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
Famous quotes containing the word problem:
“It is part of the educator’s responsibility to see equally to two things: First, that the problem grows out of the conditions of the experience being had in the present, and that it is within the range of the capacity of students; and, secondly, that it is such that it arouses in the learner an active quest for information and for production of new ideas. The new facts and new ideas thus obtained become the ground for further experiences in which new problems are presented.”
—John Dewey (1859–1952)
“The government is huge, stupid, greedy and makes nosy, officious and dangerous intrusions into the smallest corners of life—this much we can stand. But the real problem is that government is boring. We could cure or mitigate the other ills Washington visits on us if we could only bring ourselves to pay attention to Washington itself. But we cannot.”
—P.J. (Patrick Jake)