Numerical Analysis-Numerical Analysis for Complex Problems

Example

Newton-Raphson Method

Scenario

This method is used to find the roots of non-linear equations. It is particularly useful in engineering, where one might need to find the stress points on a beam under load. For instance, if an engineer is modeling the deformation of a beam, they may need to solve for when the deformation equals a certain critical value. By applying the Newton-Raphson method, the engineer can iteratively zero in on this critical point with high accuracy.

Example

Simpson’s Rule

Scenario

Simpson's Rule is used to approximate the definite integral of a function when an exact antiderivative is difficult or impossible to find. In physics, this could be used to calculate the total work done by a force over a distance when the force varies in a non-simple manner. For example, calculating the work done by varying wind forces on a sailboat over a specific trajectory.

Example

Finite Difference Method

Scenario

The Finite Difference Method (FDM) is used to numerically solve partial differential equations (PDEs) that model various physical phenomena. An example scenario is in weather forecasting, where meteorologists use FDM to solve the Navier-Stokes equations, which describe the motion of the atmosphere. These equations are too complex to solve analytically over large scales, so numerical solutions are computed over discretized grids of the atmosphere.