The Shifted Vieta Pell solution in the calculus of variations via direct parameterization technique

In this article, a new parameterization direct method using Shifted Vieta Pell (SVP) functions is proposed to solve problems in the variation calculus. An explicit general formulation of operational matrix of derivative is obtained for the SVP functions and used to find an approximate solution for the calculus of variation problem. The first step in our suggested method is to express the unknown variables in terms of the basis functions SVP with unknown coefficients. Then, the operation matrix of derivative and some important properties of the SVP functions are applied to achieve a nonlinear programming problem in terms of the unknown coefficients. Then the quadratic programming algorithm is used to calculate the unknown parameters. Two test numerical examples of different calculus of variation types are demonstrated together with approximate solutions for proving the accuracy and applicability of the suggested SVP method.