The Problem
We want to approximate the solution of the differential equation
where f is a function that maps [t0,∞) × Rd to Rd, and the initial condition y0 ∈ Rd is a given vector.
The above formulation is called an initial value problem (IVP). The Picard–Lindelöf theorem states that there is a unique solution, if f is Lipschitz continuous. In contrast, boundary value problems (BVPs) specify (components of) the solution y at more than one point. Different methods need to be used to solve BVPs, for example the shooting method (and its variants) or global methods like finite differences, Galerkin methods, or collocation methods.
Note that we restrict ourselves to first-order differential equations (meaning that only the first derivative of y appears in the equation, and no higher derivatives). This, however, does not restrict the generality of the problem, since a higher-order equation can easily be converted to a 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.
Read more about this topic: Numerical Methods For Ordinary Differential Equations
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