Generalisation of the Lagrange multipliers for variational iterations applied to systems of differential equations

In this paper, a new approach to the variational iteration method is introduced to solve systems of first-order differential equations. Since higher-order differential equations can almost always be converted into a first-order system of equations, the proposed method is still applicable to a wide range of differential equations. This generalised approach, unlike the classical method, uses restricted variations only for nonlinear terms by generalising the Lagrange multipliers. Consequently, this allows us to use the well known, but ignored, theory of linear ODEs for computing the matrix-valued Lagrange multipliers. In order to validate the newly proposed approach in solving linear and nonlinear systems of differential equations, illustrative examples are presented: it turns out that the use of the generalised Lagrange multipliers is more reliable and efficient.