Editing Equations, Formulas and Algorithms (section)

==== Fredholm Integral Equation Solvers ====
Fredholm integral equations of the second kind are solved using numerical approximation methods, iterative techniques, and spectral methods. These algorithms are employed to solve integral equations that arise in various areas, including physics, engineering, and mathematical modeling.

'''Example: Fredholm Integral Equation'''
The Fredholm integral equation of the second kind is given by:
<math mode="display"> \phi(x) = f(x) + \lambda \int_{a}^{b} K(x, s) \phi(s) \, ds </math>
where <math> \phi(x) </math> is the unknown function to be solved for, <math> f(x) </math> is a given function, <math> K(x, s) </math> is the kernel function, and <math> \lambda </math> is a parameter.

'''Source Code Example (Matlab):'''
<syntaxhighlight lang="matlab">
% Example of solving Fredholm integral equation using iterative method
a = 0; b = 1; % Define integration limits
lambda = 0.5; % Define parameter lambda

% Define kernel function
K = @(x, s) exp(x + s);

% Define function f(x)
f = @(x) x^2;

% Define initial guess for phi(x)
phi0 = @(x) 0;

% Define number of iterations
iterations = 100;

% Iterative method
phi = phi0;
for i = 1:iterations
    phi_new = @(x) f(x) + lambda * integral(@(s) K(x, s) * phi(s), a, b);
    phi = phi_new;
end

% Display result
disp('Approximated solution for phi(x):');
disp(phi(0.5));
</syntaxhighlight>