Editing Thunderstorm Generator (section)

=== Programming Tips for these Equations: ===

# '''Partial Differential Equations (PDEs) and Integral Transforms''':
#* Example:
#** <math>\frac{dy}{dx} = -2y
</math>
#* These equations involve derivatives with respect to multiple independent variables (such as time and space) and may require integration over regions or domains.
#* Writing direct functions for these equations is not feasible because they often require solving complex differential equations.
#* Instead, numerical methods such as finite difference, finite element, or spectral methods are used to approximate solutions.
#* For integral transforms like Fourier or Laplace transforms, libraries or built-in functions in programming languages can be used.
#* For these equations, we typically rely on specialized software libraries or tools for numerical simulation. Here's an example of using a hypothetical library <code>libphysics</code> to solve Maxwell's equations:<syntaxhighlight lang="c">
#include <stdio.h>
#include <math.h>

// Function to solve the differential equation dy/dx = -2y
double solve_differential_equation(double x, double y) {
    return -2 * y; // Example differential equation
}

int main() {
    double x0 = 0.0; // Initial x value
    double y0 = 1.0; // Initial y value
    double h = 0.1;  // Step size

    // Using Euler's method for numerical integration
    for (double x = x0; x <= 1.0; x += h) {
        y0 += h * solve_differential_equation(x, y0);
    }

    printf("Approximate solution at x=1: %f\n", y0);

    return 0;
}

</syntaxhighlight>
# '''Partial Differential Equations (PDEs) like Navier-Stokes, Schrödinger, Wave Equations''':
#* Example:
#** <math>\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2}
</math>
#* Equations such as the Navier-Stokes equation for fluid dynamics, the Schrödinger equation for quantum mechanics, and the wave equation for wave propagation are fundamental in physics.
#* They describe complex phenomena and cannot be directly solved with simple functions.
#* Solving these equations often requires advanced mathematical techniques or numerical methods like finite difference, finite element, or spectral methods.
#* Implementations involve discretizing the equations in space and time and solving them iteratively.<syntaxhighlight lang="c">
#include <stdio.h>

#define NX 100 // Number of grid points
#define NT 100 // Number of time steps
#define DX 0.1 // Grid spacing
#define DT 0.01 // Time step

// Function to initialize the temperature profile
void initialize_temperature(double temperature[]) {
    for (int i = 0; i < NX; i++) {
        temperature[i] = 0.0; // Initial temperature
    }
}

// Function to solve the heat equation using finite difference method
void solve_heat_equation(double temperature[]) {
    double new_temperature[NX];
    for (int t = 0; t < NT; t++) {
        for (int i = 1; i < NX - 1; i++) {
            new_temperature[i] = temperature[i] + DT * (temperature[i + 1] - 2 * temperature[i] + temperature[i - 1]) / (DX * DX);
        }
        for (int i = 1; i < NX - 1; i++) {
            temperature[i] = new_temperature[i];
        }
    }
}

int main() {
    double temperature[NX];

    // Initialize temperature profile
    initialize_temperature(temperature);

    // Solve the heat equation
    solve_heat_equation(temperature);

    // Output the results
    for (int i = 0; i < NX; i++) {
        printf("Temperature at position %d: %f\n", i, temperature[i]);
    }

    return 0;
}

</syntaxhighlight>
# '''Equations like Maxwell's Equations, Boltzmann Transport Equation''':
#* Example:
#** <math>\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}
</math>
#* These equations are sets of differential equations that describe fundamental physical principles.
#* They involve multiple variables and interactions and cannot be directly represented as simple functions.
#* Analyzing these equations typically involves numerical methods, simulations, or specific models tailored to the problem at hand.
#* Software packages or libraries dedicated to computational physics or engineering may provide tools for solving these equations.<syntaxhighlight lang="c">
#include <stdio.h>
#include "libphysics.h" // Hypothetical library for physics simulations

int main() {
    // Define parameters
    double rho = 1.0; // Charge density
    double epsilon_0 = 8.854e-12; // Permittivity of free space

    // Solve Maxwell's equations
    double electric_field = solve_maxwells_equations(rho, epsilon_0);

    // Output the result
    printf("Electric field: %f\n", electric_field);

    return 0;
}

</syntaxhighlight>

In practice, specialized software packages like COMSOL, ANSYS, or custom-built simulation tools are often used for solving complex physical equations numerically.