Ordinary Differential Equations
Suppose that the ordinary differential equation
is to be solved over the interval . Choose 0 ≤ c1< c2< … < cn ≤ 1.
The corresponding (polynomial) collocation method approximates the solution y by the polynomial p of degree n which satisfies the initial condition p(t0) = y0, and the differential equation p'(t) = f(t,p(t)) at all points, called the collocation points, t = t0 + ckh where k = 1, …, n. This gives n + 1 conditions, which matches the n + 1 parameters needed to specify a polynomial of degree n.
All these collocation methods are in fact implicit Runge–Kutta methods. The coefficient ck in the Butcher tableau of a Runge–Kutta method are the collocation points. However, not all implicit Runge–Kutta methods are collocation methods.
Read more about this topic: Collocation Method
Famous quotes containing the words ordinary and/or differential:
“If you have embraced a creed which appears to be free from the ordinary dirtiness of politics—a creed from which you yourself cannot expect to draw any material advantage—surely that proves that you are in the right?”
—George Orwell (1903–1950)
“But how is one to make a scientist understand that there is something unalterably deranged about differential calculus, quantum theory, or the obscene and so inanely liturgical ordeals of the precession of the equinoxes.”
—Antonin Artaud (1896–1948)