Collocation method

In mathematics, a collocation method is a method for the numerical solution of ordinary differential equations, partial differential equations and integral equations. The idea is to choose a finite-dimensional space of candidate solutions (usually polynomials up to a certain degree) and a number of points in the domain (called collocation points), and to select that solution which satisfies the given equation at the collocation points.

Suppose that the ordinary differential equation

is to be solved over the interval ${\displaystyle [t_{0},t_{0}+h]}$. Choose 0 ≤ c₁< c₂< … < c_n ≤ 1.

The corresponding (polynomial) collocation method approximates the solution y by the polynomial p of degree n which satisfies the initial condition ${\displaystyle p(t_{0})=y_{0}}$, and the differential equation ${\displaystyle p'(t_{k})=f(t_{k},p(t_{k}))}$

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