Classes of high-order numerical methods for solution of certain problem in calculus of variations

In this work, we extend the definition of nonic polynomial spline to non-polynomial spline function which depends on arbitrary parameter k. We derived and discussed the formulation and spline relations. Using such non-polynomial spline relations, we developed the classes of numerical methods, for the solution of the problem in calculus of variations. The proposed boundary formulas which are needed to be associated with spline methods are derived. Truncation errors and orders of accuracy of proposed methods are presented. Convergence analysis of the methods are discussed. The present methods have been tested on three examples, to illustrate practical usefulness of our method