How are mathematical models used with numerical analysis?

Mathematical models are essential for describing phenomena in science and engineering, but they are often too complex to solve explicitly. Numerical analysis provides the tools to obtain useful, approximate solutions to these models.1

In practice, numerical analysis is used to address models based on ordinary differential equations, partial differential equations, and integral equations through two primary approaches:2

• Function Approximation: The unknown function in a model is replaced by a simpler function, such as a polynomial or a spline, that closely mimics the original. The finite element method is a prominent example of this approach, widely used for stress, heat, and fluid flow analysis.21
• Equation Approximation: The model itself is simplified by approximating its derivatives or integrals. This leads to finite difference methods, which solve for values at a discrete set of points.2

Modern scientific computing integrates these numerical methods with computer technology to interpret and solve these complex real-world models efficiently.3