Application of least square method to arbitrary-order problems with separated boundary conditions

In this paper, differential equations of arbitrary order with separated boundary conditions are converted into an optimal control problem. Then a convergent approximate solution is constructed such that the exact boundary conditions are satisfied. In fact it will be shown that for every ε > 0 , there exists an approximate solution v ε of B-spline functions such that the corresponding least square error is less than ε > 0 , and also v ε satisfies the exact boundary conditions. Some examples are given and the optimal errors are obtained for the sake of comparison.