Linear versus Nonlinear Integral Equations

Integral equations are a branch of mathematics that deal with equations involving unknown functions within integrals. They are used to model a wide range of physical, engineering, and scientific phenomena where the relationship between variables is expressed through integrals rather than algebraic equations.
There are two main types of integral equations:

Volterra Equations: In these equations, the unknown function appears both inside and outside of the integral sign. They are often used to describe problems involving cumulative effects over a certain domain.

Fredholm Equations: Here, the unknown function is only found inside the integral. These equations are more common in applications where the unknown function represents a response or a distribution.

Integral equations are categorized based on whether they are linear or nonlinear, and whether the limits of integration are fixed or variable. The solutions to integral equations are often functions, and finding these solutions can be quite challenging. Various techniques, such as the method of successive approximations, kernel functions, and numerical methods, are employed to solve integral equations and obtain meaningful solutions.
Integral equations have numerous applications, including in physics (e.g., heat conduction), engineering (e.g., electrical circuits), and image processing (e.g., deblurring). They provide a powerful mathematical framework for understanding and solving complex problems involving continuous phenomena.