collocation methods

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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′), axb, an approximation can be obtained from a suitable set of orthogonal functions ∅i(x) by choosing the coefficients αi for which P″(xi) = f(xi, P(xi), P′(xi), for some set of collocation points ax1 < x2 < … < xnb Initial conditions and boundary conditions may also be incorporated into the process.

y″ = f(x,y,y′), axb,

P″(xi) = f(xi, P(xi), P′(xi),

ax1 < x2 < … < xnb

Subjects: Computing.

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