Finite Difference Method: A Numerical Approach for Solving Differential and Integro-Differential Equations
The Finite Difference Method (FDM) is a numerical technique for approximating solutions to differential equations and integro-differential equations. The core concept involves discretizing a continuous domain into a grid of finite discrete points (nodes), where continuous functions are approximated by discrete grid functions. Derivatives in the original equations and boundary conditions are replaced by difference quotients, while integrals are approximated by discrete sums, transforming the system into an algebraic finite difference equations system. Solving this system yields approximate solutions at discrete points, which can then be interpolated to obtain solutions over the entire domain. Implementation typically involves grid generation, difference operator construction, and linear system solvers.