Collocation method and residual correction using Chebyshev series

The Chebyshev collocation method has been proposed to solve the linear two-point boundary value problems. Correction of the approximated solution has been obtained using the residual function of the operator equation. The error differential equation, obtained by residual function, has been solved by a Truncated Chebyshev Series (TCS), where the order of the TCS is bigger than the order of the TCS in the Chebyshev collocation method. The obtained approximate solution for the collocation method has been corrected by the error differential equation.