How do numerical methods provide approximate solutions to differential equations?

Numerical methods provide approximate solutions to differential equations primarily through two distinct approaches.1

• Function Approximation: This method approximates the unknown function within the equation using a simpler, manageable function, such as a polynomial or a piecewise polynomial known as a spline. The finite element method is the most prominent example of this approach.1
• Equation Approximation: This method approximates the equation itself, typically by replacing derivatives or integrals with discrete approximations. These finite difference methods yield solutions at a specific, discrete set of points that serve as an approximation for the original continuous solution. This is the standard approach for most initial value problems involving ordinary differential equations and partial differential equations.1

These procedures often require solving large systems of linear or nonlinear equations, integrating approximation theory to ensure the accuracy of the final result.1