## Quick Reference

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

*for which*

_{i}*P*″(

*x*

*) =*

_{i}*f*(

*x*

*,*

_{i}*P*(

*x*

*),*

_{i}*P*′(

*x*

*), for some set of collocation points*

_{i}*a*≤

*x*

_{1}<

*x*

_{2}< … <

*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* ≤ *x*_{1} < *x*_{2} < … < *x** _{n}* ≤

*b*

*Subjects:*
Computing.