

‎Cauchy problem for ODE/methodEulera/methodEulera.sln‎

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‎Cauchy problem for ODE/methodEulera/methodEulera.vcxproj‎

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