A deficient spline function approximation to fourth-order differential equations

Abstract An approximate spline solution is developed for the initial value problem of a fourth-order ordinary differential equation. The approximation is based on deficient spline polynomials of degree m = 8 and deficiency 4. The existence and uniqueness of the solution, which satisfies a Lipschitz condition, are proved. The consistency, stability, and consequently convergence of the solution are established. Furthermore, the method is proved to be of order 9, and the errors are limited by the relation ‖S (i) (x) − y (i) (x)‖ = O(h 9 − i ), i = 0(1)8.