Finite Difference Method: A Numerical Approach for Solving Differential and Integro-Differential Equations

The Finite Difference Method (FDM) is a numerical technique for solving differential equations and integro-differential equations. Its fundamental approach involves replacing continuous solution domains with grids composed of finite discrete points, known as grid nodes. Continuous functions defined over the domain are approximated using discrete functions defined on these grid points. Subsequently, derivatives in the original equations and boundary conditions are approximated by difference quotients (e.g., forward, backward, or central differences), while integrals are approximated by discrete sums (such as Riemann sums). This transforms the original differential equations and boundary conditions into a system of algebraic equations, referred to as finite difference equations. Solving this system yields approximate solutions at discrete grid points. Through interpolation methods (like linear or polynomial interpolation), these discrete solutions can be extended to approximate the solution over the entire domain. The method is widely applicable for solving various mathematical models including partial differential equations, wave equations, and heat conduction equations, with significant implementations in engineering, physics, and mathematics. Code implementation typically involves constructing discrete grids using functions like numpy.meshgrid, calculating difference operators through matrix operations, and employing linear algebra solvers (e.g., numpy.linalg.solve) for efficient computation.