collocation methods

An important approach to the numerical solution of ordinary differential and integral equations. Approximations are obtained on the basis that the equation is satisfied exactly at a particular set of points in the given problem range. For example, for y″ = f(x,y,y′), a ≤ x ≤ b, an approximation can be obtained from a suitable set of orthogonal functions ∅_i(x) by choosing the coefficients α_i for which P″(x_i) = f(x_i, P(x_i), P′(x_i), for some set of collocation points a ≤ x₁ < x₂ < … < x_n ≤ b Initial conditions and boundary conditions may also be incorporated into the process.

y″ = f(x,y,y′), a ≤ x ≤ b,

P″(x_i) = f(x_i, P(x_i), P′(x_i),

a ≤ x₁ < x₂ < … < x_n ≤ b