Editing Equations, Formulas and Algorithms (section)

==== Volterra Integral Equation Solvers ====
Similar to Fredholm equations, algorithms for solving Volterra integral equations involve numerical approximation methods and spectral techniques. These algorithms are applied in diverse fields, such as biology, economics, and control theory, to model dynamic systems and processes.

'''Example: Volterra Integral Equation'''
The Volterra integral equation of the second kind is given by: <math> \phi(x) = f(x) + \lambda \int_{a}^{x} 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 (Julia):'''
<syntaxhighlight lang="julia">
# Example of solving Volterra integral equation using iterative method
a = 0; b = 1; # Define integration limits
lambda = 0.5; # Define parameter lambda

# Define kernel function
function K(x, s)
    return exp(x + s)
end

# Define function f(x)
function f(x)
    return x^2
end

# Define initial guess for phi(x)
function phi0(x)
    return 0
end

# Define number of iterations
iterations = 100

# Iterative method
phi = phi0
for i = 1:iterations
    phi_new(x) = f(x) + lambda * quadgk(s -> K(x, s) * phi(s), a, x)[1]
    phi = phi_new
end

# Display result
println("Approximated solution for phi(x): ", phi(0.5))
</syntaxhighlight>