How does the finite difference method approximate derivatives or integrals?

The finite difference method is a numerical technique used to solve differential equations and integral equations by approximating the continuous operators within them.1

Approximating Derivatives

In numerical analysis, the finite difference method approximates derivatives by replacing them with algebraic expressions based on the values of a function at a discrete set of points. This approach is rooted in the definition of the derivative, where the derivative at a point is the limit of the quotient $\Delta y/\Delta x$ as $\Delta x$ approaches zero. By choosing a small, finite increment for $\Delta x$ rather than taking the limit, the method transforms a complex differential equation into a system of algebraic equations that can be solved numerically.12

Approximating Integrals

While the finite difference method is primarily associated with approximating derivatives, numerical methods for solving integral equations generally follow a similar logic of discretization. These procedures approximate the continuous equation of interest by evaluating it at specific, discrete points, effectively reducing the problem to a solvable linear or nonlinear system.1