Numerical methods for ordinary differential equations explained

Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals.

Many differential equations cannot be solved exactly. For practical purposes, however – such as in engineering – a numeric approximation to the solution is often sufficient. The algorithms studied here can be used to compute such an approximation. An alternative method is to use techniques from calculus to obtain a series expansion of the solution.

Ordinary differential equations occur in many scientific disciplines, including physics, chemistry, biology, and economics.^[1] In addition, some methods in numerical partial differential equations convert the partial differential equation into an ordinary differential equation, which must then be solved.

The problem

A first-order differential equation is an Initial value problem (IVP) of the form,where

f is a function f:[t _{0,} inf ty) x \R ^{d} \to \R ^{d} , and the initial condition y _{0} \in \R ^{d} is a given vector. First-order means that only the first derivative of y appears in the equation, and higher derivatives are absent.

Without loss of generality to higher-order systems, we restrict ourselves to first-order differential equations, because a higher-order ODE can be converted into a larger system of first-order equations by introducing extra variables. For example, the second-order equation can be rewritten as two first-order equations:

Notes and References

1. Chicone, C. (2006). Ordinary differential equations with applications (Vol. 34). Springer Science & Business Media.