Editing Equations, Formulas and Algorithms (section)

== Finite Difference Methods ==
Finite difference methods discretize differential equations by approximating derivatives using finite differences. These methods are widely used for solving ordinary and partial differential equations in various domains, including fluid dynamics, heat transfer, and structural mechanics.

'''Example: Finite Difference Method'''
Consider the one-dimensional heat equation:
<math mode="display"> \frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2} </math>
where <math> u(x, t) </math> is the temperature distribution, <math> \alpha </math> is the thermal diffusivity, <math> x </math> is the spatial coordinate, and <math> t </math> is time.

'''Source Code Example (C++):'''
<syntaxhighlight lang="c++">
#include <iostream>
#include <vector>

// Function to solve the heat equation using the finite difference method
std::vector<double> solve_heat_equation(int nx, int nt, double alpha) {
    double dx = 1.0 / (nx - 1);
    double dt = 0.01;
    double r = alpha * dt / (dx * dx);

    // Initialize temperature distribution
    std::vector<double> u(nx);
    for (int i = 0; i < nx; ++i) {
        u[i] = 0.0; // Initial condition: u(x,0) = 0
    }

    // Apply finite difference method
    for (int t = 0; t < nt; ++t) {
        std::vector<double> u_new(nx);
        for (int i = 1; i < nx - 1; ++i) {
            u_new[i] = u[i] + r * (u[i+1] - 2 * u[i] + u[i-1]);
        }
        u = u_new;
    }

    return u;
}

// Example usage
int main() {
    int nx = 101; // Number of spatial points
    int nt = 1000; // Number of time steps
    double alpha = 0.01; // Thermal diffusivity

    std::vector<double> temperature = solve_heat_equation(nx, nt, alpha);

    // Print temperature distribution
    for (int i = 0; i < nx; ++i) {
        std::cout << "Temperature at x=" << i / double(nx-1) << ": " << temperature[i] << std::endl;
    }

    return 0;
}
</syntaxhighlight>