14.7.4 Approximate solution of differential equations

At its core, the Approximate Solutions of Differential Equations in numerical analysis bring mathematical models to life by leveraging computational techniques. From ordinary differential equations that describe change over time to partial differential equations that govern spatial distributions, these equations encapsulate the essence of dynamic systems. Numerical methods step in when analytical solutions are elusive, embracing a multitude of techniques to simulate, analyze, and predict behaviors. By discretizing time and space, numerical methods unfold a path of iterative refinement, progressively converging towards the sought-after solutions. Whether embracing explicit methods like Euler's with checks of Runge's rules or implicit approaches like the Crank-Nicolson scheme, the arsenal of techniques caters to diverse scenarios. From finite difference methods that slice continuous domains into grids to finite element methods that model complex geometries, we will examine how these methods translate theoretical equations into numerical approximations.