collocation methods

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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′), axb,

an approximation P(x) = !Yni=1αiφi(x)

can be obtained from a suitable set of orthogonal functions φi(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 (see boundary-value problem).