Numerical ordinary differential equations

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 is sometimes taken to mean the computation of integrals.

Many differential equations cannot be solved using symbolic computation ("analysis"). 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, for instance in physics, chemistry, biology, and economics. In addition, some methods in numerical partial differential equations convert the partial differential equation into an ordinary differential equation, which must then be solved.

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

where f is a function that maps [t₀,∞) × R^d to R^d, and the initial condition y₀ ∈ 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 y'' = −y can be rewritten as two first-order equations: y' = z and z' = −y.

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