Editorial StaffBackward Euler’s method, also known as implicit Euler’s method, is an alternative numerical method for solving ODEs.
Unlike Euler’s method, backward Euler’s method estimates the solution by considering the tangent line at the next step instead of the current step.
This makes it an implicit method as it requires solving an equation involving the unknown solution.
The formula for backward Euler’s method is as follows:
y(n+1) = y(n) + h * f(x(n+1), y(n+1))
In this case, f(x(n+1), y(n+1)) represents the derivative of the function at the next point, and the value of y(n+1) is determined by solving an equation involving the unknown y(n+1).
Backward Euler’s method is more stable and accurate compared to Euler’s method, especially for stiff systems or ODEs with rapidly changing slopes. It mitigates the issue of accumulating errors and provides better approximations.
However, it requires solving equations iteratively, which can be computationally more expensive.
#include <iostream>
#include <cmath>
// Function representing the differential equation dy/dx = f(x, y)
double f(double x, double y) {
return x - y; // Example differential equation: dy/dx = x - y
}
// Backward Euler's method implementation
double backwardEuler(double x0, double y0, double h, double x) {
double y = y0;
double xCurrent = x0;
while (xCurrent < x) {
double yNext = y + h * f(xCurrent + h, y);
y = yNext;
xCurrent += h;
}
return y;
}
int main() {
double x0, y0, h, x;
std::cout << "Enter the initial value of x: ";
std::cin >> x0;
std::cout << "Enter the initial value of y: ";
std::cin >> y0;
std::cout << "Enter the step size (h): ";
std::cin >> h;
std::cout << "Enter the target value of x: ";
std::cin >> x;
double yApprox = backwardEuler(x0, y0, h, x);
std::cout << "Approximate value of y at x = " << x << ": " << yApprox << std::endl;
return 0;
}
