What is Integral Equation?

An Integral Equation is a type of equation where an unknown function appears inside an integral. It is widely used to model various phenomena in physics, engineering, and applied mathematics, particularly when dealing with systems described by relationships involving integrals rather than direct functions.

Key Concepts:

• Kernel: A function that defines the relationship between variables in the equation.

• Volterra Integral Equations: These involve variable limits of integration, often used to model time-dependent processes.

• Fredholm Integral Equations: These have fixed limits of integration and are used in boundary value problems.

• First Kind: The equation involves only an integral term, with no direct unknown function on the left-hand side.

• Second Kind: The equation includes an integral term along with the unknown function itself.

Applications:

• Physics: Used in modeling heat transfer, wave propagation, and quantum mechanics.

• Engineering: Helps in solving problems related to potential theory and signal processing.

• Biology: Applied to population dynamics and biological modeling.

• Economics: Used in modeling financial systems and economic equilibria.

Integral equations provide essential tools for solving complex problems where solutions are based on integrals rather than direct values.