Deficient spline function approximation to second- order differential equations

An approximation with spline functions of degree m and deficiency 3 is developed for solving second-order initial-value problems. The stability of the approximate solution is investigated and it is demonstrated that the method is divergent for m 7. Convergence is shown for m = 5 and m = 6. Moreover, the method is of order (m + 1) and error bounds are of the form: boxV ]S⁽ⁱ⁾(x)−yⁱxboxV ]∞=0h^m+-i), i=01m A computational example is presented to illustrate the efficiency of the method.